Theorem itg2addnclem 33461

Description: An alternate expression for the ∫₂ integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.)

Hypothesis

Ref	Expression
itg2addnclem.1	⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}

Assertion

Ref	Expression
itg2addnclem	⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Distinct variable group:   𝑥,𝑦,𝑧,𝑔,𝐹

Allowed substitution hints:   𝐿(𝑥,𝑦,𝑧,𝑔)

Proof of Theorem itg2addnclem

Dummy variables 𝑠 𝑢 𝑓 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} = {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}
2	1	itg2val 23495	. 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
3		itg2addnclem.1	. . . 4 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}
4	3	supeq1i 8353	. . 3 ⊢ sup(𝐿, ℝ*, < ) = sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )
5		xrltso 11974	. . . . 5 ⊢ < Or ℝ*
6	5	a1i 11	. . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → < Or ℝ*)
7		simprr 796	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 = (∫1‘𝑓))
8		itg1cl 23452	. . . . . . . . . 10 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℝ)
9	8	rexrd 10089	. . . . . . . . 9 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℝ*)
10	9	adantr 481	. . . . . . . 8 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
11	7, 10	eqeltrd 2701	. . . . . . 7 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 ∈ ℝ*)
12	11	rexlimiva 3028	. . . . . 6 ⊢ (∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → 𝑥 ∈ ℝ*)
13	12	abssi 3677	. . . . 5 ⊢ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*
14		supxrcl 12145	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ*)
15	13, 14	mp1i 13	. . . 4 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ∈ ℝ*)
16		fveq1 6190	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑧) = (𝑓‘𝑧))
17	16	eqeq1d 2624	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑧) = 0 ↔ (𝑓‘𝑧) = 0))
18	16	oveq1d 6665	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑧) + 𝑦) = ((𝑓‘𝑧) + 𝑦))
19	17, 18	ifbieq2d 4111	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
20	19	mpteq2dv 4745	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
21	20	breq1d 4663	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹))
22	21	rexbidv 3052	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹))
23		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (∫1‘𝑔) = (∫1‘𝑓))
24	23	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (𝑥 = (∫1‘𝑔) ↔ 𝑥 = (∫1‘𝑓)))
25	22, 24	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
26	25	cbvrexv 3172	. . . . . . . . 9 ⊢ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑓 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)))
27		breq2 4657	. . . . . . . . . . . . . . . . 17 ⊢ (0 = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) → ((𝑓‘𝑧) ≤ 0 ↔ (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
28		breq2 4657	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓‘𝑧) + 𝑦) = if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) → ((𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦) ↔ (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
29		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) = 0 → (𝑓‘𝑧) = 0)
30		0le0 11110	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ 0
31	29, 30	syl6eqbr 4692	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) = 0 → (𝑓‘𝑧) ≤ 0)
32	31	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) ≤ 0)
33		rpge0 11845	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
34	33	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 0 ≤ 𝑦)
35		i1ff 23443	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ dom ∫1 → 𝑓:ℝ⟶ℝ)
36	35	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
37	36	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
38		rpre 11839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
39	38	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
40	37, 39	addge01d 10615	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (0 ≤ 𝑦 ↔ (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦)))
41	34, 40	mpbid 222	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦))
42	41	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) ≤ ((𝑓‘𝑧) + 𝑦))
43	27, 28, 32, 42	ifbothda 4123	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
44	43	adantlll 754	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)))
45	35	ad2antlr 763	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓:ℝ⟶ℝ)
46	45	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
47	46	rexrd 10089	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ*)
48		0re 10040	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
49	38	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → 𝑦 ∈ ℝ)
50	46, 49	readdcld 10069	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) + 𝑦) ∈ ℝ)
51		ifcl 4130	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ ((𝑓‘𝑧) + 𝑦) ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ)
52	48, 50, 51	sylancr 695	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ)
53	52	rexrd 10089	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ*)
54		iccssxr 12256	. . . . . . . . . . . . . . . . . . 19 ⊢ (0[,]+∞) ⊆ ℝ*
55		fss 6056	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ*) → 𝐹:ℝ⟶ℝ*)
56	54, 55	mpan2 707	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶ℝ*)
57	56	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹:ℝ⟶ℝ*)
58	57	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ ℝ*)
59		xrletr 11989	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑧) ∈ ℝ* ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∈ ℝ* ∧ (𝐹‘𝑧) ∈ ℝ*) → (((𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
60	47, 53, 58, 59	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) ≤ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ∧ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
61	44, 60	mpand 711	. . . . . . . . . . . . . 14 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → (if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧) → (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
62	61	ralimdva 2962	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧) → ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
63		reex 10027	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ℝ ∈ V)
65		eqidd 2623	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))))
66		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹:ℝ⟶(0[,]+∞))
67	66	feqmptd 6249	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
68	67	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝐹 = (𝑧 ∈ ℝ ↦ (𝐹‘𝑧)))
69	64, 52, 58, 65, 68	ofrfval2 6915	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦)) ≤ (𝐹‘𝑧)))
70	35	feqmptd 6249	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
71	70	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
72	64, 46, 58, 71, 68	ofrfval2 6915	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
73	62, 69, 72	3imtr4d 283	. . . . . . . . . . . 12 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑦 ∈ ℝ+) → ((𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 → 𝑓 ∘𝑟 ≤ 𝐹))
74	73	rexlimdva 3031	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 → 𝑓 ∘𝑟 ≤ 𝐹))
75	74	anim1d 588	. . . . . . . . . 10 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
76	75	reximdva 3017	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
77	26, 76	syl5bi 232	. . . . . . . 8 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) → ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))))
78	77	ss2abdv 3675	. . . . . . 7 ⊢ (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⊆ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))})
79	78	sseld 3602	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} → 𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}))
80		simp3r 1090	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 = (∫1‘𝑓))
81	9	3ad2ant2 1083	. . . . . . . . . . 11 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
82	80, 81	eqeltrd 2701	. . . . . . . . . 10 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))) → 𝑥 ∈ ℝ*)
83	82	rexlimdv3a 3033	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) → 𝑥 ∈ ℝ*))
