ovnovollem2 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnovollem2		Structured version Visualization version GIF version

Theorem ovnovollem2 45360

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 8840	. . . . . . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑_m {𝐴}))
5		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
6	4, 5	ffvelcdmd 7085	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑_m {𝐴}))
7		elmapi 8840	. . . . . 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 4662	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
11	9, 10	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
12	11	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ {𝐴})
13	8, 12	ffvelcdmd 7085	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝐴) ∈ (ℝ × ℝ))
14		ovnovollem2.f	. . . 4 ⊢ 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼‘𝑗)‘𝐴))
15	13, 14	fmptd 7111	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
16		reex 11198	. . . . . 6 ⊢ ℝ ∈ V
17	16, 16	xpex 7737	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
18		nnex 12215	. . . . 5 ⊢ ℕ ∈ V
19	17, 18	elmap 8862	. . . 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 257	. 2 ⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ) ↑_m ℕ))
22		ovnovollem2.s	. . . . . 6 ⊢ (𝜑 → (𝐵 ↑_m {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘))
23		elsni 4645	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
24	23	fveq2d 6893	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝐴} → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝐴))
25	24	adantl 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝐴))
26		elmapfun 8857	. . . . . . . . . . . . . 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 6726	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → dom (𝐼‘𝑗) = {𝐴})
29	28	eqcomd 2739	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → {𝐴} = dom (𝐼‘𝑗))
30	12, 29	eleqtrd 2836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ dom (𝐼‘𝑗))
31		fvco 6987	. . . . . . . . . . . . 13 ⊢ ((Fun (𝐼‘𝑗) ∧ 𝐴 ∈ dom (𝐼‘𝑗)) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴)))
32	27, 30, 31	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴)))
33	32	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ([,)‘((𝐼‘𝑗)‘𝐴)))
34		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
35		fvexd 6904	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → ((𝐼‘𝑗)‘𝐴) ∈ V)
36	14	fvmpt2 7007	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℕ ∧ ((𝐼‘𝑗)‘𝐴) ∈ V) → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
37	34, 35, 36	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
38	37	eqcomd 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → ((𝐼‘𝑗)‘𝐴) = (𝐹‘𝑗))
39	38	fveq2d 6893	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → ([,)‘((𝐼‘𝑗)‘𝐴)) = ([,)‘(𝐹‘𝑗)))
40	39	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘((𝐼‘𝑗)‘𝐴)) = ([,)‘(𝐹‘𝑗)))
41	15	ffund 6719	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Fun 𝐹)
42	41	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Fun 𝐹)
43	14, 13	dmmptd 6693	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom 𝐹 = ℕ)
44	43	eqcomd 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ℕ = dom 𝐹)
45	44	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ℕ = dom 𝐹)
46	5, 45	eleqtrd 2836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ dom 𝐹)
47		fvco 6987	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝑗 ∈ dom 𝐹) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹‘𝑗)))
48	42, 46, 47	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ([,)‘(𝐹‘𝑗)))
49	48	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘(𝐹‘𝑗)) = (([,) ∘ 𝐹)‘𝑗))
50	40, 49	eqtrd 2773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘((𝐼‘𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
51	50	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → ([,)‘((𝐼‘𝑗)‘𝐴)) = (([,) ∘ 𝐹)‘𝑗))
