ovolval4lem1 - Mathbox for Glauco Siliprandi

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Theorem ovolval4lem1 40863

Description: |- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) 𝑛) = (((,) ∘ 𝐹) n ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
ovolval4lem1.f	⊢ (𝜑 → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
ovolval4lem1.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
ovolval4lem1.a	⊢ 𝐴 = {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))}

**Assertion**
Ref	Expression
ovolval4lem1	⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))

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

**Proof of Theorem ovolval4lem1**
Step	Hyp	Ref	Expression
1		ioof 12271	. . . . . . . 8 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2	1	a1i 11	. . . . . . 7 ⊢ (𝜑 → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
3		ovolval4lem1.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
4		fco 6058	. . . . . . 7 ⊢ (((,):(ℝ^* × ℝ^)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
5	2, 3, 4	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
6		ffn 6045	. . . . . 6 ⊢ (((,) ∘ 𝐹):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐹) Fn ℕ)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
8		fniunfv 6505	. . . . 5 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
10	9	eqcomd 2628	. . 3 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) = ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
11		ovolval4lem1.a	. . . . . . . . 9 ⊢ 𝐴 = {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))}
12		ssrab2 3687	. . . . . . . . 9 ⊢ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))} ⊆ ℕ
13	11, 12	eqsstri 3635	. . . . . . . 8 ⊢ 𝐴 ⊆ ℕ
14		undif 4049	. . . . . . . 8 ⊢ (𝐴 ⊆ ℕ ↔ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ)
15	13, 14	mpbi 220	. . . . . . 7 ⊢ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ
16	15	eqcomi 2631	. . . . . 6 ⊢ ℕ = (𝐴 ∪ (ℕ ∖ 𝐴))
17	16	iuneq1i 39259	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛)
18		iunxun 4605	. . . . 5 ⊢ ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
19	17, 18	eqtri 2644	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
20	19	a1i 11	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
21	3	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ^* × ℝ^*))
22		xp1st 7198	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∈ (ℝ^* × ℝ^) → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^)
23	21, 22	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
24		xp2nd 7199	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ (ℝ^* × ℝ^) → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^)
25	21, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
26	25, 23	ifcld 4131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))) ∈ ℝ^*)
27	23, 26	opelxpd 5149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩ ∈ (ℝ^* × ℝ^*))
28		ovolval4lem1.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
29	27, 28	fmptd 6385	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶(ℝ^* × ℝ^*))
30		fco 6058	. . . . . . . 8 ⊢ (((,):(ℝ^* × ℝ^)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
31	2, 29, 30	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
32		ffn 6045	. . . . . . 7 ⊢ (((,) ∘ 𝐺):ℕ⟶𝒫 ℝ → ((,) ∘ 𝐺) Fn ℕ)
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
34		fniunfv 6505	. . . . . 6 ⊢ (((,) ∘ 𝐺) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ ran ((,) ∘ 𝐺))
35	33, 34	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ ran ((,) ∘ 𝐺))
36	35	eqcomd 2628	. . . 4 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐺) = ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛))
37	16	iuneq1i 39259	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛)
38		iunxun 4605	. . . . . 6 ⊢ ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
39	37, 38	eqtri 2644	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
40	39	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)))
41	29	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐺:ℕ⟶(ℝ^* × ℝ^*))
42	13	sseli 3599	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐴 → 𝑛 ∈ ℕ)
43	42	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℕ)
44		fvco3 6275	. . . . . . . 8 ⊢ ((𝐺:ℕ⟶(ℝ^* × ℝ^*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
45	41, 43, 44	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
46	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
47		fvco3 6275	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(ℝ^* × ℝ^*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
48	46, 43, 47	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
49		simpl 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝜑)
50		1st2nd2 7205	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ (ℝ^* × ℝ^*) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
51	21, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
52	49, 43, 51	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
53	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩))
54	27	elexd 3214	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩ ∈ V)
55	53, 54	fvmpt2d 6293	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
56	49, 43, 55	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
57	11	eleq2i 2693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))})
58	57	biimpi 206	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝐴 → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))})
59		rabid 3116	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))} ↔ (𝑛 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
60	58, 59	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝐴 → (𝑛 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
61	60	simprd 479	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐴 → (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)))
62	61	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)))
63	62	iftrued 4094	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))) = (2^nd ‘(𝐹‘𝑛)))