84	83	abssdv 3676	. . . . . . . 8 ⊢ (𝐹:ℝ⟶(0[,]+∞) → {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ*)
85		xrsupss 12139	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* → ∃𝑎 ∈ ℝ* (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ* (𝑏 < 𝑎 → ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠)))
86	84, 85	syl 17	. . . . . . 7 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑎 ∈ ℝ* (∀𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ* (𝑏 < 𝑎 → ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠)))
87	6, 86	supub 8365	. . . . . 6 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏))
88	79, 87	syld 47	. . . . 5 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏))
89	88	imp 445	. . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}) → ¬ sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) < 𝑏)
90		supxrlub 12155	. . . . . . . 8 ⊢ (({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))} ⊆ ℝ* ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
91	13, 90	mpan 706	. . . . . . 7 ⊢ (𝑏 ∈ ℝ* → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
92	91	adantl 482	. . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) ↔ ∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠))
93		simprrr 805	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → 𝑠 = (∫1‘𝑓))
94	93	breq2d 4665	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < 𝑠 ↔ 𝑏 < (∫1‘𝑓)))
95		simplll 798	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝐹:ℝ⟶(0[,]+∞))
96		i1f0 23454	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × {0}) ∈ dom ∫1
97		2rp 11837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ+
98	97	ne0ii 3923	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ+ ≠ ∅
99		ffvelrn 6357	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,]+∞))
100		elxrge0 12281	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑧) ∈ (0[,]+∞) ↔ ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
101	99, 100	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
102	101	simprd 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧))
103	102	ralrimiva 2966	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧))
104	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ℝ ∈ V)
105		c0ex 10034	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → 0 ∈ V)
107		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) = (𝑧 ∈ ℝ ↦ 0))
108	104, 106, 99, 107, 67	ofrfval2 6915	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ 0 ≤ (𝐹‘𝑧)))
109	103, 108	mpbird 247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹)
110	109	ralrimivw 2967	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹)
111		r19.2z 4060	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹)
112	98, 110, 111	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹)
113		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (ℝ × {0}) → (∫1‘𝑔) = (∫1‘(ℝ × {0})))
114		itg10 23455	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∫1‘(ℝ × {0})) = 0
115	113, 114	syl6req 2673	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = (ℝ × {0}) → 0 = (∫1‘𝑔))
116	115	biantrud 528	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
117		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (ℝ × {0}) → (𝑔‘𝑧) = ((ℝ × {0})‘𝑧))
118	105	fvconst2 6469	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ℝ → ((ℝ × {0})‘𝑧) = 0)
119	117, 118	sylan9eq 2676	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = 0)
120		iftrue 4092	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑔‘𝑧) = 0 → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = 0)
121	119, 120	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔 = (ℝ × {0}) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = 0)
122	121	mpteq2dva 4744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (ℝ × {0}) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ 0))
123	122	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = (ℝ × {0}) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹))
124	123	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (ℝ × {0}) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹))
125	116, 124	bitr3d 270	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (ℝ × {0}) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹))
126	125	rspcev 3309	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ℝ × {0}) ∈ dom ∫1 ∧ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ 0) ∘𝑟 ≤ 𝐹) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
127	96, 112, 126	sylancr 695	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
128		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = -∞ → 𝑏 = -∞)
129		mnflt 11957	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ ℝ → -∞ < 0)
130	48, 129	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = -∞ → -∞ < 0)
131	128, 130	eqbrtrd 4675	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = -∞ → 𝑏 < 0)
132		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 0 → (𝑎 = (∫1‘𝑔) ↔ 0 = (∫1‘𝑔)))
133	132	anbi2d 740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 0 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
134	133	rexbidv 3052	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔))))
135		breq2 4657	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → (𝑏 < 𝑎 ↔ 𝑏 < 0))
136	134, 135	anbi12d 747	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 0 → ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎) ↔ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ∧ 𝑏 < 0)))
137	105, 136	spcev 3300	. . . . . . . . . . . . . . . . . 18 ⊢ ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)) ∧ 𝑏 < 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
138	127, 131, 137	syl2an 494	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 = -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
139	95, 138	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 = -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
140		simp-4r 807	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ*)
141	8	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ)
142	141	ad3antlr 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → (∫1‘𝑓) ∈ ℝ)
143		simpllr 799	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ∈ ℝ*)
144		ngtmnft 11997	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ∈ ℝ* → (𝑏 = -∞ ↔ ¬ -∞ < 𝑏))
145	144	biimprd 238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ∈ ℝ* → (¬ -∞ < 𝑏 → 𝑏 = -∞))
146	145	necon1ad 2811	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ℝ* → (𝑏 ≠ -∞ → -∞ < 𝑏))
147	146	imp 445	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℝ* ∧ 𝑏 ≠ -∞) → -∞ < 𝑏)
148	143, 147	sylan 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → -∞ < 𝑏)
149		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → 𝑏 ∈ ℝ*)
150	9	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (∫1‘𝑓) ∈ ℝ*)
151	149, 150	anim12i 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*))
152		xrltle 11982	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*) → (𝑏 < (∫1‘𝑓) → 𝑏 ≤ (∫1‘𝑓)))
153	152	imp 445	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ*) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ≤ (∫1‘𝑓))
154	151, 153	sylan 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → 𝑏 ≤ (∫1‘𝑓))
155	154	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ≤ (∫1‘𝑓))
156		xrre 12000	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑏 ∈ ℝ* ∧ (∫1‘𝑓) ∈ ℝ) ∧ (-∞ < 𝑏 ∧ 𝑏 ≤ (∫1‘𝑓))) → 𝑏 ∈ ℝ)
157	140, 142, 148, 155, 156	syl22anc 1327	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → 𝑏 ∈ ℝ)
158	127	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 0 = (∫1‘𝑔)))
159		simplrl 800	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < (∫1‘𝑓))
160		simplrl 800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑓 ∈ dom ∫1)
161		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑓 ∈ dom ∫1)
162		cnvimass 5485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ dom 𝑓
163		fdm 6051	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓:ℝ⟶ℝ → dom 𝑓 = ℝ)
164	35, 163	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ dom ∫1 → dom 𝑓 = ℝ)
165	162, 164	syl5sseq 3653	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 ∈ dom ∫1 → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
