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Theorem hoicvrrex 44481
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 (𝜑𝑋 ∈ Fin)
hoicvrrex.y (𝜑𝑌 ⊆ (ℝ ↑m 𝑋))
Assertion
Ref Expression
hoicvrrex (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘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.
StepHypRef Expression
1 nnre 12086 . . . . . . . . 9 (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
21renegcld 11508 . . . . . . . 8 (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
3 opelxpi 5662 . . . . . . . 8 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
42, 1, 3syl2anc 585 . . . . . . 7 (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
54ad2antlr 725 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
6 eqid 2737 . . . . . 6 (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
75, 6fmptd 7049 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
8 reex 11068 . . . . . . . . 9 ℝ ∈ V
98, 8xpex 7670 . . . . . . . 8 (ℝ × ℝ) ∈ V
109a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ∈ V)
11 hoicvrrex.fi . . . . . . 7 (𝜑𝑋 ∈ Fin)
12 elmapg 8704 . . . . . . 7 (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
1310, 11, 12syl2anc 585 . . . . . 6 (𝜑 → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
1413adantr 482 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
157, 14mpbird 257 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
16 eqid 2737 . . . 4 (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
1715, 16fmptd 7049 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
18 ovex 7375 . . . 4 ((ℝ × ℝ) ↑m 𝑋) ∈ V
19 nnex 12085 . . . 4 ℕ ∈ V
2018, 19elmap 8735 . . 3 ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
2117, 20sylibr 233 . 2 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
22 hoicvrrex.y . . . 4 (𝜑𝑌 ⊆ (ℝ ↑m 𝑋))
23 eqid 2737 . . . . . 6 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))
2423, 11hoicvr 44473 . . . . 5 (𝜑 → (ℝ ↑m 𝑋) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
25 eqidd 2738 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
2625cbvmptv 5210 . . . . . . . . . . . 12 (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
2726mpteq2i 5202 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
2827a1i 11 . . . . . . . . . 10 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)))
2928fveq1d 6832 . . . . . . . . 9 (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
3029coeq2d 5809 . . . . . . . 8 (𝜑 → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
3130fveq1d 6832 . . . . . . 7 (𝜑 → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3231ixpeq2dv 8777 . . . . . 6 (𝜑X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3332iuneq2d 4975 . . . . 5 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3424, 33sseqtrd 3976 . . . 4 (𝜑 → (ℝ ↑m 𝑋) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3522, 34sstrd 3946 . . 3 (𝜑𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
36 simpr 486 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
3715elexd 3462 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
3816fvmpt2 6947 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
3936, 37, 38syl2anc 585 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
4039, 5fmpt3d 7051 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
4140adantr 482 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
42 simpr 486 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
4341, 42fvovco 43109 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = ((1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))[,)(2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))))
4439fveq1d 6832 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
4544adantr 482 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
46 simpr 486 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → 𝑘𝑋)
47 opex 5414 . . . . . . . . . . . . . . . . . 18 ⟨-𝑗, 𝑗⟩ ∈ V
4847a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → ⟨-𝑗, 𝑗⟩ ∈ V)
496fvmpt2 6947 . . . . . . . . . . . . . . . . 17 ((𝑘𝑋 ∧ ⟨-𝑗, 𝑗⟩ ∈ V) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5046, 48, 49syl2anc 585 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5150adantlr 713 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5245, 51eqtrd 2777 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5352fveq2d 6834 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (1st ‘⟨-𝑗, 𝑗⟩))
54 negex 11325 . . . . . . . . . . . . . . 15 -𝑗 ∈ V
55 vex 3446 . . . . . . . . . . . . . . 15 𝑗 ∈ V
5654, 55op1st 7912 . . . . . . . . . . . . . 14 (1st ‘⟨-𝑗, 𝑗⟩) = -𝑗
5756a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘⟨-𝑗, 𝑗⟩) = -𝑗)
5853, 57eqtrd 2777 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = -𝑗)
5952fveq2d 6834 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (2nd ‘⟨-𝑗, 𝑗⟩))
6054, 55op2nd 7913 . . . . . . . . . . . . . 14 (2nd ‘⟨-𝑗, 𝑗⟩) = 𝑗
