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Theorem hoicvrrex 44787
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 12160 . . . . . . . . 9 (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
21renegcld 11582 . . . . . . . 8 (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
3 opelxpi 5670 . . . . . . . 8 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
42, 1, 3syl2anc 584 . . . . . . 7 (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
54ad2antlr 725 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
6 eqid 2736 . . . . . 6 (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
75, 6fmptd 7062 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
8 reex 11142 . . . . . . . . 9 ℝ ∈ V
98, 8xpex 7687 . . . . . . . 8 (ℝ × ℝ) ∈ V
109a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ∈ V)
11 hoicvrrex.fi . . . . . . 7 (𝜑𝑋 ∈ Fin)
12 elmapg 8778 . . . . . . 7 (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
1310, 11, 12syl2anc 584 . . . . . 6 (𝜑 → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
1413adantr 481 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
157, 14mpbird 256 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
16 eqid 2736 . . . 4 (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
1715, 16fmptd 7062 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
18 ovex 7390 . . . 4 ((ℝ × ℝ) ↑m 𝑋) ∈ V
19 nnex 12159 . . . 4 ℕ ∈ V
2018, 19elmap 8809 . . 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 2736 . . . . . 6 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))
2423, 11hoicvr 44779 . . . . 5 (𝜑 → (ℝ ↑m 𝑋) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
25 eqidd 2737 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
2625cbvmptv 5218 . . . . . . . . . . . 12 (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
2726mpteq2i 5210 . . . . . . . . . . 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 6844 . . . . . . . . 9 (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
3029coeq2d 5818 . . . . . . . 8 (𝜑 → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
3130fveq1d 6844 . . . . . . 7 (𝜑 → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3231ixpeq2dv 8851 . . . . . 6 (𝜑X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3332iuneq2d 4983 . . . . 5 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3424, 33sseqtrd 3984 . . . 4 (𝜑 → (ℝ ↑m 𝑋) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3522, 34sstrd 3954 . . 3 (𝜑𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
36 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
3715elexd 3465 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
3816fvmpt2 6959 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
3936, 37, 38syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
4039, 5fmpt3d 7064 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
4140adantr 481 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
42 simpr 485 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
4341, 42fvovco 43403 . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
4544adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
46 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → 𝑘𝑋)
47 opex 5421 . . . . . . . . . . . . . . . . . 18 ⟨-𝑗, 𝑗⟩ ∈ V
4847a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → ⟨-𝑗, 𝑗⟩ ∈ V)
496fvmpt2 6959 . . . . . . . . . . . . . . . . 17 ((𝑘𝑋 ∧ ⟨-𝑗, 𝑗⟩ ∈ V) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5046, 48, 49syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5150adantlr 713 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5245, 51eqtrd 2776 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5352fveq2d 6846 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (1st ‘⟨-𝑗, 𝑗⟩))
54 negex 11399 . . . . . . . . . . . . . . 15 -𝑗 ∈ V
55 vex 3449 . . . . . . . . . . . . . . 15 𝑗 ∈ V
5654, 55op1st 7929 . . . . . . . . . . . . . 14 (1st ‘⟨-𝑗, 𝑗⟩) = -𝑗
5756a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘⟨-𝑗, 𝑗⟩) = -𝑗)
5853, 57eqtrd 2776 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = -𝑗)
5952fveq2d 6846 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (2nd ‘⟨-𝑗, 𝑗⟩))
6054, 55op2nd 7930 . . . . . . . . . . . . . 14 (2nd ‘⟨-𝑗, 𝑗⟩) = 𝑗
6160a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘⟨-𝑗, 𝑗⟩) = 𝑗)
6259, 61eqtrd 2776 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = 𝑗)
6358, 62oveq12d 7375 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))[,)(2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))) = (-𝑗[,)𝑗))
6443, 63eqtrd 2776 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (-𝑗[,)𝑗))
6564fveq2d 6846 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (vol‘(-𝑗[,)𝑗)))
66 volico 44214 . . . . . . . . . . . 12 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
672, 1, 66syl2anc 584 . . . . . . . . . . 11 (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
68 nnrp 12926 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℝ+)
69 neglt 43508 . . . . . . . . . . . . 13 (𝑗 ∈ ℝ+ → -𝑗 < 𝑗)
7068, 69syl 17 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → -𝑗 < 𝑗)
7170iftrued 4494 . . . . . . . . . . 11 (𝑗 ∈ ℕ → if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0) = (𝑗 − -𝑗))
721recnd 11183 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
7372, 72subnegd 11519 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (𝑗 + 𝑗))
74722timesd 12396 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (2 · 𝑗) = (𝑗 + 𝑗))
7573, 74eqtr4d 2779 . . . . . . . . . . 11 (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (2 · 𝑗))
7667, 71, 753eqtrd 2780 . . . . . . . . . 10 (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
7776ad2antlr 725 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
7865, 77eqtrd 2776 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (2 · 𝑗))
7978prodeq2dv 15806 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ∏𝑘𝑋 (2 · 𝑗))
8011adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
81 2cnd 12231 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℂ)
8272adantl 482 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
