Theorem itg1climres 23481

Description: Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is ℝ yields a sequence of simple functions whose integrals approach the integral of 𝐹. (Contributed by Mario Carneiro, 15-Aug-2014.)

Hypotheses

Ref	Expression
itg1climres.1	⊢ (𝜑 → 𝐴:ℕ⟶dom vol)
itg1climres.2	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)))
itg1climres.3	⊢ (𝜑 → ∪ ran 𝐴 = ℝ)
itg1climres.4	⊢ (𝜑 → 𝐹 ∈ dom ∫1)
itg1climres.5	⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))

Assertion

Ref	Expression
itg1climres	⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ (∫1‘𝐹))

Distinct variable groups:   𝑥,𝑛,𝐴   𝑛,𝐹,𝑥   𝜑,𝑛,𝑥

Allowed substitution hints:   𝐺(𝑥,𝑛)

Proof of Theorem itg1climres

Dummy variables 𝑗 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11723	. . 3 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 11408	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		itg1climres.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1frn 23444	. . . . 5 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
6		difss 3737	. . . 4 ⊢ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹
7		ssfi 8180	. . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ ran 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin)
8	5, 6, 7	sylancl 694	. . 3 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin)
9		1zzd 11408	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 1 ∈ ℤ)
10		i1fima 23445	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {𝑘}) ∈ dom vol)
11	3, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐹 “ {𝑘}) ∈ dom vol)
12	11	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ {𝑘}) ∈ dom vol)
13		itg1climres.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:ℕ⟶dom vol)
14	13	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ dom vol)
15	14	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ dom vol)
16		inmbl 23310	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴‘𝑛) ∈ dom vol) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ∈ dom vol)
17	12, 15, 16	syl2anc 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ∈ dom vol)
18		mblvol 23298	. . . . . . . . 9 ⊢ (((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ∈ dom vol → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
19	17, 18	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
20		inss1 3833	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘})
21	20	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘}))
22		mblss 23299	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (◡𝐹 “ {𝑘}) ⊆ ℝ)
23	12, 22	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ {𝑘}) ⊆ ℝ)
24		mblvol 23298	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘})))
25	12, 24	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘})))
26		i1fima2sn 23447	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
27	3, 26	sylan 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
28	27	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘(◡𝐹 “ {𝑘})) ∈ ℝ)
29	25, 28	eqeltrrd 2702	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘(◡𝐹 “ {𝑘})) ∈ ℝ)
30		ovolsscl 23254	. . . . . . . . 9 ⊢ ((((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘}) ∧ (◡𝐹 “ {𝑘}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑘})) ∈ ℝ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ∈ ℝ)
31	21, 23, 29, 30	syl3anc 1326	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ∈ ℝ)
32	19, 31	eqeltrd 2701	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ∈ ℝ)
33		eqid 2622	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
34	32, 33	fmptd 6385	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))):ℕ⟶ℝ)
35		itg1climres.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)))
36	35	adantlr 751	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)))
37		sslin 3839	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
38	36, 37	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
39	13	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐴:ℕ⟶dom vol)
40		peano2nn 11032	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
41		ffvelrn 6357	. . . . . . . . . . . . . 14 ⊢ ((𝐴:ℕ⟶dom vol ∧ (𝑛 + 1) ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
42	39, 40, 41	syl2an 494	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝐴‘(𝑛 + 1)) ∈ dom vol)
43		inmbl 23310	. . . . . . . . . . . . 13 ⊢ (((◡𝐹 “ {𝑘}) ∈ dom vol ∧ (𝐴‘(𝑛 + 1)) ∈ dom vol) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
44	12, 42, 43	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol)
45		mblss 23299	. . . . . . . . . . . 12 ⊢ (((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ)
46	44, 45	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ)
47		ovolss 23253	. . . . . . . . . . 11 ⊢ ((((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∧ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ⊆ ℝ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
48	38, 46, 47	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
49		mblvol 23298	. . . . . . . . . . 11 ⊢ (((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ∈ dom vol → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
50	44, 49	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
51	48, 19, 50	3brtr4d 4685	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
52	51	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
53		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (𝐴‘𝑛) = (𝐴‘𝑗))
54	53	ineq2d 3814	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))
55	54	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))))
56		fvex 6201	. . . . . . . . . . . 12 ⊢ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ∈ V
57	55, 33, 56	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))))
58		peano2nn 11032	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