6160a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘⟨-𝑗, 𝑗⟩) = 𝑗)
6259, 61eqtrd 2777 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = 𝑗)
6358, 62oveq12d 7360 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))[,)(2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))) = (-𝑗[,)𝑗))
6443, 63eqtrd 2777 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (-𝑗[,)𝑗))
6564fveq2d 6834 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (vol‘(-𝑗[,)𝑗)))
66 volico 43910 . . . . . . . . . . . 12 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
672, 1, 66syl2anc 585 . . . . . . . . . . 11 (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
68 nnrp 12847 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℝ+)
69 neglt 43208 . . . . . . . . . . . . 13 (𝑗 ∈ ℝ+ → -𝑗 < 𝑗)
7068, 69syl 17 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → -𝑗 < 𝑗)
7170iftrued 4486 . . . . . . . . . . 11 (𝑗 ∈ ℕ → if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0) = (𝑗 − -𝑗))
721recnd 11109 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
7372, 72subnegd 11445 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (𝑗 + 𝑗))
74722timesd 12322 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (2 · 𝑗) = (𝑗 + 𝑗))
7573, 74eqtr4d 2780 . . . . . . . . . . 11 (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (2 · 𝑗))
7667, 71, 753eqtrd 2781 . . . . . . . . . 10 (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
7776ad2antlr 725 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
7865, 77eqtrd 2777 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (2 · 𝑗))
7978prodeq2dv 15733 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ∏𝑘𝑋 (2 · 𝑗))
8011adantr 482 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
81 2cnd 12157 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℂ)
8272adantl 483 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
8381, 82mulcld 11101 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℂ)
84 fprodconst 15788 . . . . . . . 8 ((𝑋 ∈ Fin ∧ (2 · 𝑗) ∈ ℂ) → ∏𝑘𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
8580, 83, 84syl2anc 585 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
8679, 85eqtrd 2777 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ((2 · 𝑗)↑(♯‘𝑋)))
8786mpteq2dva 5197 . . . . 5 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))))
8887fveq2d 6834 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
8919a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
9068ssriv 3940 . . . . . . . . . 10 ℕ ⊆ ℝ+
91 ioorp 13263 . . . . . . . . . . 11 (0(,)+∞) = ℝ+
9291eqcomi 2746 . . . . . . . . . 10 + = (0(,)+∞)
9390, 92sseqtri 3972 . . . . . . . . 9 ℕ ⊆ (0(,)+∞)
94 ioossicc 13271 . . . . . . . . 9 (0(,)+∞) ⊆ (0[,]+∞)
9593, 94sstri 3945 . . . . . . . 8 ℕ ⊆ (0[,]+∞)
96 2nn 12152 . . . . . . . . . . 11 2 ∈ ℕ
9796a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℕ)
9897, 36nnmulcld 12132 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℕ)
99 hashcl 14176 . . . . . . . . . . 11 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
10011, 99syl 17 . . . . . . . . . 10 (𝜑 → (♯‘𝑋) ∈ ℕ0)
101100adantr 482 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (♯‘𝑋) ∈ ℕ0)
102 nnexpcl 13901 . . . . . . . . 9 (((2 · 𝑗) ∈ ℕ ∧ (♯‘𝑋) ∈ ℕ0) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
10398, 101, 102syl2anc 585 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
10495, 103sselid 3934 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ (0[,]+∞))
105 eqid 2737 . . . . . . 7 (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))
106104, 105fmptd 7049 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))):ℕ⟶(0[,]+∞))
10789, 106sge0xrcl 44310 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) ∈ ℝ*)
108 pnfxr 11135 . . . . . . 7 +∞ ∈ ℝ*
109108a1i 11 . . . . . 6 (𝜑 → +∞ ∈ ℝ*)
110 1nn 12090 . . . . . . . . . 10 1 ∈ ℕ
11195, 110sselii 3933 . . . . . . . . 9 1 ∈ (0[,]+∞)
112111a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
113 eqid 2737 . . . . . . . 8 (𝑗 ∈ ℕ ↦ 1) = (𝑗 ∈ ℕ ↦ 1)
114112, 113fmptd 7049 . . . . . . 7 (𝜑 → (𝑗 ∈ ℕ ↦ 1):ℕ⟶(0[,]+∞))
11589, 114sge0xrcl 44310 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) ∈ ℝ*)
116 nnnfi 13792 . . . . . . . . . 10 ¬ ℕ ∈ Fin
117116a1i 11 . . . . . . . . 9 (𝜑 → ¬ ℕ ∈ Fin)
118 1rp 12840 . . . . . . . . . 10 1 ∈ ℝ+
119118a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℝ+)
12089, 117, 119sge0rpcpnf 44346 . . . . . . . 8 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) = +∞)
121120eqcomd 2743 . . . . . . 7 (𝜑 → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ 1)))
122109, 121xreqled 43254 . . . . . 6 (𝜑 → +∞ ≤ (Σ^‘(𝑗 ∈ ℕ ↦ 1)))
123 nfv 1917 . . . . . . 7 𝑗𝜑
124114fvmptelcdm 7048 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
125103nnge1d 12127 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 1 ≤ ((2 · 𝑗)↑(♯‘𝑋)))