64	63	opeq2d 4409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩ = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
65		eqidd 2623	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩ = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
66	56, 64, 65	3eqtrd 2660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
67	52, 66	eqtr4d 2659	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐹‘𝑛) = (𝐺‘𝑛))
68	67	fveq2d 6195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((,)‘(𝐹‘𝑛)) = ((,)‘(𝐺‘𝑛)))
69	48, 68	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐺‘𝑛)))
70	45, 69	eqtr4d 2659	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
71	70	iuneq2dv 4542	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛))
72	29	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐺:ℕ⟶(ℝ^* × ℝ^*))
73		eldifi 3732	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) → 𝑛 ∈ ℕ)
74	73	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝑛 ∈ ℕ)
75	72, 74, 44	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
76		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝜑)
77	76, 74, 55	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩)
78	73	anim1i 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → (𝑛 ∈ ℕ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
79	78, 59	sylibr 224	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))})
80	79, 57	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
81	80	adantll 750	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
82		eldifn 3733	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) → ¬ 𝑛 ∈ 𝐴)
83	82	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → ¬ 𝑛 ∈ 𝐴)
84	81, 83	pm2.65da 600	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ¬ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)))
85	84	iffalsed 4097	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))) = (1^st ‘(𝐹‘𝑛)))
86	85	opeq2d 4409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ⟨(1^st ‘(𝐹‘𝑛)), if((1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛)))⟩ = ⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩)
87	77, 86	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩)
88	87	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐺‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩))
89		iooid 12203	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(1^st ‘(𝐹‘𝑛))) = ∅
90	89	eqcomi 2631	. . . . . . . . . . 11 ⊢ ∅ = ((1^st ‘(𝐹‘𝑛))(,)(1^st ‘(𝐹‘𝑛)))
91		df-ov 6653	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(1^st ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩)
92	90, 91	eqtr2i 2645	. . . . . . . . . 10 ⊢ ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩) = ∅
93	92	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (1^st ‘(𝐹‘𝑛))⟩) = ∅)
94	75, 88, 93	3eqtrd 2660	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ∅)
95	94	iuneq2dv 4542	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅)
96		iun0 4576	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅
97	96	a1i 11	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅)
98	95, 97	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∅)
99	76, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
100	99, 74, 47	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
101	76, 74, 51	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
102	101	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
103		df-ov 6653	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
104	103	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
105		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → 𝑛 ∈ (ℕ ∖ 𝐴))
106	74, 23	syldan 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
107	106	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → (1^st ‘(𝐹‘𝑛)) ∈ ℝ^*)
108	74, 25	syldan 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
109	108	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*)
110		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛)))
111	107, 109	xrltnled 39579	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → ((1^st ‘(𝐹‘𝑛)) < (2^nd ‘(𝐹‘𝑛)) ↔ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))))
112	110, 111	mpbird 247	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → (1^st ‘(𝐹‘𝑛)) < (2^nd ‘(𝐹‘𝑛)))
113	107, 109, 112	xrltled 39486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛)))
114	105, 113, 80	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
115	82	ad2antlr 763	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))) → ¬ 𝑛 ∈ 𝐴)
116	114, 115	condan 835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛)))
117		ioo0 12200	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ^* ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ^*) → (((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ∅ ↔ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))))
118	106, 108, 117	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ∅ ↔ (2^nd ‘(𝐹‘𝑛)) ≤ (1^st ‘(𝐹‘𝑛))))
119	116, 118	mpbird 247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ∅)
120	104, 119	eqtr3d 2658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩) = ∅)
121	100, 102, 120	3eqtrd 2660	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ∅)
122	121	iuneq2dv 4542	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅)
123	122, 97	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∅)
124	98, 123	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
125	71, 124	uneq12d 3768	. . . 4 ⊢ (𝜑 → (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
126	36, 40, 125	3eqtrrd 2661	. . 3 ⊢ (𝜑 → (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)) = ∪ ran ((,) ∘ 𝐺))
127	10, 20, 126	3eqtrd 2660	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺))
128		volf 23297	. . . . . 6 ⊢ vol:dom vol⟶(0[,]+∞)
129	128	a1i 11	. . . . 5 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞))
130	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
131		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
132	130, 131, 47	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
133	51	fveq2d 6195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