59		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → (𝐴‘𝑛) = (𝐴‘(𝑗 + 1)))
60	59	ineq2d 3814	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
61	60	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑗 + 1) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
62		fvex 6201	. . . . . . . . . . . . 13 ⊢ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))) ∈ V
63	61, 33, 62	fvmpt 6282	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
64	58, 63	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
65	57, 64	breq12d 4666	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))))
66	65	ralbiia 2979	. . . . . . . . 9 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) ↔ ∀𝑗 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
67		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
68	67	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → (𝐴‘(𝑛 + 1)) = (𝐴‘(𝑗 + 1)))
69	68	ineq2d 3814	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
70	69	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
71	55, 70	breq12d 4666	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → ((vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))))
72	71	cbvralv 3171	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))) ↔ ∀𝑗 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
73	66, 72	bitr4i 267	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)) ↔ ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1)))))
74	52, 73	sylibr 224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)))
75	74	r19.21bi 2932	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘(𝑗 + 1)))
76		ovolss 23253	. . . . . . . . . . 11 ⊢ ((((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ (◡𝐹 “ {𝑘}) ∧ (◡𝐹 “ {𝑘}) ⊆ ℝ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘(◡𝐹 “ {𝑘})))
77	20, 23, 76	sylancr 695	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol*‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol*‘(◡𝐹 “ {𝑘})))
78	77, 19, 25	3brtr4d 4685	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})))
79	78	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})))
80	57	breq1d 4663	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘}))))
81	80	ralbiia 2979	. . . . . . . . 9 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ ∀𝑗 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘})))
82	55	breq1d 4663	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → ((vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘}))))
83	82	cbvralv 3171	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ ∀𝑗 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))) ≤ (vol‘(◡𝐹 “ {𝑘})))
84	81, 83	bitr4i 267	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})) ↔ ∀𝑛 ∈ ℕ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) ≤ (vol‘(◡𝐹 “ {𝑘})))
85	79, 84	sylibr 224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘})))
86		breq2 4657	. . . . . . . . 9 ⊢ (𝑥 = (vol‘(◡𝐹 “ {𝑘})) → (((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥 ↔ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘}))))
87	86	ralbidv 2986	. . . . . . . 8 ⊢ (𝑥 = (vol‘(◡𝐹 “ {𝑘})) → (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘}))))
88	87	rspcev 3309	. . . . . . 7 ⊢ (((vol‘(◡𝐹 “ {𝑘})) ∈ ℝ ∧ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ (vol‘(◡𝐹 “ {𝑘}))) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥)
89	27, 85, 88	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥)
90	1, 9, 34, 75, 89	climsup 14400	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⇝ sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ))
91		eqid 2622	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))
92	17, 91	fmptd 6385	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))):ℕ⟶dom vol)
93	38	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
94		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑗) ∈ V
95	94	inex2 4800	. . . . . . . . . . . 12 ⊢ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ∈ V
96	54, 91, 95	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))
97		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (𝐴‘(𝑗 + 1)) ∈ V
98	97	inex2 4800	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))) ∈ V
99	60, 91, 98	fvmpt 6282	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
100	58, 99	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
101	96, 100	sseq12d 3634	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) ↔ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
102	101	ralbiia 2979	. . . . . . . . 9 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) ↔ ∀𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
103	54, 69	sseq12d 3634	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1)))))
104	103	cbvralv 3171	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))) ↔ ∀𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑗 + 1))))
105	102, 104	bitr4i 267	. . . . . . . 8 ⊢ (∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)) ↔ ∀𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ⊆ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘(𝑛 + 1))))
106	93, 105	sylibr 224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1)))
107		volsup 23324	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))):ℕ⟶dom vol ∧ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) ⊆ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘(𝑗 + 1))) → (vol‘∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ))
108	92, 106, 107	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ))
109	96	iuneq2i 4539	. . . . . . . . . 10 ⊢ ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ∪ 𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))
110	54	cbviunv 4559	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ∪ 𝑗 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗))
