hoicvrrex - Mathbox for Glauco Siliprandi

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Theorem hoicvrrex 46511

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 12270	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
2	1	renegcld 11687	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
3		opelxpi 5725	. . . . . . . 8 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
4	2, 1, 3	syl2anc 584	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
5	4	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
6		eqid 2734	. . . . . 6 ⊢ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
7	5, 6	fmptd 7133	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
8		reex 11243	. . . . . . . . 9 ⊢ ℝ ∈ V
9	8, 8	xpex 7771	. . . . . . . 8 ⊢ (ℝ × ℝ) ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
11		hoicvrrex.fi	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Fin)
12		elmapg 8877	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
15	7, 14	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋))
16		eqid 2734	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
17	15, 16	fmptd 7133	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
18		ovex 7463	. . . 4 ⊢ ((ℝ × ℝ) ↑_m 𝑋) ∈ V
19		nnex 12269	. . . 4 ⊢ ℕ ∈ V
20	18, 19	elmap 8909	. . 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 234	. 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 2734	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
24	23, 11	hoicvr 46503	. . . . 5 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
25		eqidd 2735	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
26	25	cbvmptv 5260	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
27	26	mpteq2i 5252	. . . . . . . . . . 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 6908	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
30	29	coeq2d 5875	. . . . . . . 8 ⊢ (𝜑 → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
31	30	fveq1d 6908	. . . . . . 7 ⊢ (𝜑 → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
32	31	ixpeq2dv 8951	. . . . . 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 5026	. . . . 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 4035	. . . 4 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
35	22, 34	sstrd 4005	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
36		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
37	15	elexd 3501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
38	16	fvmpt2 7026	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
39	36, 37, 38	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
40	39, 5	fmpt3d 7135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
41	40	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
42		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
43	41, 42	fvovco 45135	. . . . . . . . . . 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 6908	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
45	44	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
46		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
47		opex 5474	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨-𝑗, 𝑗⟩ ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ V)
49	6	fvmpt2 7026	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑋 ∧ ⟨-𝑗, 𝑗⟩ ∈ V) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
50	46, 48, 49	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
51	50	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
52	45, 51	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ⟨-𝑗, 𝑗⟩)
53	52	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (1^st ‘⟨-𝑗, 𝑗⟩))
54		negex 11503	. . . . . . . . . . . . . . 15 ⊢ -𝑗 ∈ V
55		vex 3481	. . . . . . . . . . . . . . 15 ⊢ 𝑗 ∈ V
56	54, 55	op1st 8020	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = -𝑗)
59	52	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (2^nd ‘⟨-𝑗, 𝑗⟩))
60	54, 55	op2nd 8021	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = 𝑗)
63	58, 62	oveq12d 7448	. . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (-𝑗[,)𝑗))
65	64	fveq2d 6910	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (vol‘(-𝑗[,)𝑗)))
66		volico 45938	. . . . . . . . . . . 12 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
67	2, 1, 66	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
68		nnrp 13043	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ⁺)
69		neglt 45234	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℝ⁺ → -𝑗 < 𝑗)
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → -𝑗 < 𝑗)
71	70	iftrued 4538	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0) = (𝑗 − -𝑗))
72	1	recnd 11286	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
73	72, 72	subnegd 11624	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (𝑗 + 𝑗))
74	72	2timesd 12506	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (2 · 𝑗) = (𝑗 + 𝑗))
75	73, 74	eqtr4d 2777	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (2 · 𝑗))
76	67, 71, 75	3eqtrd 2778	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
77	76	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
78	65, 77	eqtrd 2774	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (2 · 𝑗))
79	78	prodeq2dv 15954	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (2 · 𝑗))
80	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
81		2cnd 12341	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℂ)
82	72	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
83	81, 82	mulcld 11278	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℂ)
84		fprodconst 16010	. . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ (2 · 𝑗) ∈ ℂ) → ∏𝑘 ∈ 𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
85	80, 83, 84	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
86	79, 85	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ((2 · 𝑗)↑(♯‘𝑋)))
87	86	mpteq2dva 5247	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))))
