Theorem ovolval4lem1 46634

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 ↦ ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩)
ovolval4lem1.a	⊢ 𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(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 13350	. . . . . . . 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 6676	. . . . . . 7 ⊢ (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
5	2, 3, 4	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶𝒫 ℝ)
6	5	ffnd 6653	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
7		fniunfv 7183	. . . . 5 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ ran ((,) ∘ 𝐹))
9	8	eqcomd 2735	. . 3 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) = ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛))
10		ovolval4lem1.a	. . . . . . . . 9 ⊢ 𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))}
11		ssrab2 4031	. . . . . . . . 9 ⊢ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))} ⊆ ℕ
12	10, 11	eqsstri 3982	. . . . . . . 8 ⊢ 𝐴 ⊆ ℕ
13		undif 4433	. . . . . . . 8 ⊢ (𝐴 ⊆ ℕ ↔ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ)
14	12, 13	mpbi 230	. . . . . . 7 ⊢ (𝐴 ∪ (ℕ ∖ 𝐴)) = ℕ
15	14	eqcomi 2738	. . . . . 6 ⊢ ℕ = (𝐴 ∪ (ℕ ∖ 𝐴))
16	15	iuneq1i 45067	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛)
17		iunxun 5043	. . . . 5 ⊢ ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
18	16, 17	eqtri 2752	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
19	18	a1i 11	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
20	3	ffvelcdmda 7018	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ* × ℝ*))
21		xp1st 7956	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑛) ∈ (ℝ* × ℝ*) → (1st ‘(𝐹‘𝑛)) ∈ ℝ*)
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st ‘(𝐹‘𝑛)) ∈ ℝ*)
23		xp2nd 7957	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ (ℝ* × ℝ*) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ*)
24	20, 23	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ*)
25	24, 22	ifcld 4523	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))) ∈ ℝ*)
26	22, 25	opelxpd 5658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩ ∈ (ℝ* × ℝ*))
27		ovolval4lem1.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩)
28	26, 27	fmptd 7048	. . . . . . . 8 ⊢ (𝜑 → 𝐺:ℕ⟶(ℝ* × ℝ*))
29		fco 6676	. . . . . . . 8 ⊢ (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
30	2, 28, 29	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
31	30	ffnd 6653	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐺) Fn ℕ)
32		fniunfv 7183	. . . . . 6 ⊢ (((,) ∘ 𝐺) Fn ℕ → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ ran ((,) ∘ 𝐺))
33	31, 32	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ ran ((,) ∘ 𝐺))
34	33	eqcomd 2735	. . . 4 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐺) = ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛))
35	15	iuneq1i 45067	. . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛)
36		iunxun 5043	. . . . . 6 ⊢ ∪ 𝑛 ∈ (𝐴 ∪ (ℕ ∖ 𝐴))(((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
37	35, 36	eqtri 2752	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛))
38	37	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)))
39	28	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐺:ℕ⟶(ℝ* × ℝ*))
40	12	sseli 3931	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐴 → 𝑛 ∈ ℕ)
41	40	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℕ)
42		fvco3 6922	. . . . . . . 8 ⊢ ((𝐺:ℕ⟶(ℝ* × ℝ*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
43	39, 41, 42	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
44	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐹:ℕ⟶(ℝ* × ℝ*))
45		fvco3 6922	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶(ℝ* × ℝ*) ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
46	44, 41, 45	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
47		simpl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝜑)
48		1st2nd2 7963	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑛) ∈ (ℝ* × ℝ*) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
49	20, 48	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
50	47, 41, 49	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
51	27	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩))
52	26	elexd 3460	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩ ∈ V)
53	51, 52	fvmpt2d 6943	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩)
54	47, 41, 53	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐺‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩)
55	10	eleq2i 2820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))})
56	55	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝐴 → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))})
57		rabid 3416	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))} ↔ (𝑛 ∈ ℕ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
58	56, 57	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝐴 → (𝑛 ∈ ℕ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
59	58	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐴 → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
60	59	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
61	60	iftrued 4484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))) = (2nd ‘(𝐹‘𝑛)))
62	61	opeq2d 4831	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩ = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
63		eqidd 2730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩ = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
64	54, 62, 63	3eqtrd 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐺‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
65	50, 64	eqtr4d 2767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐹‘𝑛) = (𝐺‘𝑛))
66	65	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((,)‘(𝐹‘𝑛)) = ((,)‘(𝐺‘𝑛)))
67	46, 66	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐺‘𝑛)))
68	43, 67	eqtr4d 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
69	68	iuneq2dv 4966	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛))
