Theorem ovnovollem2 47108

Description: if 𝐼 is a cover of (𝐵 ↑_m {𝐴}) in ℝ^1, then 𝐹 is the corresponding cover in the reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ovnovollem2.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ovnovollem2.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
ovnovollem2.i	⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ))
ovnovollem2.s	⊢ (𝜑 → (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘))
ovnovollem2.z	⊢ (𝜑 → 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
ovnovollem2.f	⊢ 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼‘𝑗)‘𝐴))

Assertion

Ref	Expression
ovnovollem2	⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))

Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑓   𝑓,𝐹   𝑗,𝐹,𝑘   𝑘,𝐼   𝑘,𝑉   𝑓,𝑍   𝜑,𝑗,𝑘

Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑗,𝑘)   𝐼(𝑓,𝑗)   𝑉(𝑓,𝑗)   𝑊(𝑓,𝑗,𝑘)   𝑍(𝑗,𝑘)

Proof of Theorem ovnovollem2

Step	Hyp	Ref	Expression
1		ovnovollem2.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ))
2		elmapi 8787	. . . . . . . . 9 ⊢ (𝐼 ∈ (((ℝ × ℝ) ↑m {𝐴}) ↑m ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴}))
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴}))
4	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑m {𝐴}))
5		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
6	4, 5	ffvelcdmd 7027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m {𝐴}))
7		elmapi 8787	. . . . . 6 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m {𝐴}) → (𝐼‘𝑗):{𝐴}⟶(ℝ × ℝ))
8	6, 7	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):{𝐴}⟶(ℝ × ℝ))
9		ovnovollem2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		snidg 4593	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
11	9, 10	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
12	11	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ {𝐴})
13	8, 12	ffvelcdmd 7027	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝐴) ∈ (ℝ × ℝ))
14		ovnovollem2.f	. . . 4 ⊢ 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼‘𝑗)‘𝐴))
15	13, 14	fmptd 7056	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
16		reex 11121	. . . . . 6 ⊢ ℝ ∈ V
17	16, 16	xpex 7697	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
18		nnex 12172	. . . . 5 ⊢ ℕ ∈ V
19	17, 18	elmap 8810	. . . 4 ⊢ (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ↔ 𝐹:ℕ⟶(ℝ × ℝ))
20	19	a1i 11	. . 3 ⊢ (𝜑 → (𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ↔ 𝐹:ℕ⟶(ℝ × ℝ)))
21	15, 20	mpbird 258	. 2 ⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
22		ovnovollem2.s	. . . . . 6 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘))
23		elsni 4573	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
24	23	fveq2d 6832	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝐴} → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝐴))
25	24	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝐴))
26		elmapfun 8804	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m {𝐴}) → Fun (𝐼‘𝑗))
27	6, 26	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Fun (𝐼‘𝑗))
28	8	fdmd 6666	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → dom (𝐼‘𝑗) = {𝐴})
29	28	eqcomd 2745	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → {𝐴} = dom (𝐼‘𝑗))
30	12, 29	eleqtrd 2841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼‘𝑗))
31		fvco 6926	. . . . . . . . . . . . 13 ⊢ ((Fun (𝐼‘𝑗) ∧ 𝐴 ∈ dom (𝐼‘𝑗)) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴)))
32	27, 30, 31	syl2anc 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴)))
33	32	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴)))
34		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
35		fvexd 6843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → ((𝐼‘𝑗)‘𝐴) ∈ V)
36	14	fvmpt2 6948	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐼‘𝑗)‘𝐴) ∈ V) → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
37	34, 35, 36	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
38	37	eqcomd 2745	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → ((𝐼‘𝑗)‘𝐴) = (𝐹‘𝑗))
39	38	fveq2d 6832	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → ([,)‘((𝐼‘𝑗)‘𝐴)) = ([,)‘(𝐹‘𝑗)))
40	39	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘((𝐼‘𝑗)‘𝐴)) = ([,)‘(𝐹‘𝑗)))
41	15	ffund 6660	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Fun 𝐹)
42	41	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Fun 𝐹)
43	14, 13	dmmptd 6631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝐹 = ℕ)
44	43	eqcomd 2745	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ = dom 𝐹)
45	44	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ℕ = dom 𝐹)
46	5, 45	eleqtrd 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
47		fvco 6926	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹‘𝑗)))
48	42, 46, 47	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹‘𝑗)))
49	48	eqcomd 2745	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘(𝐹‘𝑗)) = (([,) ∘ 𝐹)‘𝑗))
50	40, 49	eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘((𝐼‘𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
51	50	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼‘𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
52	25, 33, 51	3eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗))
53	52	ixpeq2dva 8851	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗))
54		snex 5369	. . . . . . . . . . 11 ⊢ {𝐴} ∈ V
55		fvex 6841	. . . . . . . . . . 11 ⊢ (([,) ∘ 𝐹)‘𝑗) ∈ V
56	54, 55	ixpconst 8846	. . . . . . . . . 10 ⊢ X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴})
57	56	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗) = ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
58	53, 57	eqtrd 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
59	58	iuneq2dv 4947	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
60		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
61	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
62		fvexd 6843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V)
63	60, 61, 62, 9	iunmapsn 45670	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
64	59, 63	eqtrd 2774	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
65	22, 64	sseqtrd 3951	. . . . 5 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ⊆ (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
66		ovnovollem2.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
67	18, 55	iunex 7911	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V
68	67	a1i 11	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ∈ V)
69	54	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝐴} ∈ V)
70	11	ne0d 4271	. . . . . 6 ⊢ (𝜑 → {𝐴} ≠ ∅)
71	66, 68, 69, 70	mapss2 45659	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵 ↑m {𝐴}) ⊆ (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴})))