52	25, 33, 51	3eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ {𝐴}) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ 𝐹)‘𝑗))
53	52	ixpeq2dva 8903	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = X𝑘 ∈ {𝐴} (([,) ∘ 𝐹)‘𝑗))
54		snex 5431	. . . . . . . . . . 11 ⊢ {𝐴} ∈ V
55		fvex 6902	. . . . . . . . . . 11 ⊢ (([,) ∘ 𝐹)‘𝑗) ∈ V
56	54, 55	ixpconst 8898	. . . . . . . . . 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 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((([,) ∘ 𝐹)‘𝑗) ↑_m {𝐴}))
59	58	iuneq2dv 5021	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑_m {𝐴}))
60		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑗𝜑
61	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ∈ V)
62		fvexd 6904	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) ∈ V)
63	60, 61, 62, 9	iunmapsn 43902	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ ((([,) ∘ 𝐹)‘𝑗) ↑_m {𝐴}) = (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑_m {𝐴}))
64	59, 63	eqtrd 2773	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘) = (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑_m {𝐴}))
65	22, 64	sseqtrd 4022	. . . . 5 ⊢ (𝜑 → (𝐵 ↑_m {𝐴}) ⊆ (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑_m {𝐴}))
66		ovnovollem2.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
67	18, 55	iunex 7952	. . . . . . 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 4335	. . . . . 6 ⊢ (𝜑 → {𝐴} ≠ ∅)
71	66, 68, 69, 70	mapss2 43890	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↔ (𝐵 ↑_m {𝐴}) ⊆ (∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗) ↑_m {𝐴})))
72	65, 71	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝑗 ∈ ℕ (([,) ∘ 𝐹)‘𝑗))
73		icof 43904	. . . . . . . 8 ⊢ [,):(ℝ^* × ℝ^)⟶𝒫 ℝ^
74	73	a1i 11	. . . . . . 7 ⊢ (𝜑 → [,):(ℝ^* × ℝ^)⟶𝒫 ℝ^)
75		rexpssxrxp 11256	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
76	75	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*))
77	74, 76, 15	fcoss 43895	. . . . . 6 ⊢ (𝜑 → ([,) ∘ 𝐹):ℕ⟶𝒫 ℝ^*)
78	77	ffnd 6716	. . . . 5 ⊢ (𝜑 → ([,) ∘ 𝐹) Fn ℕ)
79		fniunfv 7243	. . . . 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 4022	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ([,) ∘ 𝐹))
82		ovnovollem2.z	. . . 4 ⊢ (𝜑 → 𝑍 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
83		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑗𝐹
84		ressxr 11255	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ^*
85		xpss2 5696	. . . . . . . . . 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 6733	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ^*))
89	83, 88	volicofmpt 44700	. . . . . 6 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ (vol‘((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗))))))
90	9	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ 𝑉)
91		fvexd 6904	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝐴) ∈ V)
92	5, 91, 36	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = ((𝐼‘𝑗)‘𝐴))
93	92, 13	eqeltrd 2834	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) ∈ (ℝ × ℝ))
94		1st2nd2 8011	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) → (𝐹‘𝑗) = ⟨(1^st ‘(𝐹‘𝑗)), (2^nd ‘(𝐹‘𝑗))⟩)
95	93, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘𝑗) = ⟨(1^st ‘(𝐹‘𝑗)), (2^nd ‘(𝐹‘𝑗))⟩)
96	95	fveq2d 6893	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘(𝐹‘𝑗)) = ([,)‘⟨(1^st ‘(𝐹‘𝑗)), (2^nd ‘(𝐹‘𝑗))⟩))
97		df-ov 7409	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗))) = ([,)‘⟨(1^st ‘(𝐹‘𝑗)), (2^nd ‘(𝐹‘𝑗))⟩)
98	97	eqcomi 2742	. . . . . . . . . . . . . . 15 ⊢ ([,)‘⟨(1^st ‘(𝐹‘𝑗)), (2^nd ‘(𝐹‘𝑗))⟩) = ((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗)))
99	98	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ([,)‘⟨(1^st ‘(𝐹‘𝑗)), (2^nd ‘(𝐹‘𝑗))⟩) = ((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗))))
100	48, 96, 99	3eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ 𝐹)‘𝑗) = ((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗))))