166	165	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
167		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0)
168	163	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:ℝ⟶ℝ → ℝ = dom 𝑓)
169		ffun 6048	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑓:ℝ⟶ℝ → Fun 𝑓)
170		difpreima 6343	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (Fun 𝑓 → (◡𝑓 “ (ran 𝑓 ∖ {0})) = ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})))
171	169, 170	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})))
172		cnvimarndm 5486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (◡𝑓 “ ran 𝑓) = dom 𝑓
173	172	difeq1i 3724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) = (dom 𝑓 ∖ (◡𝑓 “ {0}))
174	171, 173	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = (dom 𝑓 ∖ (◡𝑓 “ {0})))
175	168, 174	difeq12d 3729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓:ℝ⟶ℝ → (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) = (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))))
176		cnvimass 5485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (◡𝑓 “ {0}) ⊆ dom 𝑓
177		dfss4 3858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ {0}) ⊆ dom 𝑓 ↔ (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))) = (◡𝑓 “ {0}))
178	176, 177	mpbi 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (dom 𝑓 ∖ (dom 𝑓 ∖ (◡𝑓 “ {0}))) = (◡𝑓 “ {0})
179	175, 178	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓:ℝ⟶ℝ → (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) = (◡𝑓 “ {0}))
180	179	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 𝑧 ∈ (◡𝑓 “ {0})))
181		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓:ℝ⟶ℝ → 𝑓 Fn ℝ)
182		fniniseg 6338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) ↔ (𝑧 ∈ ℝ ∧ (𝑓‘𝑧) = 0)))
183		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑧 ∈ ℝ ∧ (𝑓‘𝑧) = 0) → (𝑓‘𝑧) = 0)
184	182, 183	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
185	181, 184	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
186	180, 185	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0))
187	35, 186	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ dom ∫1 → (𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0))
188	187	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓‘𝑧) = 0)
189	188	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) ∧ 𝑧 ∈ (ℝ ∖ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (𝑓‘𝑧) = 0)
190	161, 166, 167, 189	itg10a 23477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 ∈ dom ∫1 ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1‘𝑓) = 0)
191	160, 190	sylan 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → (∫1‘𝑓) = 0)
192	159, 191	breqtrd 4679	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → 𝑏 < 0)
193	158, 192, 137	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
194		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → 𝑓 ∈ dom ∫1)
195		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
196	194, 195	anim12i 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ))
197	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
198		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓‘𝑢) ∈ V
199	198	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑓‘𝑢) ∈ V)
200		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V
201	200, 105	ifex 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
202	201	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V)
203	35	feqmptd 6249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 = (𝑢 ∈ ℝ ↦ (𝑓‘𝑢)))
204	203	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑢 ∈ ℝ ↦ (𝑓‘𝑢)))
205		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
206	197, 199, 202, 204, 205	offval2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
207		ovif2 6738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑢) − 0))
208	172, 163	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ ran 𝑓) = ℝ)
209	208	difeq1d 3727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) = (ℝ ∖ (◡𝑓 “ {0})))
210	171, 209	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓:ℝ⟶ℝ → (◡𝑓 “ (ran 𝑓 ∖ {0})) = (ℝ ∖ (◡𝑓 “ {0})))
211	210	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓:ℝ⟶ℝ → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
212	35, 211	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
213	212	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
214		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → 𝑢 ∈ ℝ)
215	214	biantrurd 529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (◡𝑓 “ {0}))))
216		eldif 3584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0})) ↔ (𝑢 ∈ ℝ ∧ ¬ 𝑢 ∈ (◡𝑓 “ {0})))
217	215, 216	syl6bbr 278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ {0}) ↔ 𝑢 ∈ (ℝ ∖ (◡𝑓 “ {0}))))
218	213, 217	bitr4d 271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ 𝑢 ∈ (◡𝑓 “ {0})))
219	218	con2bid 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ {0}) ↔ ¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
220		fniniseg 6338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓 Fn ℝ → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
221	35, 181, 220	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
222	221	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (𝑢 ∈ (◡𝑓 “ {0}) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
223	219, 222	bitr3d 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0)))
224		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓‘𝑢) = 0 → ((𝑓‘𝑢) − 0) = (0 − 0))
225		0m0e0 11130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (0 − 0) = 0
226	224, 225	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑓‘𝑢) = 0 → ((𝑓‘𝑢) − 0) = 0)
227	226	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑢 ∈ ℝ ∧ (𝑓‘𝑢) = 0) → ((𝑓‘𝑢) − 0) = 0)
228	223, 227	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑢) − 0) = 0))
229	228	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) ∧ ¬ 𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑢) − 0) = 0)
230	229	ifeq2da 4117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑢) − 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
231	207, 230	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
232	231	mpteq2dva 4744	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ ((𝑓‘𝑢) − if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
233	206, 232	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
234		simpll 790	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 ∈ dom ∫1)
235	200	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
236		1ex 10035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ V
237	236, 105	ifex 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
238	237	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑢 ∈ ℝ) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
239		fconstmpt 5163	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
240	239	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑢 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
241		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))
242	197, 235, 238, 240, 241	offval2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))))
243		ovif2 6738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0))
244		resubcl 10345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((∫1‘𝑓) ∈ ℝ ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
245	8, 244	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
246	245	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
247		2re 11090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 2 ∈ ℝ
248		i1fima 23445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓 ∈ dom ∫1 → (◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol)
249		mblvol 23298	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
250	248, 249	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
251		neldifsn 4321	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ¬ 0 ∈ (ran 𝑓 ∖ {0})
252		i1fima2 23446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓 ∈ dom ∫1 ∧ ¬ 0 ∈ (ran 𝑓 ∖ {0})) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
253	251, 252	mpan2 707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