70	28	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
71		eldifi 4082	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) → 𝑛 ∈ ℕ)
72	71	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝑛 ∈ ℕ)
73	70, 72, 42	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ((,)‘(𝐺‘𝑛)))
74		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝜑)
75	74, 72, 53	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩)
76	71	anim1i 615	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))) → (𝑛 ∈ ℕ ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))))
77	76, 57	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))) → 𝑛 ∈ {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))})
78	77, 55	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (ℕ ∖ 𝐴) ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
79	78	adantll 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
80		eldifn 4083	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) → ¬ 𝑛 ∈ 𝐴)
81	80	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))) → ¬ 𝑛 ∈ 𝐴)
82	79, 81	pm2.65da 816	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ¬ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
83	82	iffalsed 4487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))) = (1st ‘(𝐹‘𝑛)))
84	83	opeq2d 4831	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩ = ⟨(1st ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))⟩)
85	75, 84	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐺‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))⟩)
86	85	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐺‘𝑛)) = ((,)‘⟨(1st ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))⟩))
87		iooid 13276	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑛))(,)(1st ‘(𝐹‘𝑛))) = ∅
88	87	eqcomi 2738	. . . . . . . . . . 11 ⊢ ∅ = ((1st ‘(𝐹‘𝑛))(,)(1st ‘(𝐹‘𝑛)))
89		df-ov 7352	. . . . . . . . . . 11 ⊢ ((1st ‘(𝐹‘𝑛))(,)(1st ‘(𝐹‘𝑛))) = ((,)‘⟨(1st ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))⟩)
90	88, 89	eqtr2i 2753	. . . . . . . . . 10 ⊢ ((,)‘⟨(1st ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))⟩) = ∅
91	90	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1st ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛))⟩) = ∅)
92	73, 86, 91	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) = ∅)
93	92	iuneq2dv 4966	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅)
94		iun0 5011	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅
95	94	a1i 11	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅ = ∅)
96	93, 95	eqtrd 2764	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∅)
97	74, 3	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
98	97, 72, 45	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
99	74, 72, 49	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
100	99	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩))
101		df-ov 7352	. . . . . . . . . . 11 ⊢ ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
102	101	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩))
103		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → 𝑛 ∈ (ℕ ∖ 𝐴))
104	72, 22	syldan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (1st ‘(𝐹‘𝑛)) ∈ ℝ*)
105	104	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → (1st ‘(𝐹‘𝑛)) ∈ ℝ*)
106	72, 24	syldan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ*)
107	106	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ*)
108		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛)))
109	105, 107	xrltnled 11183	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → ((1st ‘(𝐹‘𝑛)) < (2nd ‘(𝐹‘𝑛)) ↔ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))))
110	108, 109	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → (1st ‘(𝐹‘𝑛)) < (2nd ‘(𝐹‘𝑛)))
111	105, 107, 110	xrltled 13052	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)))
112	103, 111, 78	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → 𝑛 ∈ 𝐴)
113	80	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) ∧ ¬ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))) → ¬ 𝑛 ∈ 𝐴)
114	112, 113	condan 817	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛)))
115		ioo0 13273	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝐹‘𝑛)) ∈ ℝ* ∧ (2nd ‘(𝐹‘𝑛)) ∈ ℝ*) → (((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) = ∅ ↔ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))))
116	104, 106, 115	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) = ∅ ↔ (2nd ‘(𝐹‘𝑛)) ≤ (1st ‘(𝐹‘𝑛))))
117	114, 116	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) = ∅)
118	102, 117	eqtr3d 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ((,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩) = ∅)
119	98, 100, 118	3eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = ∅)
120	119	iuneq2dv 4966	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)∅)
121	120, 95	eqtrd 2764	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛) = ∅)
122	96, 121	eqtr4d 2767	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛))
123	69, 122	uneq12d 4120	. . . 4 ⊢ (𝜑 → (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐺)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐺)‘𝑛)) = (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)))
124	34, 38, 123	3eqtrrd 2769	. . 3 ⊢ (𝜑 → (∪ 𝑛 ∈ 𝐴 (((,) ∘ 𝐹)‘𝑛) ∪ ∪ 𝑛 ∈ (ℕ ∖ 𝐴)(((,) ∘ 𝐹)‘𝑛)) = ∪ ran ((,) ∘ 𝐺))
125	9, 19, 124	3eqtrd 2768	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺))
126		volf 25428	. . . . . 6 ⊢ vol:dom vol⟶(0[,]+∞)
127	126	a1i 11	. . . . 5 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞))
128	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶(ℝ* × ℝ*))
129		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
130	128, 129, 45	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((,)‘(𝐹‘𝑛)))
131	49	fveq2d 6826	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩))
132	101	eqcomi 2738	. . . . . . . . . . 11 ⊢ ((,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩) = ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛)))