8381, 82mulcld 11175 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℂ)
84 fprodconst 15861 . . . . . . . 8 ((𝑋 ∈ Fin ∧ (2 · 𝑗) ∈ ℂ) → ∏𝑘𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
8580, 83, 84syl2anc 584 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(♯‘𝑋)))
8679, 85eqtrd 2776 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ((2 · 𝑗)↑(♯‘𝑋)))
8786mpteq2dva 5205 . . . . 5 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))))
8887fveq2d 6846 . . . 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 3948 . . . . . . . . . 10 ℕ ⊆ ℝ+
91 ioorp 13342 . . . . . . . . . . 11 (0(,)+∞) = ℝ+
9291eqcomi 2745 . . . . . . . . . 10 + = (0(,)+∞)
9390, 92sseqtri 3980 . . . . . . . . 9 ℕ ⊆ (0(,)+∞)
94 ioossicc 13350 . . . . . . . . 9 (0(,)+∞) ⊆ (0[,]+∞)
9593, 94sstri 3953 . . . . . . . 8 ℕ ⊆ (0[,]+∞)
96 2nn 12226 . . . . . . . . . . 11 2 ∈ ℕ
9796a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℕ)
9897, 36nnmulcld 12206 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℕ)
99 hashcl 14256 . . . . . . . . . . 11 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
10011, 99syl 17 . . . . . . . . . 10 (𝜑 → (♯‘𝑋) ∈ ℕ0)
101100adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (♯‘𝑋) ∈ ℕ0)
102 nnexpcl 13980 . . . . . . . . 9 (((2 · 𝑗) ∈ ℕ ∧ (♯‘𝑋) ∈ ℕ0) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
10398, 101, 102syl2anc 584 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
10495, 103sselid 3942 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ (0[,]+∞))
105 eqid 2736 . . . . . . 7 (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))
106104, 105fmptd 7062 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))):ℕ⟶(0[,]+∞))
10789, 106sge0xrcl 44616 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) ∈ ℝ*)
108 pnfxr 11209 . . . . . . 7 +∞ ∈ ℝ*
109108a1i 11 . . . . . 6 (𝜑 → +∞ ∈ ℝ*)
110 1nn 12164 . . . . . . . . . 10 1 ∈ ℕ
11195, 110sselii 3941 . . . . . . . . 9 1 ∈ (0[,]+∞)
112111a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
113 eqid 2736 . . . . . . . 8 (𝑗 ∈ ℕ ↦ 1) = (𝑗 ∈ ℕ ↦ 1)
114112, 113fmptd 7062 . . . . . . 7 (𝜑 → (𝑗 ∈ ℕ ↦ 1):ℕ⟶(0[,]+∞))
11589, 114sge0xrcl 44616 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) ∈ ℝ*)
116 nnnfi 13871 . . . . . . . . . 10 ¬ ℕ ∈ Fin
117116a1i 11 . . . . . . . . 9 (𝜑 → ¬ ℕ ∈ Fin)
118 1rp 12919 . . . . . . . . . 10 1 ∈ ℝ+
119118a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℝ+)
12089, 117, 119sge0rpcpnf 44652 . . . . . . . 8 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) = +∞)
121120eqcomd 2742 . . . . . . 7 (𝜑 → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ 1)))
122109, 121xreqled 43554 . . . . . 6 (𝜑 → +∞ ≤ (Σ^‘(𝑗 ∈ ℕ ↦ 1)))
123 nfv 1917 . . . . . . 7 𝑗𝜑
124114fvmptelcdm 7061 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
125103nnge1d 12201 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 1 ≤ ((2 · 𝑗)↑(♯‘𝑋)))
126123, 89, 124, 104, 125sge0lempt 44641 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
127109, 115, 107, 122, 126xrletrd 13081 . . . . 5 (𝜑 → +∞ ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
128107, 127xrgepnfd 43555 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) = +∞)
129 eqidd 2737 . . . 4 (𝜑 → +∞ = +∞)
13088, 128, 1293eqtrrd 2781 . . 3 (𝜑 → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))
13135, 130jca 512 . 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 2907 . . . . . . 7 𝑗𝑖
133 nfmpt1 5213 . . . . . . 7 𝑗(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
134132, 133nfeq 2920 . . . . . 6 𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
135 nfcv 2907 . . . . . . . . 9 𝑘𝑖
136 nfcv 2907 . . . . . . . . . 10 𝑘
137 nfmpt1 5213 . . . . . . . . . 10 𝑘(𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
138136, 137nfmpt 5212 . . . . . . . . 9 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
139135, 138nfeq 2920 . . . . . . . 8 𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
140 fveq1 6841 . . . . . . . . . . 11 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑖𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
141140coeq2d 5818 . . . . . . . . . 10 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
142141fveq1d 6844 . . . . . . . . 9 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
143142adantr 481 . . . . . . . 8 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
144139, 143ixpeq2d 43266 . . . . . . 7 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
145144adantr 481 . . . . . 6 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
146134, 145iuneq2df 43244 . . . . 5 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
147146sseq2d 3976 . . . 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 6846 . . . . . . . . . . 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 3240 . . . . . . . . 9 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
151150adantr 481 . . . . . . . 8 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
152151prodeq2d 15805 . . . . . . 7 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
153134, 152mpteq2da 5203 . . . . . 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 6846 . . . . 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 2747 . . . 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 631 . . 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 3581 . 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 584 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 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  ifcif 4486  cop 4592   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  Xcixp 8835  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  *cxr 11188   < clt 11189  cmin 11385  -cneg 11386  cn 12153  2c2 12208  0cn0 12413  +crp 12915  (,)cioo 13264  [,)cico 13266  [,]cicc 13267  cexp 13967  chash 14230  cprod 15788  volcvol 24827  Σ^csumge0 44593
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-prod 15789  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-sumge0 44594
This theorem is referenced by:  ovnpnfelsup  44790
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