88	87	fveq2d 6910	. . . 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 3998	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ⁺
91		ioorp 13461	. . . . . . . . . . 11 ⊢ (0(,)+∞) = ℝ⁺
92	91	eqcomi 2743	. . . . . . . . . 10 ⊢ ℝ⁺ = (0(,)+∞)
93	90, 92	sseqtri 4031	. . . . . . . . 9 ⊢ ℕ ⊆ (0(,)+∞)
94		ioossicc 13469	. . . . . . . . 9 ⊢ (0(,)+∞) ⊆ (0[,]+∞)
95	93, 94	sstri 4004	. . . . . . . 8 ⊢ ℕ ⊆ (0[,]+∞)
96		2nn 12336	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
97	96	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℕ)
98	97, 36	nnmulcld 12316	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℕ)
99		hashcl 14391	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
100	11, 99	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ₀)
101	100	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘𝑋) ∈ ℕ₀)
102		nnexpcl 14111	. . . . . . . . 9 ⊢ (((2 · 𝑗) ∈ ℕ ∧ (♯‘𝑋) ∈ ℕ₀) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
103	98, 101, 102	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
104	95, 103	sselid 3992	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ (0[,]+∞))
105		eqid 2734	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))
106	104, 105	fmptd 7133	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))):ℕ⟶(0[,]+∞))
107	89, 106	sge0xrcl 46340	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) ∈ ℝ^*)
108		pnfxr 11312	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
109	108	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
110		1nn 12274	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
111	95, 110	sselii 3991	. . . . . . . . 9 ⊢ 1 ∈ (0[,]+∞)
112	111	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
113		eqid 2734	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ 1) = (𝑗 ∈ ℕ ↦ 1)
114	112, 113	fmptd 7133	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ 1):ℕ⟶(0[,]+∞))
115	89, 114	sge0xrcl 46340	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) ∈ ℝ^*)
116		nnnfi 14003	. . . . . . . . . 10 ⊢ ¬ ℕ ∈ Fin
117	116	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ¬ ℕ ∈ Fin)
118		1rp 13035	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
119	118	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ⁺)
120	89, 117, 119	sge0rpcpnf 46376	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) = +∞)
121	120	eqcomd 2740	. . . . . . 7 ⊢ (𝜑 → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)))
122	109, 121	xreqled 45279	. . . . . 6 ⊢ (𝜑 → +∞ ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)))
123		nfv 1911	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
124	114	fvmptelcdm 7132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
125	103	nnge1d 12311	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ≤ ((2 · 𝑗)↑(♯‘𝑋)))
126	123, 89, 124, 104, 125	sge0lempt 46365	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
127	109, 115, 107, 122, 126	xrletrd 13200	. . . . 5 ⊢ (𝜑 → +∞ ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
128	107, 127	xrgepnfd 45280	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) = +∞)
129		eqidd 2735	. . . 4 ⊢ (𝜑 → +∞ = +∞)
130	88, 128, 129	3eqtrrd 2779	. . 3 ⊢ (𝜑 → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))
131	35, 130	jca 511	. 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 2902	. . . . . . 7 ⊢ Ⅎ𝑗𝑖
133		nfmpt1 5255	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
134	132, 133	nfeq 2916	. . . . . 6 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
135		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑖
136		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑘ℕ
137		nfmpt1 5255	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
138	136, 137	nfmpt 5254	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
139	135, 138	nfeq 2916	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
140		fveq1 6905	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑖‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
141	140	coeq2d 5875	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
142	141	fveq1d 6908	. . . . . . . . 9 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
143	142	adantr 480	. . . . . . . 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 45007	. . . . . . 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 480	. . . . . 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 44985	. . . . 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 4027	. . . 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 6910	. . . . . . . . . . 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 3254	. . . . . . . . 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 480	. . . . . . . 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 15953	. . . . . . 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 5245	. . . . . 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 6910	. . . . 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 2745	. . . 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 3621	. 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 584	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 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 Vcvv 3477 ⊆ wss 3962 ifcif 4530 ⟨cop 4636 ∪ ciun 4995 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 1^st c1st 8010 2^nd c2nd 8011 ↑_m cmap 8864 Xcixp 8935 Fincfn 8983 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 +∞cpnf 11289 ℝ^*cxr 11291 < clt 11292 − cmin 11489 -cneg 11490 ℕcn 12263 2c2 12318 ℕ₀cn0 12523 ℝ⁺crp 13031 (,)cioo 13383 [,)cico 13385 [,]cicc 13386 ↑cexp 14098 ♯chash 14365 ∏cprod 15935 volcvol 25511 Σ^{^}csumge0 46317

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-prod 15936 df-rest 17468 df-topgen 17489 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-top 22915 df-topon 22932 df-bases 22968 df-cmp 23410 df-ovol 25512 df-vol 25513 df-sumge0 46318

This theorem is referenced by: ovnpnfelsup 46514

W3C validator