hoicvrrex - Mathbox for Glauco Siliprandi

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Theorem hoicvrrex 46561

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 12200	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
2	1	renegcld 11612	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
3		opelxpi 5678	. . . . . . . 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 2730	. . . . . 6 ⊢ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
7	5, 6	fmptd 7089	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
8		reex 11166	. . . . . . . . 9 ⊢ ℝ ∈ V
9	8, 8	xpex 7732	. . . . . . . 8 ⊢ (ℝ × ℝ) ∈ V
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ × ℝ) ∈ V)
11		hoicvrrex.fi	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Fin)
12		elmapg 8815	. . . . . . 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 2730	. . . 4 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
17	15, 16	fmptd 7089	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
18		ovex 7423	. . . 4 ⊢ ((ℝ × ℝ) ↑_m 𝑋) ∈ V
19		nnex 12199	. . . 4 ⊢ ℕ ∈ V
20	18, 19	elmap 8847	. . 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 2730	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
24	23, 11	hoicvr 46553	. . . . 5 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
25		eqidd 2731	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
26	25	cbvmptv 5214	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
27	26	mpteq2i 5206	. . . . . . . . . . 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 6863	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
30	29	coeq2d 5829	. . . . . . . 8 ⊢ (𝜑 → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
31	30	fveq1d 6863	. . . . . . 7 ⊢ (𝜑 → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
32	31	ixpeq2dv 8889	. . . . . 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 4989	. . . . 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 3986	. . . 4 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
35	22, 34	sstrd 3960	. . 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 3474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
38	16	fvmpt2 6982	. . . . . . . . . . . . . . 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 7091	. . . . . . . . . . . . 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 45194	. . . . . . . . . . 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 6863	. . . . . . . . . . . . . . . 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 5427	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨-𝑗, 𝑗⟩ ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ V)
49	6	fvmpt2 6982	. . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ⟨-𝑗, 𝑗⟩)
53	52	fveq2d 6865	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (1^st ‘⟨-𝑗, 𝑗⟩))
54		negex 11426	. . . . . . . . . . . . . . 15 ⊢ -𝑗 ∈ V
55		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑗 ∈ V
56	54, 55	op1st 7979	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = -𝑗)
59	52	fveq2d 6865	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (2^nd ‘⟨-𝑗, 𝑗⟩))
60	54, 55	op2nd 7980	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = 𝑗)
63	58, 62	oveq12d 7408	. . . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (-𝑗[,)𝑗))
65	64	fveq2d 6865	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (vol‘(-𝑗[,)𝑗)))
66		volico 45988	. . . . . . . . . . . 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 12970	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ⁺)
69		neglt 12978	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℝ⁺ → -𝑗 < 𝑗)
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → -𝑗 < 𝑗)
71	70	iftrued 4499	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0) = (𝑗 − -𝑗))
72	1	recnd 11209	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
73	72, 72	subnegd 11547	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (𝑗 + 𝑗))
74	72	2timesd 12432	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (2 · 𝑗) = (𝑗 + 𝑗))
75	73, 74	eqtr4d 2768	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (2 · 𝑗))
76	67, 71, 75	3eqtrd 2769	. . . . . . . . . 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 2765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (2 · 𝑗))
79	78	prodeq2dv 15895	. . . . . . 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 12271	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℂ)
82	72	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
83	81, 82	mulcld 11201	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℂ)
84		fprodconst 15951	. . . . . . . 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 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ((2 · 𝑗)↑(♯‘𝑋)))
87	86	mpteq2dva 5203	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))))
88	87	fveq2d 6865	. . . 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 3953	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ⁺
91		ioorp 13393	. . . . . . . . . . 11 ⊢ (0(,)+∞) = ℝ⁺
92	91	eqcomi 2739	. . . . . . . . . 10 ⊢ ℝ⁺ = (0(,)+∞)
93	90, 92	sseqtri 3998	. . . . . . . . 9 ⊢ ℕ ⊆ (0(,)+∞)
94		ioossicc 13401	. . . . . . . . 9 ⊢ (0(,)+∞) ⊆ (0[,]+∞)