111		iunin2 4584	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
112	109, 110, 111	3eqtr2i 2650	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛))
113		ffn 6045	. . . . . . . . . . . . . 14 ⊢ (𝐴:ℕ⟶dom vol → 𝐴 Fn ℕ)
114		fniunfv 6505	. . . . . . . . . . . . . 14 ⊢ (𝐴 Fn ℕ → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ∪ ran 𝐴)
115	13, 113, 114	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ∪ ran 𝐴)
116		itg1climres.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ ran 𝐴 = ℝ)
117	115, 116	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ℝ)
118	117	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) = ℝ)
119	118	ineq2d 3814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((◡𝐹 “ {𝑘}) ∩ ℝ))
120	11	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ∈ dom vol)
121	120, 22	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ⊆ ℝ)
122		df-ss 3588	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑘}) ⊆ ℝ ↔ ((◡𝐹 “ {𝑘}) ∩ ℝ) = (◡𝐹 “ {𝑘}))
123	121, 122	sylib 208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((◡𝐹 “ {𝑘}) ∩ ℝ) = (◡𝐹 “ {𝑘}))
124	119, 123	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((◡𝐹 “ {𝑘}) ∩ ∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = (◡𝐹 “ {𝑘}))
125	112, 124	syl5eq 2668	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = (◡𝐹 “ {𝑘}))
126		ffn 6045	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))):ℕ⟶dom vol → (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) Fn ℕ)
127		fniunfv 6505	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) Fn ℕ → ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
128	92, 126, 127	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))‘𝑗) = ∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
129	125, 128	eqtr3d 2658	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) = ∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
130	129	fveq2d 6195	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = (vol‘∪ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
131		frn 6053	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))):ℕ⟶ℝ → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⊆ ℝ)
132	34, 131	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⊆ ℝ)
133		fdm 6051	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))):ℕ⟶ℝ → dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ℕ)
134	34, 133	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ℕ)
135		1nn 11031	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
136		ne0i 3921	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
137	135, 136	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ℕ ≠ ∅)
138	134, 137	eqnetrd 2861	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
139		dm0rn0 5342	. . . . . . . . . 10 ⊢ (dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ∅)
140	139	necon3bii 2846	. . . . . . . . 9 ⊢ (dom (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
141	138, 140	sylib 208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅)
142		ffn 6045	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))):ℕ⟶ℝ → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) Fn ℕ)
143		breq1 4656	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) → (𝑧 ≤ 𝑥 ↔ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
144	143	ralrn 6362	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
145	34, 142, 144	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
146	145	rexbidv 3052	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ≤ 𝑥))
147	89, 146	mpbird 247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥)
148		supxrre 12157	. . . . . . . 8 ⊢ ((ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))𝑧 ≤ 𝑥) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ))
149	132, 141, 147, 148	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ))
150		rnco2 5642	. . . . . . . . 9 ⊢ ran (vol ∘ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
151		eqidd 2623	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
152		volf 23297	. . . . . . . . . . . . 13 ⊢ vol:dom vol⟶(0[,]+∞)
153	152	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → vol:dom vol⟶(0[,]+∞))
154	153	feqmptd 6249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → vol = (𝑦 ∈ dom vol ↦ (vol‘𝑦)))
155		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) → (vol‘𝑦) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
156	17, 151, 154, 155	fmptco 6396	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol ∘ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
157	156	rneqd 5353	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (vol ∘ (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
158	150, 157	syl5reqr 2671	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
159	158	supeq1d 8352	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ))
160	149, 159	eqtr3d 2658	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ) = sup((vol “ ran (𝑛 ∈ ℕ ↦ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ*, < ))
161	108, 130, 160	3eqtr4d 2666	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = sup(ran (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))), ℝ, < ))
162	90, 161	breqtrrd 4681	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ⇝ (vol‘(◡𝐹 “ {𝑘})))
163		i1ff 23443	. . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
164		frn 6053	. . . . . . . 8 ⊢ (𝐹:ℝ⟶ℝ → ran 𝐹 ⊆ ℝ)
165	3, 163, 164	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
166	165	ssdifssd 3748	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
167	166	sselda 3603	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ)
168	167	recnd 10068	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ)
169		nnex 11026	. . . . . 6 ⊢ ℕ ∈ V
170	169	mptex 6486	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) ∈ V
