hoicvrrex - Mathbox for Glauco Siliprandi

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Theorem hoicvrrex 43561

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 11681	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
2	1	renegcld 11105	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
3		opelxpi 5561	. . . . . . . 8 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
4	2, 1, 3	syl2anc 587	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
5	4	ad2antlr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
6		eqid 2758	. . . . . 6 ⊢ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
7	5, 6	fmptd 6869	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
8		reex 10666	. . . . . . . . 9 ⊢ ℝ ∈ V
9	8, 8	xpex 7474	. . . . . . . 8 ⊢ (ℝ × ℝ) ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
11		hoicvrrex.fi	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Fin)
12		elmapg 8429	. . . . . . 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 587	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
14	13	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
15	7, 14	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑_m 𝑋))
16		eqid 2758	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
17	15, 16	fmptd 6869	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
18		ovex 7183	. . . 4 ⊢ ((ℝ × ℝ) ↑_m 𝑋) ∈ V
19		nnex 11680	. . . 4 ⊢ ℕ ∈ V
20	18, 19	elmap 8453	. . 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 237	. 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 2758	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
24	23, 11	hoicvr 43553	. . . . 5 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
25		eqidd 2759	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
26	25	cbvmptv 5135	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
27	26	mpteq2i 5124	. . . . . . . . . . 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 6660	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
30	29	coeq2d 5702	. . . . . . . 8 ⊢ (𝜑 → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
31	30	fveq1d 6660	. . . . . . 7 ⊢ (𝜑 → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
32	31	ixpeq2dv 8495	. . . . . 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 4912	. . . . 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 3932	. . . 4 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
35	22, 34	sstrd 3902	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
36		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
37	15	elexd 3430	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
38	16	fvmpt2 6770	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
39	36, 37, 38	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
40	39, 5	fmpt3d 6871	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
41	40	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
42		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
43	41, 42	fvovco 42191	. . . . . . . . . . 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 6660	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
45	44	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
46		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
47		opex 5324	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨-𝑗, 𝑗⟩ ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ V)
49	6	fvmpt2 6770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑋 ∧ ⟨-𝑗, 𝑗⟩ ∈ V) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
50	46, 48, 49	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
51	50	adantlr 714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
52	45, 51	eqtrd 2793	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ⟨-𝑗, 𝑗⟩)
53	52	fveq2d 6662	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (1^st ‘⟨-𝑗, 𝑗⟩))
54		negex 10922	. . . . . . . . . . . . . . 15 ⊢ -𝑗 ∈ V
55		vex 3413	. . . . . . . . . . . . . . 15 ⊢ 𝑗 ∈ V
56	54, 55	op1st 7701	. . . . . . . . . . . . . 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 2793	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = -𝑗)
59	52	fveq2d 6662	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (2^nd ‘⟨-𝑗, 𝑗⟩))
60	54, 55	op2nd 7702	. . . . . . . . . . . . . 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 2793	. . . . . . . . . . . 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 2793	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (-𝑗[,)𝑗))
65	64	fveq2d 6662	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (vol‘(-𝑗[,)𝑗)))
66		volico 42991	. . . . . . . . . . . 12 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
67	2, 1, 66	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
68		nnrp 12441	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ⁺)
69		neglt 42283	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℝ⁺ → -𝑗 < 𝑗)
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → -𝑗 < 𝑗)
71	70	iftrued 4428	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0) = (𝑗 − -𝑗))
72	1	recnd 10707	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
73	72, 72	subnegd 11042	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (𝑗 + 𝑗))
74	72	2timesd 11917	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (2 · 𝑗) = (𝑗 + 𝑗))
75	73, 74	eqtr4d 2796	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (2 · 𝑗))
76	67, 71, 75	3eqtrd 2797	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
77	76	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
78	65, 77	eqtrd 2793	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (2 · 𝑗))
79	78	prodeq2dv 15325	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (2 · 𝑗))
80	11	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
81		2cnd 11752	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℂ)
82	72	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
83	81, 82	mulcld 10699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℂ)
84		fprodconst 15380	. . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ (2 · 𝑗) ∈ ℂ) → ∏𝑘 ∈ 𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
85	80, 83, 84	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
86	79, 85	eqtrd 2793	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ((2 · 𝑗)↑(♯‘𝑋)))
87	86	mpteq2dva 5127	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))))
88	87	fveq2d 6662	. . . 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 3896	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ⁺
91		ioorp 12857	. . . . . . . . . . 11 ⊢ (0(,)+∞) = ℝ⁺