254	250, 253	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓 ∈ dom ∫1 → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
255		remulcl 10021	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((2 ∈ ℝ ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
256	247, 254, 255	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑓 ∈ dom ∫1 → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
257	256	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
258		2cnd 11093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ∈ ℂ)
259	254	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
260	259	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
261		2ne0 11113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ 2 ≠ 0
262	261	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 2 ≠ 0)
263		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)
264	258, 260, 262, 263	mulne0d 10679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ≠ 0)
265	246, 257, 264	redivcld 10853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
266	265	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
267	266	mulid1d 10057	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1) = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
268	266	mul01d 10235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0) = 0)
269	267, 268	ifeq12d 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
270	243, 269	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
271	270	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
272	242, 271	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
273		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
274	273	i1f1 23457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
275	248, 253, 274	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
276	275	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
277	276, 265	i1fmulc 23470	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) ∈ dom ∫1)
278	272, 277	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1)
279		i1fsub 23475	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (𝑓 ∘𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
280	234, 278, 279	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
281	233, 280	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
282		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
283		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
284	283	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 = 𝑧 → ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
285	282, 284	ifbieq1d 4109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑢 = 𝑧 → if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
286		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
287		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
288	287, 105	ifex 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
289	285, 286, 288	fvmpt 6282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∈ ℝ → ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
290	289	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ ℝ → (0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) ↔ 0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
291	290, 289	ifbieq1d 4109	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ ℝ → if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
292		iftrue 4092	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
293		iftrue 4092	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
294	293	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
295	294, 293	ifbieq1d 4109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
296		iftrue 4092	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
297	295, 296	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
298	292, 297	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
299		iffalse 4095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
300		ianor 509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))))
301	294	ifbid 4108	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
302		iffalse 4095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
303	301, 302	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
304	303	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
305		iffalse 4095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
306		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 0 = 0
307		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) → (if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ↔ if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
308		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (0 = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) → (0 = 0 ↔ if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0))
309	307, 308	ifboth 4124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ∧ 0 = 0) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
310	305, 306, 309	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
311	304, 310	pm2.61d1 171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
312	311, 310	jaoi 394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
313	300, 312	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0) = 0)
314	299, 313	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0))
315	298, 314	pm2.61i 176	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0), 0)
316	291, 315	syl6reqr 2675	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ ℝ → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0))
317	316	mpteq2ia 4740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if(0 ≤ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧), 0))
318	317	i1fpos 23473	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑢 ∈ ℝ ↦ if(𝑢 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((𝑓‘𝑢) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
319	281, 318	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
320	196, 319	sylan 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
321	196, 265	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
322	8	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (∫1‘𝑓) ∈ ℝ)
323	322, 195, 244	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
324	323	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℝ)
325	256	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
326	325	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℝ)
327		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 < (∫1‘𝑓))
328		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 𝑏 ∈ ℝ)
329	141	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (∫1‘𝑓) ∈ ℝ)
330	328, 329	posdifd 10614	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → (𝑏 < (∫1‘𝑓) ↔ 0 < ((∫1‘𝑓) − 𝑏)))
331	327, 330	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → 0 < ((∫1‘𝑓) − 𝑏))
332	331	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < ((∫1‘𝑓) − 𝑏))
333	254	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
334	333	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ)
335		mblss 23299	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol → (◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ)
336		ovolge0 23249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((◡𝑓 “ (ran 𝑓 ∖ {0})) ⊆ ℝ → 0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
337	248, 335, 336	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → 0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
338		ltlen 10138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((0 ∈ ℝ ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)))
339	48, 254, 338	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → (0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ (0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0)))
340	339	biimprd 238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → ((0 ≤ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
341	337, 340	mpand 711	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
342	341	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0 → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
343	342	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
344	343	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