171	170	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) ∈ V)
172	34	ffvelrnda 6359	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ∈ ℝ)
173	172	recnd 10068	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗) ∈ ℂ)
174	55	oveq2d 6666	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
175		eqid 2622	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) = (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
176		ovex 6678	. . . . . . 7 ⊢ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))) ∈ V
177	174, 175, 176	fvmpt 6282	. . . . . 6 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
178	57	oveq2d 6666	. . . . . 6 ⊢ (𝑗 ∈ ℕ → (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗)) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
179	177, 178	eqtr4d 2659	. . . . 5 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗)))
180	179	adantl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = (𝑘 · ((𝑛 ∈ ℕ ↦ (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))‘𝑗)))
181	1, 9, 162, 168, 171, 173, 180	climmulc2 14367	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) ⇝ (𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
182	169	mptex 6486	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ∈ V
183	182	a1i 11	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ∈ V)
184	167	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → 𝑘 ∈ ℝ)
185	184, 32	remulcld 10070	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑛 ∈ ℕ) → (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) ∈ ℝ)
186	185, 175	fmptd 6385	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))):ℕ⟶ℝ)
187	186	ffvelrnda 6359	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℝ)
188	187	recnd 10068	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℂ)
189	188	anasss 679	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (ran 𝐹 ∖ {0}) ∧ 𝑗 ∈ ℕ)) → ((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) ∈ ℂ)
190	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ dom ∫1)
191		itg1climres.5	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
192	191	i1fres 23472	. . . . . . . . 9 ⊢ ((𝐹 ∈ dom ∫1 ∧ (𝐴‘𝑛) ∈ dom vol) → 𝐺 ∈ dom ∫1)
193	190, 14, 192	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 ∈ dom ∫1)
194	8	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ∈ Fin)
195		ffn 6045	. . . . . . . . . . . . . 14 ⊢ (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
196	3, 163, 195	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn ℝ)
197	196	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 Fn ℝ)
198		fnfvelrn 6356	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn ℝ ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran 𝐹)
199	197, 198	sylan 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ran 𝐹)
200		i1f0rn 23449	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹)
201	3, 200	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ran 𝐹)
202	201	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran 𝐹)
203	199, 202	ifcld 4131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ ran 𝐹)
204	203, 191	fmptd 6385	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺:ℝ⟶ran 𝐹)
205		frn 6053	. . . . . . . . 9 ⊢ (𝐺:ℝ⟶ran 𝐹 → ran 𝐺 ⊆ ran 𝐹)
206		ssdif 3745	. . . . . . . . 9 ⊢ (ran 𝐺 ⊆ ran 𝐹 → (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}))
207	204, 205, 206	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}))
208	165	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran 𝐹 ⊆ ℝ)
209	208	ssdifd 3746	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran 𝐹 ∖ {0}) ⊆ (ℝ ∖ {0}))
210		itg1val2 23451	. . . . . . . 8 ⊢ ((𝐺 ∈ dom ∫1 ∧ ((ran 𝐹 ∖ {0}) ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ (ran 𝐹 ∖ {0}) ∧ (ran 𝐹 ∖ {0}) ⊆ (ℝ ∖ {0}))) → (∫1‘𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐺 “ {𝑘}))))
211	193, 194, 207, 209, 210	syl13anc 1328	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫1‘𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐺 “ {𝑘}))))
212		fvex 6201	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹‘𝑥) ∈ V
213		c0ex 10034	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
214	212, 213	ifex 4156	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ V
215	191	fvmpt2 6291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ∈ V) → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
216	214, 215	mpan2 707	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
217	216	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
218	217	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑥) = 𝑘 ↔ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘))
219		eldifsni 4320	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (ran 𝐹 ∖ {0}) → 𝑘 ≠ 0)
220	219	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ≠ 0)
221		neeq1 2856	. . . . . . . . . . . . . . . . . . . 20 ⊢ (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ≠ 0 ↔ 𝑘 ≠ 0))
222	220, 221	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ≠ 0))
223		iffalse 4095	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑥 ∈ (𝐴‘𝑛) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 0)
224	223	necon1ai 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) ≠ 0 → 𝑥 ∈ (𝐴‘𝑛))
225	222, 224	syl6 35	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 → 𝑥 ∈ (𝐴‘𝑛)))
226	225	pm4.71rd 667	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 ↔ (𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘)))
227	218, 226	bitrd 268	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑥) = 𝑘 ↔ (𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘)))
228		iftrue 4092	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐴‘𝑛) → if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = (𝐹‘𝑥))