92	91	eqcomi 2767	. . . . . . . . . 10 ⊢ ℝ⁺ = (0(,)+∞)
93	90, 92	sseqtri 3928	. . . . . . . . 9 ⊢ ℕ ⊆ (0(,)+∞)
94		ioossicc 12865	. . . . . . . . 9 ⊢ (0(,)+∞) ⊆ (0[,]+∞)
95	93, 94	sstri 3901	. . . . . . . 8 ⊢ ℕ ⊆ (0[,]+∞)
96		2nn 11747	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
97	96	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℕ)
98	97, 36	nnmulcld 11727	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℕ)
99		hashcl 13767	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
100	11, 99	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ₀)
101	100	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (♯‘𝑋) ∈ ℕ₀)
102		nnexpcl 13492	. . . . . . . . 9 ⊢ (((2 · 𝑗) ∈ ℕ ∧ (♯‘𝑋) ∈ ℕ₀) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
103	98, 101, 102	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
104	95, 103	sseldi 3890	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ (0[,]+∞))
105		eqid 2758	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))
106	104, 105	fmptd 6869	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))):ℕ⟶(0[,]+∞))
107	89, 106	sge0xrcl 43390	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) ∈ ℝ^*)
108		pnfxr 10733	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
109	108	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
110		1nn 11685	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
111	95, 110	sselii 3889	. . . . . . . . 9 ⊢ 1 ∈ (0[,]+∞)
112	111	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
113		eqid 2758	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ 1) = (𝑗 ∈ ℕ ↦ 1)
114	112, 113	fmptd 6869	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ 1):ℕ⟶(0[,]+∞))
115	89, 114	sge0xrcl 43390	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) ∈ ℝ^*)
116		nnnfi 13383	. . . . . . . . . 10 ⊢ ¬ ℕ ∈ Fin
117	116	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ¬ ℕ ∈ Fin)
118		1rp 12434	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
119	118	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ⁺)
120	89, 117, 119	sge0rpcpnf 43426	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) = +∞)
121	120	eqcomd 2764	. . . . . . 7 ⊢ (𝜑 → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)))
122	109, 121	xreqled 42330	. . . . . 6 ⊢ (𝜑 → +∞ ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)))
123		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
124	114	fvmptelrn 6868	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
125	103	nnge1d 11722	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ≤ ((2 · 𝑗)↑(♯‘𝑋)))
126	123, 89, 124, 104, 125	sge0lempt 43415	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
127	109, 115, 107, 122, 126	xrletrd 12596	. . . . 5 ⊢ (𝜑 → +∞ ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
128	107, 127	xrgepnfd 42331	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) = +∞)
129		eqidd 2759	. . . 4 ⊢ (𝜑 → +∞ = +∞)
130	88, 128, 129	3eqtrrd 2798	. . 3 ⊢ (𝜑 → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))
131	35, 130	jca 515	. 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 2919	. . . . . . 7 ⊢ Ⅎ𝑗𝑖
133		nfmpt1 5130	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
134	132, 133	nfeq 2932	. . . . . 6 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
135		nfcv 2919	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑖
136		nfcv 2919	. . . . . . . . . 10 ⊢ Ⅎ𝑘ℕ
137		nfmpt1 5130	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
138	136, 137	nfmpt 5129	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
139	135, 138	nfeq 2932	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
140		fveq1 6657	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑖‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
141	140	coeq2d 5702	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
142	141	fveq1d 6660	. . . . . . . . 9 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
143	142	adantr 484	. . . . . . . 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 42075	. . . . . . 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 484	. . . . . 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 42053	. . . . 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 3924	. . . 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 6662	. . . . . . . . . . 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 3144	. . . . . . . . 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 484	. . . . . . . 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 15324	. . . . . . 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 5126	. . . . . 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 6662	. . . . 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 2769	. . . 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 633	. . 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 3541	. 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 587	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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3070 ∃wrex 3071 Vcvv 3409 ⊆ wss 3858 ifcif 4420 ⟨cop 4528 ∪ ciun 4883 class class class wbr 5032 ↦ cmpt 5112 × cxp 5522 ∘ ccom 5528 ⟶wf 6331 ‘cfv 6335 (class class class)co 7150 1^st c1st 7691 2^nd c2nd 7692 ↑_m cmap 8416 Xcixp 8479 Fincfn 8527 ℂcc 10573 ℝcr 10574 0cc0 10575 1c1 10576 + caddc 10578 · cmul 10580 +∞cpnf 10710 ℝ^*cxr 10712 < clt 10713 − cmin 10908 -cneg 10909 ℕcn 11674 2c2 11729 ℕ₀cn0 11934 ℝ⁺crp 12430 (,)cioo 12779 [,)cico 12781 [,]cicc 12782 ↑cexp 13479 ♯chash 13740 ∏cprod 15307 volcvol 24163 Σ^{^}csumge0 43367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7405 df-om 7580 df-1st 7693 df-2nd 7694 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-2o 8113 df-er 8299 df-map 8418 df-pm 8419 df-ixp 8480 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-fi 8908 df-sup 8939 df-inf 8940 df-oi 9007 df-dju 9363 df-card 9401 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-n0 11935 df-z 12021 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-ioo 12783 df-ico 12785 df-icc 12786 df-fz 12940 df-fzo 13083 df-fl 13211 df-seq 13419 df-exp 13480 df-hash 13741 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-clim 14893 df-rlim 14894 df-sum 15091 df-prod 15308 df-rest 16754 df-topgen 16775 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-top 21594 df-topon 21611 df-bases 21646 df-cmp 22087 df-ovol 24164 df-vol 24165 df-sumge0 43368

This theorem is referenced by: ovnpnfelsup 43564

W3C validator