345		2pos 11112	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 < 2
346		mulgt0 10115	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2 ∈ ℝ ∧ 0 < 2) ∧ ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
347	247, 345, 346	mpanl12 718	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ ∧ 0 < (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
348	334, 344, 347	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
349	324, 326, 332, 348	divgt0d 10959	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 0 < (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
350	321, 349	elrpd 11869	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ+)
351		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 ∘𝑟 ≤ 𝐹)
352	351	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 ∘𝑟 ≤ 𝐹)
353		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹:ℝ⟶(0[,]+∞) → 𝐹 Fn ℝ)
354	35, 181	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → 𝑓 Fn ℝ)
355	354	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 Fn ℝ)
356		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝑓 Fn ℝ)
357		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → 𝐹 Fn ℝ)
358	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → ℝ ∈ V)
359		inidm 3822	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℝ ∩ ℝ) = ℝ
360		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) = (𝑓‘𝑧))
361		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) = (𝐹‘𝑧))
362	356, 357, 358, 358, 359, 360, 361	ofrfval 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹 Fn ℝ ∧ 𝑓 Fn ℝ) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
363	353, 355, 362	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
364	363	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧)))
365		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → 𝑓 ∈ dom ∫1)
366	365	anim2i 593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1))
367	366, 195	anim12i 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ))
368		breq1 4656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (0 ≤ (𝐹‘𝑧) ↔ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
369		breq1 4656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
370		simplll 798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝐹:ℝ⟶(0[,]+∞))
371	370	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (0[,]+∞))
372	371, 100	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧) ∈ ℝ* ∧ 0 ≤ (𝐹‘𝑧)))
373	372	simprd 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ≤ (𝐹‘𝑧))
374	373	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) → 0 ≤ (𝐹‘𝑧))
375		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
376	375	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → ((((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
377		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (0 = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
378	377	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) → ((0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧) ↔ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
379	35	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
380	379	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
381	380	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℂ)
382	245	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − 𝑏) ∈ ℂ)
383	382	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − 𝑏) ∈ ℂ)
384	256	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 ∈ dom ∫1 → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
385	384	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) ∈ ℂ)
386	383, 385, 264	divcld 10801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
387	386	adantlll 754	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
388	387	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℂ)
389	381, 388	npcand 10396	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (𝑓‘𝑧))
390	389	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (𝑓‘𝑧))
391		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (𝑓‘𝑧) ≤ (𝐹‘𝑧))
392	390, 391	eqbrtrd 4675	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
393	392	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) ∧ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
394	299	pm2.24d 147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧)))
395	394	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0 ∧ ¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
396	395	adantll 750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) ∧ ¬ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))) → (0 + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
397	376, 378, 393, 396	ifbothda 4123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) ∧ ¬ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0) → (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ (𝐹‘𝑧))
398	368, 369, 374, 397	ifbothda 4123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ (𝑓‘𝑧) ≤ (𝐹‘𝑧)) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧))
399	398	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑓 ∈ dom ∫1) ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) ≤ (𝐹‘𝑧) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
400	367, 399	sylanl1 682	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) ≤ (𝐹‘𝑧) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
401	400	ralimdva 2962	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∀𝑧 ∈ ℝ (𝑓‘𝑧) ≤ (𝐹‘𝑧) → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
402	364, 401	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑟 ≤ 𝐹 → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
403	352, 402	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧))
404		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ V
405	105, 404	ifex 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ∈ V
406	405	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑧 ∈ ℝ) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ∈ V)
407		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))))
408	104, 406, 99, 407, 67	ofrfval2 6915	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:ℝ⟶(0[,]+∞) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
409	408	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹 ↔ ∀𝑧 ∈ ℝ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) ≤ (𝐹‘𝑧)))
410	403, 409	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹)
411		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
412	411	ifeq2d 4105	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦)) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
413	412	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))))
414	413	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) → ((𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹))
415	414	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ+ ∧ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))) ∘𝑟 ≤ 𝐹) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹)
416	350, 410, 415	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹)
417		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = 𝑔 → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))
418	417	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))
419	418	biantrud 528	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
420		nfmpt1 4747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎ𝑧(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
421	420	nfeq2 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑧 𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
422		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (𝑔‘𝑧) = ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧))
423	287, 105	ifex 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V
424		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
425	424	fvmpt2 6291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑧 ∈ ℝ ∧ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V) → ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
426	423, 425	mpan2 707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ ℝ → ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
427	422, 426	sylan9eq 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → (𝑔‘𝑧) = if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