101	32, 50, 100	3eqtrd 2777	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (𝐼‘𝑗))‘𝐴) = ((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗))))
102	101	fveq2d 6893	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) = (vol‘((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗)))))
103		xp1st 8004	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) → (1^st ‘(𝐹‘𝑗)) ∈ ℝ)
104	93, 103	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (1^st ‘(𝐹‘𝑗)) ∈ ℝ)
105		xp2nd 8005	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑗) ∈ (ℝ × ℝ) → (2^nd ‘(𝐹‘𝑗)) ∈ ℝ)
106	93, 105	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2^nd ‘(𝐹‘𝑗)) ∈ ℝ)
107		volicore 45284	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐹‘𝑗)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑗)) ∈ ℝ) → (vol‘((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗)))) ∈ ℝ)
108	104, 106, 107	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗)))) ∈ ℝ)
109	102, 108	eqeltrd 2834	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℝ)
110	109	recnd 11239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℂ)
111		2fveq3 6894	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
112	111	prodsn 15903	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)) ∈ ℂ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
113	90, 110, 112	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝐴)))
114	113, 102	eqtr2d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗)))) = ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))
115	114	mpteq2dva 5248	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((1^st ‘(𝐹‘𝑗))[,)(2^nd ‘(𝐹‘𝑗))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
116	89, 115	eqtrd 2773	. . . . 5 ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))
117	116	fveq2d 6893	. . . 4 ⊢ (𝜑 → (Σ^{^}‘((vol ∘ [,)) ∘ 𝐹)) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))
118	82, 117	eqtr4d 2776	. . 3 ⊢ (𝜑 → 𝑍 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝐹)))
119	81, 118	jca 513	. 2 ⊢ (𝜑 → (𝐵 ⊆ ∪ ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝐹))))
120		coeq2 5857	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ([,) ∘ 𝑓) = ([,) ∘ 𝐹))
121	120	rneqd 5936	. . . . . 6 ⊢ (𝑓 = 𝐹 → ran ([,) ∘ 𝑓) = ran ([,) ∘ 𝐹))
122	121	unieqd 4922	. . . . 5 ⊢ (𝑓 = 𝐹 → ∪ ran ([,) ∘ 𝑓) = ∪ ran ([,) ∘ 𝐹))
123	122	sseq2d 4014	. . . 4 ⊢ (𝑓 = 𝐹 → (𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ↔ 𝐵 ⊆ ∪ ran ([,) ∘ 𝐹)))
124		coeq2 5857	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((vol ∘ [,)) ∘ 𝑓) = ((vol ∘ [,)) ∘ 𝐹))
125	124	fveq2d 6893	. . . . 5 ⊢ (𝑓 = 𝐹 → (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)) = (Σ^{^}‘((vol ∘ [,)) ∘ 𝐹)))
126	125	eqeq2d 2744	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑍 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓)) ↔ 𝑍 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝐹))))
127	123, 126	anbi12d 632	. . 3 ⊢ (𝑓 = 𝐹 → ((𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓))) ↔ (𝐵 ⊆ ∪ ran ([,) ∘ 𝐹) ∧ 𝑍 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝐹)))))
128	127	rspcev 3613	. 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 585	1 ⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑_m ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^{^}‘((vol ∘ [,)) ∘ 𝑓))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 Vcvv 3475 ⊆ wss 3948 𝒫 cpw 4602 {csn 4628 ⟨cop 4634 ∪ cuni 4908 ∪ ciun 4997 ↦ cmpt 5231 × cxp 5674 dom cdm 5676 ran crn 5677 ∘ ccom 5680 Fun wfun 6535 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 1^st c1st 7970 2^nd c2nd 7971 ↑_m cmap 8817 Xcixp 8888 ℂcc 11105 ℝcr 11106 ℝ^*cxr 11244 ℕcn 12209 [,)cico 13323 ∏cprod 15846 volcvol 24972 Σ^{^}csumge0 45065

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630 df-prod 15847 df-rest 17365 df-topgen 17386 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-top 22388 df-topon 22405 df-bases 22441 df-cmp 22883 df-ovol 24973 df-vol 24974

This theorem is referenced by: ovnovollem3 45361

W3C validator