Theorem ovnovollem2 44195

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 8637	. . . . . . . . 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	ffvelrnd 6962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m {𝐴}))
7		elmapi 8637	. . . . . 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 4595	. . . . . . 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	ffvelrnd 6962	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝐴) ∈ (ℝ × ℝ))
14		ovnovollem2.f	. . . 4 ⊢ 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼‘𝑗)‘𝐴))
15	13, 14	fmptd 6988	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
16		reex 10962	. . . . . 6 ⊢ ℝ ∈ V
17	16, 16	xpex 7603	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
18		nnex 11979	. . . . 5 ⊢ ℕ ∈ V
19	17, 18	elmap 8659	. . . 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 256	. 2 ⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ) ↑m ℕ))
22		ovnovollem2.s	. . . . . 6 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘))
23		elsni 4578	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
24	23	fveq2d 6778	. . . . . . . . . . . 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 8654	. . . . . . . . . . . . . 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 6611	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → dom (𝐼‘𝑗) = {𝐴})
29	28	eqcomd 2744	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → {𝐴} = dom (𝐼‘𝑗))
30	12, 29	eleqtrd 2841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼‘𝑗))
31		fvco 6866	. . . . . . . . . . . . 13 ⊢ ((Fun (𝐼‘𝑗) ∧ 𝐴 ∈ dom (𝐼‘𝑗)) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴)))
32	27, 30, 31	syl2anc 584	. . . . . . . . . . . 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 6789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → ((𝐼‘𝑗)‘𝐴) ∈ V)
36	14	fvmpt2 6886	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐼‘𝑗)‘𝐴) ∈ V) → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
37	34, 35, 36	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
38	37	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → ((𝐼‘𝑗)‘𝐴) = (𝐹‘𝑗))
39	38	fveq2d 6778	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → ([,)‘((𝐼‘𝑗)‘𝐴)) = ([,)‘(𝐹‘𝑗)))
40	39	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘((𝐼‘𝑗)‘𝐴)) = ([,)‘(𝐹‘𝑗)))
41	15	ffund 6604	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Fun 𝐹)
42	41	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Fun 𝐹)
43	14, 13	dmmptd 6578	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝐹 = ℕ)
44	43	eqcomd 2744	. . . . . . . . . . . . . . . . 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 6866	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹‘𝑗)))
48	42, 46, 47	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹‘𝑗)))
49	48	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘(𝐹‘𝑗)) = (([,) ∘ 𝐹)‘𝑗))
50	40, 49	eqtrd 2778	. . . . . . . . . . . 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 2782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗))
53	52	ixpeq2dva 8700	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗))
54		snex 5354	. . . . . . . . . . 11 ⊢ {𝐴} ∈ V
55		fvex 6787	. . . . . . . . . . 11 ⊢ (([,) ∘ 𝐹)‘𝑗) ∈ V
56	54, 55	ixpconst 8695	. . . . . . . . . 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 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
59	58	iuneq2dv 4948	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
60		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
61	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
62		fvexd 6789	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V)
63	60, 61, 62, 9	iunmapsn 42757	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}) = (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
64	59, 63	eqtrd 2778	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
65	22, 64	sseqtrd 3961	. . . . 5 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ⊆ (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴}))
66		ovnovollem2.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
67	18, 55	iunex 7811	. . . . . . 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 4269	. . . . . 6 ⊢ (𝜑 → {𝐴} ≠ ∅)
71	66, 68, 69, 70	mapss2 42745	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵 ↑m {𝐴}) ⊆ (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑m {𝐴})))
72	65, 71	mpbird 256	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
73		icof 42759	. . . . . . . 8 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
74	73	a1i 11	. . . . . . 7 ⊢ (𝜑 → [,):(ℝ* × ℝ*)⟶𝒫 ℝ*)
75		rexpssxrxp 11020	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
76	75	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℝ* × ℝ*))
77	74, 76, 15	fcoss 42750	. . . . . 6 ⊢ (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ*)
78	77	ffnd 6601	. . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
79		fniunfv 7120	. . . . 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 3961	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ([,) ∘ 𝐹))
82		ovnovollem2.z	. . . 4 ⊢ (𝜑 → 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
83		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑗𝐹
84		ressxr 11019	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
85		xpss2 5609	. . . . . . . . . 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 6618	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ*))
89	83, 88	volicofmpt 43538	. . . . . 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 6789	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝐴) ∈ V)
92	5, 91, 36	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
93	92, 13	eqeltrd 2839	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) ∈ (ℝ × ℝ))
94		1st2nd2 7870	. . . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘(𝐹‘𝑗)) = ([,)‘⟨(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))⟩))
97		df-ov 7278	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))) = ([,)‘⟨(1st ‘(𝐹‘𝑗)), (2nd ‘(𝐹‘𝑗))⟩)
98	97	eqcomi 2747	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))
101	32, 50, 100	3eqtrd 2782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))
102	101	fveq2d 6778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) = (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))))
103		xp1st 7863	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘𝑗)) ∈ ℝ)
104	93, 103	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐹‘𝑗)) ∈ ℝ)
105		xp2nd 7864	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘𝑗)) ∈ ℝ)
106	93, 105	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐹‘𝑗)) ∈ ℝ)
107		volicore 44119	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝐹‘𝑗)) ∈ ℝ ∧ (2nd ‘(𝐹‘𝑗)) ∈ ℝ) → (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) ∈ ℝ)
108	104, 106, 107	syl2anc 584	. . . . . . . . . . 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 11003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℂ)
111		2fveq3 6779	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
112	111	prodsn 15672	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
113	90, 110, 112	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
114	113, 102	eqtr2d 2779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
115	114	mpteq2dva 5174	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1st ‘(𝐹‘𝑗))[,)(2nd ‘(𝐹‘𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
116	89, 115	eqtrd 2778	. . . . 5 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
117	116	fveq2d 6778	. . . 4 ⊢ (𝜑 → (Σ^‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
118	82, 117	eqtr4d 2781	. . 3 ⊢ (𝜑 → 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
119	81, 118	jca 512	. 2 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
120		coeq2 5767	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹))
121	120	rneqd 5847	. . . . . 6 ⊢ (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
122	121	unieqd 4853	. . . . 5 ⊢ (𝑓 = 𝐹 → ∪ ran ([,) ∘ 𝑓) = ∪ ran ([,) ∘ 𝐹))
123	122	sseq2d 3953	. . . 4 ⊢ (𝑓 = 𝐹 → (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ↔ 𝐵 ⊆ ∪ ran ([,) ∘ 𝐹)))
124		coeq2 5767	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝐹))
125	124	fveq2d 6778	. . . . 5 ⊢ (𝑓 = 𝐹 → (Σ^‘((vol ∘ [,)) ∘ 𝑓)) = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))
126	125	eqeq2d 2749	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹))))
127	123, 126	anbi12d 631	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ⊆ ∪ ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))))
128	127	rspcev 3561	. 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 584	1 ⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561  ⟨cop 4567  ∪ cuni 4839  ∪ ciun 4924   ↦ cmpt 5157   × cxp 5587  dom cdm 5589  ran crn 5590   ∘ ccom 5593  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830   ↑_m cmap 8615  Xcixp 8685  ℂcc 10869  ℝcr 10870  ℝ^*cxr 11008  ℕcn 11973  [,)cico 13081  ∏cprod 15615  volcvol 24627  Σ^{^}csumge0 43900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-prod 15616  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538  df-ovol 24628  df-vol 24629

This theorem is referenced by:  ovnovollem3  44196