428	427	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) = 0 ↔ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0))
429	427	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → ((𝑔‘𝑧) + 𝑦) = (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))
430	428, 429	ifbieq2d 4111	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∧ 𝑧 ∈ ℝ) → if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦)) = if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦)))
431	421, 430	mpteq2da 4743	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) = (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))))
432	431	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → ((𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹))
433	432	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹))
434	419, 433	bitr3d 270	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ↔ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹))
435	434	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 ∧ ∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if(if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0, 0, (if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) + 𝑦))) ∘𝑟 ≤ 𝐹) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
436	320, 416, 435	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
437		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℝ)
438	200	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ V)
439	236, 105	ifex 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V
440	439	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0) ∈ V)
441		fconstmpt 5163	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
442	441	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) = (𝑧 ∈ ℝ ↦ (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
443		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))
444	197, 438, 440, 442, 443	offval2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))))
445		ovif2 6738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0))
446	267, 268	ifeq12d 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 1), ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
447	445, 446	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))
448	447	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
449	444, 448	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
450		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))
451	450	i1f1 23457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
452	248, 253, 451	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
453	452	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)) ∈ dom ∫1)
454	453, 265	i1fmulc 23470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) ∈ dom ∫1)
455	449, 454	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1)
456		i1fsub 23475	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
457	234, 455, 456	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
458		itg1cl 23452	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1 → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
459	457, 458	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
460	459	adantlrl 756	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ∈ ℝ)
461	319	adantlrl 756	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1)
462		itg1cl 23452	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ ℝ)
463	461, 462	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ ℝ)
464		simplrl 800	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘𝑓))
465		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
466	8	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (∫1‘𝑓) ∈ ℝ)
467	97	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → 2 ∈ ℝ+)
468	465, 466, 467	ltdiv1d 11917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 < (∫1‘𝑓) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
469		recn 10026	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑏 ∈ ℝ → 𝑏 ∈ ℂ)
470	469	2halvesd 11278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℝ → ((𝑏 / 2) + (𝑏 / 2)) = 𝑏)
471	470	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 − (𝑏 / 2)))
472	469	halfcld 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℂ)
473	472, 472	pncand 10393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑏 ∈ ℝ → (((𝑏 / 2) + (𝑏 / 2)) − (𝑏 / 2)) = (𝑏 / 2))
474	471, 473	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ ℝ → (𝑏 − (𝑏 / 2)) = (𝑏 / 2))
475	474	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 ∈ ℝ → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
476	475	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ (𝑏 / 2) < ((∫1‘𝑓) / 2)))
477		rehalfcl 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ∈ ℝ → (𝑏 / 2) ∈ ℝ)
478	477	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℝ)
479	8	rehalfcld 11279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) / 2) ∈ ℝ)
480	479	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) / 2) ∈ ℝ)
481	465, 478, 480	ltsubaddd 10623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((𝑏 − (𝑏 / 2)) < ((∫1‘𝑓) / 2) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
482	468, 476, 481	3bitr2d 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
483	482	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
484	483	adantlrl 756	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑏 < (∫1‘𝑓) ↔ 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2))))
485	464, 484	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (((∫1‘𝑓) / 2) + (𝑏 / 2)))
486	453, 265	itg1mulc 23471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))))
487	449	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘((ℝ × {(((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))}) ∘𝑓 · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
488	450	itg11 23458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((◡𝑓 “ (ran 𝑓 ∖ {0})) ∈ dom vol ∧ (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℝ) → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
489	248, 253, 488	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0))) = (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
490	489	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
491	490	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
492	253	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
493	492	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ∈ ℂ)
494	266, 493	mulcomd 10061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) = ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
495	250	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) = (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))
496	495	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)))
497	260, 383	mulcomd 10061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = (((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
498	496, 497	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) = (((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))
499	498	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = ((((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
500	493, 383, 385, 264	divassd 10836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · ((∫1‘𝑓) − 𝑏)) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
501	383, 258, 260, 262, 263	divcan5rd 10828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) = (((∫1‘𝑓) − 𝑏) / 2))
502	499, 500, 501	3eqtr3d 2664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((vol‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) · (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) = (((∫1‘𝑓) − 𝑏) / 2))
503	491, 494, 502	3eqtrd 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) · (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), 1, 0)))) = (((∫1‘𝑓) − 𝑏) / 2))
504	486, 487, 503	3eqtr3d 2664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (((∫1‘𝑓) − 𝑏) / 2))
505	504	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)))
506		itg1sub 23476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ dom ∫1) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
