hoicvrrex - Mathbox for Glauco Siliprandi

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Theorem hoicvrrex 42832

Description: Any subset of the multidimensional reals can be covered by a countable set of half-open intervals, see Definition 115A (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
hoicvrrex.fi	⊢ (𝜑 → 𝑋 ∈ Fin)
hoicvrrex.y	⊢ (𝜑 → 𝑌 ⊆ (ℝ ↑_m 𝑋))

**Assertion**
Ref	Expression
hoicvrrex	⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))

Distinct variable groups: 𝑖,𝑋,𝑗,𝑘 𝑖,𝑌 𝜑,𝑗,𝑘 Allowed substitution hints: 𝜑(𝑖) 𝑌(𝑗,𝑘)

Proof of Theorem hoicvrrex Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnre 11639	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
2	1	renegcld 11061	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
3		opelxpi 5586	. . . . . . . 8 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
4	2, 1, 3	syl2anc 586	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
5	4	ad2antlr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
6		eqid 2821	. . . . . 6 ⊢ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
7	5, 6	fmptd 6872	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
8		reex 10622	. . . . . . . . 9 ⊢ ℝ ∈ V
9	8, 8	xpex 7470	. . . . . . . 8 ⊢ (ℝ × ℝ) ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
11		hoicvrrex.fi	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Fin)
12		elmapg 8413	. . . . . . 7 ⊢ (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
13	10, 11, 12	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
14	13	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
15	7, 14	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋))
16		eqid 2821	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
17	15, 16	fmptd 6872	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
18		ovex 7183	. . . 4 ⊢ ((ℝ × ℝ) ↑_m 𝑋) ∈ V
19		nnex 11638	. . . 4 ⊢ ℕ ∈ V
20	18, 19	elmap 8429	. . 3 ⊢ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
21	17, 20	sylibr 236	. 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ))
22		hoicvrrex.y	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ (ℝ ↑_m 𝑋))
23		eqid 2821	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
24	23, 11	hoicvr 42824	. . . . 5 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
25		eqidd 2822	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
26	25	cbvmptv 5161	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
27	26	mpteq2i 5150	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)))
29	28	fveq1d 6666	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
30	29	coeq2d 5727	. . . . . . . 8 ⊢ (𝜑 → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
31	30	fveq1d 6666	. . . . . . 7 ⊢ (𝜑 → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
32	31	ixpeq2dv 8471	. . . . . 6 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
33	32	iuneq2d 4940	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
34	24, 33	sseqtrd 4006	. . . 4 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
35	22, 34	sstrd 3976	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
36		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
37	15	elexd 3514	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
38	16	fvmpt2 6773	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
39	36, 37, 38	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
40	39, 5	fmpt3d 6874	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
41	40	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
42		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
43	41, 42	fvovco 41448	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = ((1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))[,)(2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))))
44	39	fveq1d 6666	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
45	44	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
46		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
47		opex 5348	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨-𝑗, 𝑗⟩ ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ V)
49	6	fvmpt2 6773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑋 ∧ ⟨-𝑗, 𝑗⟩ ∈ V) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
50	46, 48, 49	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
51	50	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
52	45, 51	eqtrd 2856	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ⟨-𝑗, 𝑗⟩)
53	52	fveq2d 6668	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (1^st ‘⟨-𝑗, 𝑗⟩))
54		negex 10878	. . . . . . . . . . . . . . 15 ⊢ -𝑗 ∈ V
55		vex 3497	. . . . . . . . . . . . . . 15 ⊢ 𝑗 ∈ V
56	54, 55	op1st 7691	. . . . . . . . . . . . . 14 ⊢ (1^st ‘⟨-𝑗, 𝑗⟩) = -𝑗
57	56	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘⟨-𝑗, 𝑗⟩) = -𝑗)
58	53, 57	eqtrd 2856	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = -𝑗)
59	52	fveq2d 6668	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (2^nd ‘⟨-𝑗, 𝑗⟩))
60	54, 55	op2nd 7692	. . . . . . . . . . . . . 14 ⊢ (2^nd ‘⟨-𝑗, 𝑗⟩) = 𝑗
61	60	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘⟨-𝑗, 𝑗⟩) = 𝑗)
62	59, 61	eqtrd 2856	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = 𝑗)
63	58, 62	oveq12d 7168	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))[,)(2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))) = (-𝑗[,)𝑗))
64	43, 63	eqtrd 2856	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (-𝑗[,)𝑗))
65	64	fveq2d 6668	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (vol‘(-𝑗[,)𝑗)))
66		volico 42262	. . . . . . . . . . . 12 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
67	2, 1, 66	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
68		nnrp 12394	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ⁺)
69		neglt 41543	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℝ⁺ → -𝑗 < 𝑗)
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → -𝑗 < 𝑗)
71	70	iftrued 4474	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0) = (𝑗 − -𝑗))
72	1	recnd 10663	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
73	72, 72	subnegd 10998	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (𝑗 + 𝑗))
74	72	2timesd 11874	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (2 · 𝑗) = (𝑗 + 𝑗))
75	73, 74	eqtr4d 2859	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (2 · 𝑗))
76	67, 71, 75	3eqtrd 2860	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
77	76	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
78	65, 77	eqtrd 2856	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (2 · 𝑗))
79	78	prodeq2dv 15271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (2 · 𝑗))
80	11	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
81		2cnd 11709	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℂ)
82	72	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
83	81, 82	mulcld 10655	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℂ)
84		fprodconst 15326	. . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ (2 · 𝑗) ∈ ℂ) → ∏𝑘 ∈ 𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