133	132	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((,)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩) = ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))))
134	130, 131, 133	3eqtrd 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))))
135		ioombl 25464	. . . . . . . . . 10 ⊢ ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) ∈ dom vol
136	135	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ‘(𝐹‘𝑛))(,)(2nd ‘(𝐹‘𝑛))) ∈ dom vol)
137	134, 136	eqeltrd 2828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
138	137	ralrimiva 3121	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
139	6, 138	jca 511	. . . . . 6 ⊢ (𝜑 → (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
140		ffnfv 7053	. . . . . 6 ⊢ (((,) ∘ 𝐹):ℕ⟶dom vol ↔ (((,) ∘ 𝐹) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐹)‘𝑛) ∈ dom vol))
141	139, 140	sylibr 234	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐹):ℕ⟶dom vol)
142		fco 6676	. . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐹):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
143	127, 141, 142	syl2anc 584	. . . 4 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)):ℕ⟶(0[,]+∞))
144	143	ffnd 6653	. . 3 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)) Fn ℕ)
145	68	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) = (((,) ∘ 𝐹)‘𝑛))
146	137	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) ∈ dom vol)
147	145, 146	eqeltrd 2828	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
148		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → 𝜑)
149		eldif 3913	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℕ ∖ 𝐴) ↔ (𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴))
150	149	bicomi 224	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴) ↔ 𝑛 ∈ (ℕ ∖ 𝐴))
151	150	biimpi 216	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ ¬ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
152	151	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℕ ∖ 𝐴))
153	117, 135	eqeltrrdi 2837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → ∅ ∈ dom vol)
154	92, 153	eqeltrd 2828	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
155	148, 152, 154	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
156	147, 155	pm2.61dan 812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
157	156	ralrimiva 3121	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol)
158	31, 157	jca 511	. . . . . 6 ⊢ (𝜑 → (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
159		ffnfv 7053	. . . . . 6 ⊢ (((,) ∘ 𝐺):ℕ⟶dom vol ↔ (((,) ∘ 𝐺) Fn ℕ ∧ ∀𝑛 ∈ ℕ (((,) ∘ 𝐺)‘𝑛) ∈ dom vol))
160	158, 159	sylibr 234	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐺):ℕ⟶dom vol)
161		fco 6676	. . . . 5 ⊢ ((vol:dom vol⟶(0[,]+∞) ∧ ((,) ∘ 𝐺):ℕ⟶dom vol) → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
162	127, 160, 161	syl2anc 584	. . . 4 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐺)):ℕ⟶(0[,]+∞))
163	162	ffnd 6653	. . 3 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐺)) Fn ℕ)
164	145	eqcomd 2735	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
165	119, 92	eqtr4d 2767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ 𝐴)) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
166	148, 152, 165	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 ∈ 𝐴) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
167	164, 166	pm2.61dan 812	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑛) = (((,) ∘ 𝐺)‘𝑛))
168	167	fveq2d 6826	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol‘(((,) ∘ 𝐹)‘𝑛)) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
169		fnfun 6582	. . . . . . 7 ⊢ (((,) ∘ 𝐹) Fn ℕ → Fun ((,) ∘ 𝐹))
170	6, 169	syl 17	. . . . . 6 ⊢ (𝜑 → Fun ((,) ∘ 𝐹))
171	170	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐹))
172	5	fdmd 6662	. . . . . . . 8 ⊢ (𝜑 → dom ((,) ∘ 𝐹) = ℕ)
173	172	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → ℕ = dom ((,) ∘ 𝐹))
174	173	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐹))
175	129, 174	eleqtrd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐹))
176		fvco 6921	. . . . 5 ⊢ ((Fun ((,) ∘ 𝐹) ∧ 𝑛 ∈ dom ((,) ∘ 𝐹)) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
177	171, 175, 176	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = (vol‘(((,) ∘ 𝐹)‘𝑛)))
178		fnfun 6582	. . . . . . 7 ⊢ (((,) ∘ 𝐺) Fn ℕ → Fun ((,) ∘ 𝐺))
179	31, 178	syl 17	. . . . . 6 ⊢ (𝜑 → Fun ((,) ∘ 𝐺))
180	179	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun ((,) ∘ 𝐺))
181	30	fdmd 6662	. . . . . . . 8 ⊢ (𝜑 → dom ((,) ∘ 𝐺) = ℕ)
182	181	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → ℕ = dom ((,) ∘ 𝐺))
183	182	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom ((,) ∘ 𝐺))
184	129, 183	eleqtrd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom ((,) ∘ 𝐺))
185		fvco 6921	. . . . 5 ⊢ ((Fun ((,) ∘ 𝐺) ∧ 𝑛 ∈ dom ((,) ∘ 𝐺)) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
186	180, 184, 185	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐺))‘𝑛) = (vol‘(((,) ∘ 𝐺)‘𝑛)))
187	168, 177, 186	3eqtr4d 2774	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((vol ∘ ((,) ∘ 𝐹))‘𝑛) = ((vol ∘ ((,) ∘ 𝐺))‘𝑛))
188	144, 163, 187	eqfnfvd 6968	. 2 ⊢ (𝜑 → (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺)))
189	125, 188	jca 511	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3394  Vcvv 3436   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  ifcif 4476  𝒫 cpw 4551  ⟨cop 4583  ∪ cuni 4858  ∪ ciun 4941   class class class wbr 5092   ↦ cmpt 5173   × cxp 5617  dom cdm 5619  ran crn 5620   ∘ ccom 5623  Fun wfun 6476   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  ℝcr 11008  0cc0 11009  +∞cpnf 11146  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150  ℕcn 12128  (,)cioo 13248  [,]cicc 13251  volcvol 25362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xadd 13015  df-ioo 13252  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-xmet 21254  df-met 21255  df-ovol 25363  df-vol 25364

This theorem is referenced by:  ovolval4lem2  46635