507	234, 455, 506	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = ((∫1‘𝑓) − (∫1‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
508	8	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → (∫1‘𝑓) ∈ ℂ)
509	508	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘𝑓) ∈ ℂ)
510	469	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 ∈ ℂ)
511	509, 510, 258, 262	divsubdird 10840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − 𝑏) / 2) = (((∫1‘𝑓) / 2) − (𝑏 / 2)))
512	511	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)) = ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))))
513	508	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (∫1‘𝑓) ∈ ℂ)
514	513	halfcld 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) / 2) ∈ ℂ)
515	472	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → (𝑏 / 2) ∈ ℂ)
516	513, 514, 515	subsubd 10420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) → ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))) = (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)))
517	516	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((∫1‘𝑓) − (((∫1‘𝑓) / 2) − (𝑏 / 2))) = (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)))
518	508	2halvesd 11278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → (((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) = (∫1‘𝑓))
519	518	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) − ((∫1‘𝑓) / 2)))
520	508	halfcld 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) / 2) ∈ ℂ)
521	520, 520	pncand 10393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓 ∈ dom ∫1 → ((((∫1‘𝑓) / 2) + ((∫1‘𝑓) / 2)) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) / 2))
522	519, 521	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑓 ∈ dom ∫1 → ((∫1‘𝑓) − ((∫1‘𝑓) / 2)) = ((∫1‘𝑓) / 2))
523	522	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ dom ∫1 → (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
524	523	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) − ((∫1‘𝑓) / 2)) + (𝑏 / 2)) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
525	512, 517, 524	3eqtrrd 2661	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (((∫1‘𝑓) / 2) + (𝑏 / 2)) = ((∫1‘𝑓) − (((∫1‘𝑓) − 𝑏) / 2)))
526	505, 507, 525	3eqtr4d 2666	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
527	526	adantlrl 756	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) = (((∫1‘𝑓) / 2) + (𝑏 / 2)))
528	485, 527	breqtrrd 4681	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))))
529	457	adantlrl 756	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1)
530		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
531	530	adantlrl 756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0))
532	234, 36	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ ℝ)
533	265	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))) ∈ ℝ)
534	532, 533	resubcld 10458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
535	534	leidd 10594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
536	535	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
537	296	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
538	537	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
539	536, 538	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
540	534	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∈ ℝ)
541	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → 0 ∈ ℝ)
542	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → 0 ∈ ℝ)
543	534, 542	ltnled 10184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) < 0 ↔ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
544	543	biimpar 502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) < 0)
545	540, 541, 544	ltled 10185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0)
546		iffalse 4095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
547	546	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0))
548	547	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → (((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ 0))
549	545, 548	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
550	539, 549	pm2.61dan 832	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑓 ∈ dom ∫1 ∧ 𝑏 ∈ ℝ) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
551	531, 550	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
552	551	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
553		iftrue 4092	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))
554	553	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))))
555		iba 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ↔ (0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})))))
556	555	bicomd 213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) ↔ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))))))
557	556	ifbid 4108	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
558	554, 557	breq12d 4666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
559	558	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ≤ if(0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
560	552, 559	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
561	35	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓:ℝ⟶ℝ)
562	171	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ 𝑧 ∈ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0}))))
563		eldif 3584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑧 ∈ ((◡𝑓 “ ran 𝑓) ∖ (◡𝑓 “ {0})) ↔ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})))
564	562, 563	syl6bb 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
565	564	notbid 308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑓:ℝ⟶ℝ → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
566	565	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) ↔ ¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0}))))
567		pm4.53 513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})) ↔ (¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (◡𝑓 “ {0})))
568	208	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ ran 𝑓) ↔ 𝑧 ∈ ℝ))
569	568	biimpar 502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ (◡𝑓 “ ran 𝑓))
570	569	pm2.24d 147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) → (𝑓‘𝑧) = 0))
571	182	simplbda 654	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑓 Fn ℝ ∧ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0)
572	571	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 Fn ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
573	181, 572	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑓:ℝ⟶ℝ → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
574	573	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ (◡𝑓 “ {0}) → (𝑓‘𝑧) = 0))
575	570, 574	jaod 395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → ((¬ 𝑧 ∈ (◡𝑓 “ ran 𝑓) ∨ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0))
576	567, 575	syl5bi 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ (𝑧 ∈ (◡𝑓 “ ran 𝑓) ∧ ¬ 𝑧 ∈ (◡𝑓 “ {0})) → (𝑓‘𝑧) = 0))
577	566, 576	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (𝑓‘𝑧) = 0))
578	577	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑓:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0)
579	561, 578	sylanl1 682	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (𝑓‘𝑧) = 0)
580	579	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) = (0 − 0))
581	580, 225	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) = 0)
582	581, 30	syl6eqbr 4692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − 0) ≤ 0)
583		iffalse 4095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) = 0)
584	583	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = ((𝑓‘𝑧) − 0))
585	300, 299	sylbir 225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((¬ 0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∨ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
586	585	olcs 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) = 0)
587	584, 586	breq12d 4666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − 0) ≤ 0))
588	587	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → (((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ↔ ((𝑓‘𝑧) − 0) ≤ 0))
589	582, 588	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ ¬ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