126123, 89, 124, 104, 125sge0lempt 44335 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
127109, 115, 107, 122, 126xrletrd 13002 . . . . 5 (𝜑 → +∞ ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
128107, 127xrgepnfd 43255 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) = +∞)
129 eqidd 2738 . . . 4 (𝜑 → +∞ = +∞)
13088, 128, 1293eqtrrd 2782 . . 3 (𝜑 → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))
13135, 130jca 513 . 2 (𝜑 → (𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))))))
132 nfcv 2905 . . . . . . 7 𝑗𝑖
133 nfmpt1 5205 . . . . . . 7 𝑗(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
134132, 133nfeq 2918 . . . . . 6 𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
135 nfcv 2905 . . . . . . . . 9 𝑘𝑖
136 nfcv 2905 . . . . . . . . . 10 𝑘
137 nfmpt1 5205 . . . . . . . . . 10 𝑘(𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
138136, 137nfmpt 5204 . . . . . . . . 9 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
139135, 138nfeq 2918 . . . . . . . 8 𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
140 fveq1 6829 . . . . . . . . . . 11 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑖𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
141140coeq2d 5809 . . . . . . . . . 10 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
142141fveq1d 6832 . . . . . . . . 9 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
143142adantr 482 . . . . . . . 8 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
144139, 143ixpeq2d 42986 . . . . . . 7 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
145144adantr 482 . . . . . 6 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
146134, 145iuneq2df 42964 . . . . 5 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
147146sseq2d 3968 . . . 4 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
148142fveq2d 6834 . . . . . . . . . . 11 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
149148a1d 25 . . . . . . . . . 10 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑘𝑋 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))))
150139, 149ralrimi 3237 . . . . . . . . 9 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
151150adantr 482 . . . . . . . 8 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
152151prodeq2d 15732 . . . . . . 7 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
153134, 152mpteq2da 5195 . . . . . 6 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))))
154153fveq2d 6834 . . . . 5 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))
155154eqeq2d 2748 . . . 4 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (+∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))))))
156147, 155anbi12d 632 . . 3 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ((𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))))
157156rspcev 3574 . 2 (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
15821, 131, 157syl2anc 585 1 (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1541  wcel 2106  wral 3062  wrex 3071  Vcvv 3442  wss 3902  ifcif 4478  cop 4584   ciun 4946   class class class wbr 5097  cmpt 5180   × cxp 5623  ccom 5629  wf 6480  cfv 6484  (class class class)co 7342  1st c1st 7902  2nd c2nd 7903  m cmap 8691  Xcixp 8761  Fincfn 8809  cc 10975  cr 10976  0cc0 10977  1c1 10978   + caddc 10980   · cmul 10982  +∞cpnf 11112  *cxr 11114   < clt 11115  cmin 11311  -cneg 11312  cn 12079  2c2 12134  0cn0 12339  +crp 12836  (,)cioo 13185  [,)cico 13187  [,]cicc 13188  cexp 13888  chash 14150  cprod 15715  volcvol 24733  Σ^csumge0 44287
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-inf2 9503  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054  ax-pre-sup 11055
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-se 5581  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-isom 6493  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-of 7600  df-om 7786  df-1st 7904  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-2o 8373  df-er 8574  df-map 8693  df-pm 8694  df-ixp 8762  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-fi 9273  df-sup 9304  df-inf 9305  df-oi 9372  df-dju 9763  df-card 9801  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-div 11739  df-nn 12080  df-2 12142  df-3 12143  df-n0 12340  df-z 12426  df-uz 12689  df-q 12795  df-rp 12837  df-xneg 12954  df-xadd 12955  df-xmul 12956  df-ioo 13189  df-ico 13191  df-icc 13192  df-fz 13346  df-fzo 13489  df-fl 13618  df-seq 13828  df-exp 13889  df-hash 14151  df-cj 14910  df-re 14911  df-im 14912  df-sqrt 15046  df-abs 15047  df-clim 15297  df-rlim 15298  df-sum 15498  df-prod 15716  df-rest 17231  df-topgen 17252  df-psmet 20695  df-xmet 20696  df-met 20697  df-bl 20698  df-mopn 20699  df-top 22149  df-topon 22166  df-bases 22202  df-cmp 22644  df-ovol 24734  df-vol 24735  df-sumge0 44288
This theorem is referenced by:  ovnpnfelsup  44484
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