229	228	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴‘𝑛) → (if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘 ↔ (𝐹‘𝑥) = 𝑘))
230	229	pm5.32i 669	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘) ↔ (𝑥 ∈ (𝐴‘𝑛) ∧ (𝐹‘𝑥) = 𝑘))
231		ancom 466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝐴‘𝑛) ∧ (𝐹‘𝑥) = 𝑘) ↔ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛)))
232	230, 231	bitri 264	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝐴‘𝑛) ∧ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0) = 𝑘) ↔ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛)))
233	227, 232	syl6bb 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑥) = 𝑘 ↔ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛))))
234	233	pm5.32da 673	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘) ↔ (𝑥 ∈ ℝ ∧ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛)))))
235		anass 681	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛)) ↔ (𝑥 ∈ ℝ ∧ ((𝐹‘𝑥) = 𝑘 ∧ 𝑥 ∈ (𝐴‘𝑛))))
236	234, 235	syl6bbr 278	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘) ↔ ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛))))
237		i1ff 23443	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
238		ffn 6045	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
239	193, 237, 238	3syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 Fn ℝ)
240	239	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐺 Fn ℝ)
241		fniniseg 6338	. . . . . . . . . . . . . 14 ⊢ (𝐺 Fn ℝ → (𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘)))
242	240, 241	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐺‘𝑥) = 𝑘)))
243		elin 3796	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ↔ (𝑥 ∈ (◡𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴‘𝑛)))
244	197	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐹 Fn ℝ)
245		fniniseg 6338	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)))
246	244, 245	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)))
247	246	anbi1d 741	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ (◡𝐹 “ {𝑘}) ∧ 𝑥 ∈ (𝐴‘𝑛)) ↔ ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛))))
248	243, 247	syl5bb 272	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ↔ ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) ∧ 𝑥 ∈ (𝐴‘𝑛))))
249	236, 242, 248	3bitr4d 300	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
250	249	alrimiv 1855	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ∀𝑥(𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
251		nfmpt1 4747	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))
252	191, 251	nfcxfr 2762	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐺
253	252	nfcnv 5301	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥◡𝐺
254		nfcv 2764	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑘}
255	253, 254	nfima 5474	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(◡𝐺 “ {𝑘})
256		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))
257	255, 256	cleqf 2790	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑘}) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)) ↔ ∀𝑥(𝑥 ∈ (◡𝐺 “ {𝑘}) ↔ 𝑥 ∈ ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
258	250, 257	sylibr 224	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐺 “ {𝑘}) = ((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))
259	258	fveq2d 6195	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐺 “ {𝑘})) = (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))
260	259	oveq2d 6666	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐺 “ {𝑘}))) = (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
261	260	sumeq2dv 14433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐺 “ {𝑘}))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
262	211, 261	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫1‘𝐺) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
263	262	mpteq2dva 4744	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))))
264	263	fveq1d 6193	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (∫1‘𝐺))‘𝑗) = ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗))
265	174	sumeq2sdv 14435	. . . . . 6 ⊢ (𝑛 = 𝑗 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
266		eqid 2622	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛))))) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))
267		sumex 14418	. . . . . 6 ⊢ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))) ∈ V
268	265, 266, 267	fvmpt 6282	. . . . 5 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
269	177	sumeq2sdv 14435	. . . . 5 ⊢ (𝑗 ∈ ℕ → Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑗)))))
270	268, 269	eqtr4d 2659	. . . 4 ⊢ (𝑗 ∈ ℕ → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗))
271	264, 270	sylan9eq 2676	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫1‘𝐺))‘𝑗) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})((𝑛 ∈ ℕ ↦ (𝑘 · (vol‘((◡𝐹 “ {𝑘}) ∩ (𝐴‘𝑛)))))‘𝑗))
272	1, 2, 8, 181, 183, 189, 271	climfsum 14552	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
273		itg1val 23450	. . 3 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
274	3, 273	syl 17	. 2 ⊢ (𝜑 → (∫1‘𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))))
275	272, 274	breqtrrd 4681	1 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ (∫1‘𝐹))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  supcsup 8346  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  ℕcn 11020  [,]cicc 12178   ⇝ cli 14215  Σcsu 14416  vol*covol 23231  volcvol 23232  ∫₁citg1 23384

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-cc 9257  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-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-rlim 14220  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

This theorem is referenced by:  itg2monolem1  23517