134	103	eqcomi 2631	. . . . . . . . . . 11 ⊢ ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))
135	134	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
136	132, 133, 135	3eqtrd 2660	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
137		ioombl 23333	. . . . . . . . . 10 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ∈ dom vol
138	137	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) ∈ dom vol)
139	136, 138	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
140	139	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
141	7, 140	jca 554	. . . . . 6 ⊢ (𝜑 → (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
142		ffnfv 6388	. . . . . 6 ⊢ (((,) ∘ 𝐹):ℕ⟶dom vol ↔ (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
143	141, 142	sylibr 224	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶dom vol)
144		fco 6058	. . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐹):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
145	129, 143, 144	syl2anc 693	. . . 4 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
146		ffn 6045	. . . 4 ⊢ ((vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞) → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
147	145, 146	syl 17	. . 3 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
148	70	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
149	139	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
150	148, 149	eqeltrd 2701	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
151		simpll 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → 𝜑)
152		eldif 3584	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) ↔ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴))
153	152	bicomi 214	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴) ↔ 𝑛 ∈ (ℕ ∖ 𝐴))
154	153	biimpi 206	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
155	154	adantll 750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
156	119, 137	syl6eqelr 2710	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ∅ ∈ dom vol)
157	94, 156	eqeltrd 2701	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
158	151, 155, 157	syl2anc 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
159	150, 158	pm2.61dan 832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
160	159	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
161	33, 160	jca 554	. . . . . 6 ⊢ (𝜑 → (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
162		ffnfv 6388	. . . . . 6 ⊢ (((,) ∘ 𝐺):ℕ⟶dom vol ↔ (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
163	161, 162	sylibr 224	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐺):ℕ⟶dom vol)
164		fco 6058	. . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐺):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
165	129, 163, 164	syl2anc 693	. . . 4 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
166		ffn 6045	. . . 4 ⊢ ((vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞) → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
167	165, 166	syl 17	. . 3 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
168	148	eqcomd 2628	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
169	121, 94	eqtr4d 2659	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
170	151, 155, 169	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
171	168, 170	pm2.61dan 832	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
172	171	fveq2d 6195	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(((,) ∘ 𝐹)‘𝑛)) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
173		fnfun 5988	. . . . . . 7 ⊢ (((,) ∘ 𝐹) Fn ℕ → Fun ((,) ∘ 𝐹))
174	7, 173	syl 17	. . . . . 6 ⊢ (𝜑 → Fun ((,) ∘ 𝐹))
175	174	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐹))
176		fdm 6051	. . . . . . . . 9 ⊢ (((,) ∘ 𝐹):ℕ⟶dom vol → dom ((,) ∘ 𝐹) = ℕ)
177	143, 176	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom ((,) ∘ 𝐹) = ℕ)
178	177	eqcomd 2628	. . . . . . 7 ⊢ (𝜑 → ℕ = dom ((,) ∘ 𝐹))
179	178	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐹))
180	131, 179	eleqtrd 2703	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐹))
181		fvco 6274	. . . . 5 ⊢ ((Fun ((,) ∘ 𝐹) ∧ 𝑛 ∈ dom ((,) ∘ 𝐹)) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
182	175, 180, 181	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
183		fnfun 5988	. . . . . . 7 ⊢ (((,) ∘ 𝐺) Fn ℕ → Fun ((,) ∘ 𝐺))
184	33, 183	syl 17	. . . . . 6 ⊢ (𝜑 → Fun ((,) ∘ 𝐺))
185	184	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐺))
186		fdm 6051	. . . . . . . . 9 ⊢ (((,) ∘ 𝐺):ℕ⟶dom vol → dom ((,) ∘ 𝐺) = ℕ)
187	163, 186	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom ((,) ∘ 𝐺) = ℕ)
188	187	eqcomd 2628	. . . . . . 7 ⊢ (𝜑 → ℕ = dom ((,) ∘ 𝐺))
189	188	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐺))
190	131, 189	eleqtrd 2703	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐺))
191		fvco 6274	. . . . 5 ⊢ ((Fun ((,) ∘ 𝐺) ∧ 𝑛 ∈ dom ((,) ∘ 𝐺)) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
192	185, 190, 191	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
193	172, 182, 192	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = ((vol ∘ ((,) ∘ 𝐺))‘𝑛))
194	147, 167, 193	eqfnfvd 6314	. 2 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺)))
195	127, 194	jca 554	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 ifcif 4086 𝒫 cpw 4158 ⟨cop 4183 ∪ cuni 4436 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ran crn 5115 ∘ ccom 5118 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 ℝcr 9935 0cc0 9936 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 (,)cioo 12175 [,]cicc 12178 volcvol 23232

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

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-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-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-xadd 11947 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-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234

This theorem is referenced by: ovolval4lem2 40864

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