72	65, 71	mpbird 258	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
73		icof 45672	. . . . . . . 8 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
74	73	a1i 11	. . . . . . 7 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
75		rexpssxrxp 11182	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
76	75	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
77	74, 76, 15	fcoss 45663	. . . . . 6 ⊢ (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
78	77	ffnd 6657	. . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
79		fniunfv 7192	. . . . 5 ⊢ (([,) ∘ 𝐹) Fn ℕ → ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ∪ ran ([,) ∘ 𝐹))
80	78, 79	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) = ∪ ran ([,) ∘ 𝐹))
81	72, 80	sseqtrd 3951	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ([,) ∘ 𝐹))
82		ovnovollem2.z	. . . 4 ⊢ (𝜑 → 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
83		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑗𝐹
84		ressxr 11181	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
85		xpss2 5639	. . . . . . . . . 10 ⊢ (ℝ ⊆ ℝ* → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
86	84, 85	ax-mp 5	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ × ℝ*)
87	86	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℝ × ℝ*))
88	15, 87	fssd 6673	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ*))
89	83, 88	volicofmpt 46448	. . . . . 6 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))))
90	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ 𝑉)
91		fvexd 6843	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝐴) ∈ V)
92	5, 91, 36	syl2anc 590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
93	92, 13	eqeltrd 2839	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) ∈ (ℝ × ℝ))
94		1st2nd2 7971	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) → (𝐹‘𝑗) = ⟨(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))⟩)
95	93, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = ⟨(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))⟩)
96	95	fveq2d 6832	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘(𝐹‘𝑗)) = ([,)‘⟨(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))⟩))
97		df-ov 7360	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))) = ([,)‘⟨(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))⟩)
98	97	eqcomi 2748	. . . . . . . . . . . . . . 15 ⊢ ([,)‘⟨(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))⟩) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))
99	98	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘⟨(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))⟩) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))
100	48, 96, 99	3eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))
101	32, 50, 100	3eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))
102	101	fveq2d 6832	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))))
103		xp1st 7964	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑗)) ∈ ℝ)
104	93, 103	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐹‘𝑗)) ∈ ℝ)
105		xp2nd 7965	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘𝑗)) ∈ ℝ)
106	93, 105	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐹‘𝑗)) ∈ ℝ)
107		volicore 47032	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝐹‘𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) ∈ ℝ)
108	104, 106, 107	syl2anc 590	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) ∈ ℝ)
109	102, 108	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℝ)
110	109	recnd 11165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℂ)
111		2fveq3 6833	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
112	111	prodsn 15919	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
113	90, 110, 112	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
114	113, 102	eqtr2d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
115	114	mpteq2dva 5166	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
116	89, 115	eqtrd 2774	. . . . 5 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
117	116	fveq2d 6832	. . . 4 ⊢ (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
118	82, 117	eqtr4d 2777	. . 3 ⊢ (𝜑 → 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
119	81, 118	jca 516	. 2 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
120		coeq2 5801	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹))
121	120	rneqd 5881	. . . . . 6 ⊢ (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
122	121	unieqd 4852	. . . . 5 ⊢ (𝑓 = 𝐹 → ∪ ran ([,) ∘ 𝑓) = ∪ ran ([,) ∘ 𝐹))
123	122	sseq2d 3947	. . . 4 ⊢ (𝑓 = 𝐹 → (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ↔ 𝐵 ⊆ ∪ ran ([,) ∘ 𝐹)))
124		coeq2 5801	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝐹))
125	124	fveq2d 6832	. . . . 5 ⊢ (𝑓 = 𝐹 → (Σ^‘((vol ∘ [,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
126	125	eqeq2d 2750	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
127	123, 126	anbi12d 638	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ⊆ ∪ ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))))
128	127	rspcev 3560	. 2 ⊢ ((𝐹 ∈ ((ℝ × ℝ) ↑m ℕ) ∧ (𝐵 ⊆ ∪ ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))) → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
129	21, 119, 128	syl2anc 590	1 ⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  𝒫 cpw 4530  {csn 4556  ⟨cop 4562  ∪ cuni 4839  ∪ ciun 4922   ↦ cmpt 5154   × cxp 5617  dom cdm 5619  ran crn 5620   ∘ ccom 5623  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  1^st c1st 7930  2^nd c2nd 7931   ↑_m cmap 8764  Xcixp 8836  ℂcc 11028  ℝcr 11029  ℝ^*cxr 11170  ℕcn 12166  [,)cico 13292  ∏cprod 15860  volcvol 25449  Σ^{^}csumge0 46813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  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 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7621  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-map 8766  df-pm 8767  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fi 9315  df-sup 9346  df-inf 9347  df-oi 9416  df-dju 9817  df-card 9855  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12167  df-2 12236  df-3 12237  df-n0 12430  df-z 12517  df-uz 12781  df-q 12891  df-rp 12935  df-xneg 13055  df-xadd 13056  df-xmul 13057  df-ioo 13294  df-ico 13296  df-icc 13297  df-fz 13454  df-fzo 13601  df-fl 13743  df-seq 13956  df-exp 14016  df-hash 14285  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15442  df-rlim 15443  df-sum 15641  df-prod 15861  df-rest 17377  df-topgen 17398  df-psmet 21340  df-xmet 21341  df-met 21342  df-bl 21343  df-mopn 21344  df-top 22878  df-topon 22895  df-bases 22930  df-cmp 23371  df-ovol 25450  df-vol 25451

This theorem is referenced by:  ovnovollem3  47109