85	80, 83, 84	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
86	79, 85	eqtrd 2856	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ((2 · 𝑗)↑(♯‘𝑋)))
87	86	mpteq2dva 5153	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))))
88	87	fveq2d 6668	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
89	19	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
90	68	ssriv 3970	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ⁺
91		ioorp 12808	. . . . . . . . . . 11 ⊢ (0(,)+∞) = ℝ⁺
92	91	eqcomi 2830	. . . . . . . . . 10 ⊢ ℝ⁺ = (0(,)+∞)
93	90, 92	sseqtri 4002	. . . . . . . . 9 ⊢ ℕ ⊆ (0(,)+∞)
94		ioossicc 12816	. . . . . . . . 9 ⊢ (0(,)+∞) ⊆ (0[,]+∞)
95	93, 94	sstri 3975	. . . . . . . 8 ⊢ ℕ ⊆ (0[,]+∞)
96		2nn 11704	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
97	96	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℕ)
98	97, 36	nnmulcld 11684	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℕ)
99		hashcl 13711	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
100	11, 99	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ₀)
101	100	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘𝑋) ∈ ℕ₀)
102		nnexpcl 13436	. . . . . . . . 9 ⊢ (((2 · 𝑗) ∈ ℕ ∧ (♯‘𝑋) ∈ ℕ₀) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
103	98, 101, 102	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
104	95, 103	sseldi 3964	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ (0[,]+∞))
105		eqid 2821	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))
106	104, 105	fmptd 6872	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))):ℕ⟶(0[,]+∞))
107	89, 106	sge0xrcl 42661	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) ∈ ℝ^*)
108		pnfxr 10689	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
109	108	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
110		1nn 11643	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
111	95, 110	sselii 3963	. . . . . . . . 9 ⊢ 1 ∈ (0[,]+∞)
112	111	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
113		eqid 2821	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ 1) = (𝑗 ∈ ℕ ↦ 1)
114	112, 113	fmptd 6872	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ 1):ℕ⟶(0[,]+∞))
115	89, 114	sge0xrcl 42661	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) ∈ ℝ^*)
116		nnnfi 13328	. . . . . . . . . 10 ⊢ ¬ ℕ ∈ Fin
117	116	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ¬ ℕ ∈ Fin)
118		1rp 12387	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
119	118	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ⁺)
120	89, 117, 119	sge0rpcpnf 42697	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) = +∞)
121	120	eqcomd 2827	. . . . . . 7 ⊢ (𝜑 → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)))
122	109, 121	xreqled 41591	. . . . . 6 ⊢ (𝜑 → +∞ ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)))
123		nfv 1911	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
124	114	fvmptelrn 6871	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
125	103	nnge1d 11679	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ≤ ((2 · 𝑗)↑(♯‘𝑋)))
126	123, 89, 124, 104, 125	sge0lempt 42686	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
127	109, 115, 107, 122, 126	xrletrd 12549	. . . . 5 ⊢ (𝜑 → +∞ ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
128	107, 127	xrgepnfd 41592	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) = +∞)
129		eqidd 2822	. . . 4 ⊢ (𝜑 → +∞ = +∞)
130	88, 128, 129	3eqtrrd 2861	. . 3 ⊢ (𝜑 → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))
131	35, 130	jca 514	. 2 ⊢ (𝜑 → (𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) ∧ +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))))))
132		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑗𝑖
133		nfmpt1 5156	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
134	132, 133	nfeq 2991	. . . . . 6 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
135		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑖
136		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑘ℕ
137		nfmpt1 5156	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
138	136, 137	nfmpt 5155	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
139	135, 138	nfeq 2991	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
140		fveq1 6663	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑖‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
141	140	coeq2d 5727	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
142	141	fveq1d 6666	. . . . . . . . 9 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
143	142	adantr 483	. . . . . . . 8 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
144	139, 143	ixpeq2d 41323	. . . . . . 7 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
145	144	adantr 483	. . . . . 6 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
146	134, 145	iuneq2df 41301	. . . . 5 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
147	146	sseq2d 3998	. . . 4 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
148	142	fveq2d 6668	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
149	148	a1d 25	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑘 ∈ 𝑋 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))))
150	139, 149	ralrimi 3216	. . . . . . . . 9 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
151	150	adantr 483	. . . . . . . 8 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
152	151	prodeq2d 15270	. . . . . . 7 ⊢ ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
153	134, 152	mpteq2da 5152	. . . . . 6 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))))
154	153	fveq2d 6668	. . . . 5 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))
155	154	eqeq2d 2832	. . . 4 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (+∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))))))
156	147, 155	anbi12d 632	. . 3 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ((𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ (𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) ∧ +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))))
157	156	rspcev 3622	. 2 ⊢ (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) ∧ +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
158	21, 131, 157	syl2anc 586	1 ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3935 ifcif 4466 ⟨cop 4566 ∪ ciun 4911 class class class wbr 5058 ↦ cmpt 5138 × cxp 5547 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 1^st c1st 7681 2^nd c2nd 7682 ↑_m cmap 8400 Xcixp 8455 Fincfn 8503 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 +∞cpnf 10666 ℝ^*cxr 10668 < clt 10669 − cmin 10864 -cneg 10865 ℕcn 11632 2c2 11686 ℕ₀cn0 11891 ℝ⁺crp 12383 (,)cioo 12732 [,)cico 12734 [,]cicc 12735 ↑cexp 13423 ♯chash 13684 ∏cprod 15253 volcvol 24058 Σ^{^}csumge0 42638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-prod 15254 df-rest 16690 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-cmp 21989 df-ovol 24059 df-vol 24060 df-sumge0 42639

This theorem is referenced by: ovnpnfelsup 42835

W3C validator