95	93, 94	sstri 3959	. . . . . . . 8 ⊢ ℕ ⊆ (0[,]+∞)
96		2nn 12266	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
97	96	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 2 ∈ ℕ)
98	97, 36	nnmulcld 12246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℕ)
99		hashcl 14328	. . . . . . . . . . 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 14046	. . . . . . . . 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 3947	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ (0[,]+∞))
105		eqid 2730	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))
106	104, 105	fmptd 7089	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))):ℕ⟶(0[,]+∞))
107	89, 106	sge0xrcl 46390	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) ∈ ℝ^*)
108		pnfxr 11235	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
109	108	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ^*)
110		1nn 12204	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
111	95, 110	sselii 3946	. . . . . . . . 9 ⊢ 1 ∈ (0[,]+∞)
112	111	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
113		eqid 2730	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ ↦ 1) = (𝑗 ∈ ℕ ↦ 1)
114	112, 113	fmptd 7089	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ 1):ℕ⟶(0[,]+∞))
115	89, 114	sge0xrcl 46390	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) ∈ ℝ^*)
116		nnnfi 13938	. . . . . . . . . 10 ⊢ ¬ ℕ ∈ Fin
117	116	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ¬ ℕ ∈ Fin)
118		1rp 12962	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
119	118	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ⁺)
120	89, 117, 119	sge0rpcpnf 46426	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) = +∞)
121	120	eqcomd 2736	. . . . . . 7 ⊢ (𝜑 → +∞ = (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)))
122	109, 121	xreqled 45333	. . . . . 6 ⊢ (𝜑 → +∞ ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)))
123		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
124	114	fvmptelcdm 7088	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
125	103	nnge1d 12241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ≤ ((2 · 𝑗)↑(♯‘𝑋)))
126	123, 89, 124, 104, 125	sge0lempt 46415	. . . . . 6 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ 1)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
127	109, 115, 107, 122, 126	xrletrd 13129	. . . . 5 ⊢ (𝜑 → +∞ ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
128	107, 127	xrgepnfd 45334	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) = +∞)
129		eqidd 2731	. . . 4 ⊢ (𝜑 → +∞ = +∞)
130	88, 128, 129	3eqtrrd 2770	. . 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 2892	. . . . . . 7 ⊢ Ⅎ𝑗𝑖
133		nfmpt1 5209	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
134	132, 133	nfeq 2906	. . . . . 6 ⊢ Ⅎ𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
135		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑖
136		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑘ℕ
137		nfmpt1 5209	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
138	136, 137	nfmpt 5208	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
139	135, 138	nfeq 2906	. . . . . . . 8 ⊢ Ⅎ𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
140		fveq1 6860	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑖‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
141	140	coeq2d 5829	. . . . . . . . . 10 ⊢ (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
142	141	fveq1d 6863	. . . . . . . . 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 45069	. . . . . . 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 45048	. . . . 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 3982	. . . 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 6865	. . . . . . . . . . 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 3236	. . . . . . . . 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 15894	. . . . . . 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 5202	. . . . . 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 6865	. . . . 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 2741	. . . 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 3591	. 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 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ⊆ wss 3917 ifcif 4491 ⟨cop 4598 ∪ ciun 4958 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ∘ ccom 5645 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 ↑_m cmap 8802 Xcixp 8873 Fincfn 8921 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 +∞cpnf 11212 ℝ^*cxr 11214 < clt 11215 − cmin 11412 -cneg 11413 ℕcn 12193 2c2 12248 ℕ₀cn0 12449 ℝ⁺crp 12958 (,)cioo 13313 [,)cico 13315 [,]cicc 13316 ↑cexp 14033 ♯chash 14302 ∏cprod 15876 volcvol 25371 Σ^{^}csumge0 46367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-prod 15877 df-rest 17392 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-top 22788 df-topon 22805 df-bases 22840 df-cmp 23281 df-ovol 25372 df-vol 25373 df-sumge0 46368

This theorem is referenced by: ovnpnfelsup 46564

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