590	560, 589	pm2.61dan 832	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
591	590	ralrimiva 2966	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∀𝑧 ∈ ℝ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))
592	63	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ℝ ∈ V)
593		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V
594	593	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ∈ V)
595	423	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0) ∈ V)
596		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓‘𝑧) ∈ V
597	596	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑓‘𝑧) ∈ V)
598	200, 105	ifex 4156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V
599	598	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0) ∈ V)
600	70	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑓 = (𝑧 ∈ ℝ ↦ (𝑓‘𝑧)))
601		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))
602	592, 597, 599, 600, 601	offval2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) = (𝑧 ∈ ℝ ↦ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))))
603		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) = (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
604	592, 594, 595, 602, 603	ofrfval2 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ((𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ↔ ∀𝑧 ∈ ℝ ((𝑓‘𝑧) − if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)) ≤ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
605	591, 604	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))
606		itg1le 23480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∈ dom ∫1 ∧ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)) ∈ dom ∫1 ∧ (𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0))) ∘𝑟 ≤ (𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ≤ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
607	529, 461, 605, 606	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → (∫1‘(𝑓 ∘𝑓 − (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0})), (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))))), 0)))) ≤ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
608	437, 460, 463, 528, 607	ltletrd 10197	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 ∈ dom ∫1 ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
609	608	adantllr 755	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
610	609	adantlll 754	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))
611		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) ∈ V
612		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (𝑎 = (∫1‘𝑔) ↔ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)))
613	612	anbi2d 740	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
614	613	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔))))
615		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → (𝑏 < 𝑎 ↔ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))))
616	614, 615	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) → ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎) ↔ (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ∧ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))))))
617	611, 616	spcev 3300	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0))) = (∫1‘𝑔)) ∧ 𝑏 < (∫1‘(𝑧 ∈ ℝ ↦ if((0 ≤ ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))) ∧ 𝑧 ∈ (◡𝑓 “ (ran 𝑓 ∖ {0}))), ((𝑓‘𝑧) − (((∫1‘𝑓) − 𝑏) / (2 · (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0})))))), 0)))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
618	436, 610, 617	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) ∧ (vol*‘(◡𝑓 “ (ran 𝑓 ∖ {0}))) ≠ 0) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
619	193, 618	pm2.61dane 2881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ (𝑏 < (∫1‘𝑓) ∧ 𝑏 ∈ ℝ)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
620	619	expr 643	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
621	620	adantllr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
622	621	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → (𝑏 ∈ ℝ → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
623	157, 622	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) ∧ 𝑏 ≠ -∞) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
624	139, 623	pm2.61dane 2881	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < (∫1‘𝑓)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
625	624	ex 450	. . . . . . . . . . . . . 14 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < (∫1‘𝑓) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
626	94, 625	sylbid 230	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → (𝑏 < 𝑠 → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
627	626	imp 445	. . . . . . . . . . . 12 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
628	627	an32s 846	. . . . . . . . . . 11 ⊢ ((((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ 𝑏 < 𝑠) ∧ (𝑓 ∈ dom ∫1 ∧ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
629	628	rexlimdvaa 3032	. . . . . . . . . 10 ⊢ (((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) ∧ 𝑏 < 𝑠) → (∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
630	629	expimpd 629	. . . . . . . . 9 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → ((𝑏 < 𝑠 ∧ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
631	630	ancomsd 470	. . . . . . . 8 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → ((∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
632	631	exlimdv 1861	. . . . . . 7 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (∃𝑠(∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠) → ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎)))
633		eqeq1 2626	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → (𝑥 = (∫1‘𝑓) ↔ 𝑠 = (∫1‘𝑓)))
634	633	anbi2d 740	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → ((𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ (𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))))
635	634	rexbidv 3052	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓)) ↔ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓))))
636	635	rexab 3369	. . . . . . 7 ⊢ (∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠 ↔ ∃𝑠(∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑠 = (∫1‘𝑓)) ∧ 𝑏 < 𝑠))
637		eqeq1 2626	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑥 = (∫1‘𝑔) ↔ 𝑎 = (∫1‘𝑔)))
638	637	anbi2d 740	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ (∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔))))
639	638	rexbidv 3052	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔))))
640	639	rexab 3369	. . . . . . 7 ⊢ (∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎 ↔ ∃𝑎(∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑎 = (∫1‘𝑔)) ∧ 𝑏 < 𝑎))
641	632, 636, 640	3imtr4g 285	. . . . . 6 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (∃𝑠 ∈ {𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}𝑏 < 𝑠 → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎))
642	92, 641	sylbid 230	. . . . 5 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝑏 ∈ ℝ*) → (𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ) → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎))
643	642	impr 649	. . . 4 ⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ (𝑏 ∈ ℝ* ∧ 𝑏 < sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))) → ∃𝑎 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}𝑏 < 𝑎)
644	6, 15, 89, 643	eqsupd 8363	. . 3 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
645	4, 644	syl5eq 2668	. 2 ⊢ (𝐹:ℝ⟶(0[,]+∞) → sup(𝐿, ℝ*, < ) = sup({𝑥 ∣ ∃𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑓))}, ℝ*, < ))
646	2, 645	eqtr4d 2659	1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177   class class class wbr 4653   ↦ cmpt 4729   Or wor 5034   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   ∘_𝑟 cofr 6896  supcsup 8346  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  +∞cpnf 10071  -∞cmnf 10072  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  2c2 11070  ℝ⁺crp 11832  [,]cicc 12178  vol*covol 23231  volcvol 23232  ∫₁citg1 23384  ∫₂citg2 23385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390

This theorem is referenced by:  itg2addnc  33464