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Theorem hoicvrrex 46571
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 12273 . . . . . . . . 9 (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
21renegcld 11690 . . . . . . . 8 (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
3 opelxpi 5722 . . . . . . . 8 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
42, 1, 3syl2anc 584 . . . . . . 7 (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
54ad2antlr 727 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
6 eqid 2737 . . . . . 6 (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
75, 6fmptd 7134 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
8 reex 11246 . . . . . . . . 9 ℝ ∈ V
98, 8xpex 7773 . . . . . . . 8 (ℝ × ℝ) ∈ V
109a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ∈ V)
11 hoicvrrex.fi . . . . . . 7 (𝜑𝑋 ∈ Fin)
12 elmapg 8879 . . . . . . 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 480 . . . . 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 7134 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
18 ovex 7464 . . . 4 ((ℝ × ℝ) ↑m 𝑋) ∈ V
19 nnex 12272 . . . 4 ℕ ∈ V
2018, 19elmap 8911 . . 3 ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑m 𝑋))
2117, 20sylibr 234 . 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 46563 . . . . 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 5255 . . . . . . . . . . . 12 (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
2726mpteq2i 5247 . . . . . . . . . . 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 6908 . . . . . . . . 9 (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
3029coeq2d 5873 . . . . . . . 8 (𝜑 → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
3130fveq1d 6908 . . . . . . 7 (𝜑 → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3231ixpeq2dv 8953 . . . . . 6 (𝜑X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3332iuneq2d 5022 . . . . 5 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3424, 33sseqtrd 4020 . . . 4 (𝜑 → (ℝ ↑m 𝑋) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3522, 34sstrd 3994 . . 3 (𝜑𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
36 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
3715elexd 3504 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
3816fvmpt2 7027 . . . . . . . . . . . . . . 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 7136 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
4140adantr 480 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗):𝑋⟶(ℝ × ℝ))
42 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
4341, 42fvovco 45198 . . . . . . . . . . 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 6908 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
4544adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘))
46 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → 𝑘𝑋)
47 opex 5469 . . . . . . . . . . . . . . . . . 18 ⟨-𝑗, 𝑗⟩ ∈ V
4847a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → ⟨-𝑗, 𝑗⟩ ∈ V)
496fvmpt2 7027 . . . . . . . . . . . . . . . . 17 ((𝑘𝑋 ∧ ⟨-𝑗, 𝑗⟩ ∈ V) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5046, 48, 49syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5150adantlr 715 . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (1st ‘⟨-𝑗, 𝑗⟩))
54 negex 11506 . . . . . . . . . . . . . . 15 -𝑗 ∈ V
55 vex 3484 . . . . . . . . . . . . . . 15 𝑗 ∈ V
5654, 55op1st 8022 . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (2nd ‘⟨-𝑗, 𝑗⟩))
6054, 55op2nd 8023 . . . . . . . . . . . . . 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 7449 . . . . . . . . . . 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 6910 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (vol‘(-𝑗[,)𝑗)))
66 volico 45998 . . . . . . . . . . . 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 13046 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℝ+)
69 neglt 45296 . . . . . . . . . . . . 13 (𝑗 ∈ ℝ+ → -𝑗 < 𝑗)
7068, 69syl 17 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → -𝑗 < 𝑗)
7170iftrued 4533 . . . . . . . . . . 11 (𝑗 ∈ ℕ → if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0) = (𝑗 − -𝑗))
721recnd 11289 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
7372, 72subnegd 11627 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (𝑗 + 𝑗))
74722timesd 12509 . . . . . . . . . . . 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 727 . . . . . . . . 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 15958 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ∏𝑘𝑋 (2 · 𝑗))
8011adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
81 2cnd 12344 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℂ)
8272adantl 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
8381, 82mulcld 11281 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℂ)
84 fprodconst 16014 . . . . . . . 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 2777 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ((2 · 𝑗)↑(♯‘𝑋)))
8786mpteq2dva 5242 . . . . 5 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))))
8887fveq2d 6910 . . . 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 3987 . . . . . . . . . 10 ℕ ⊆ ℝ+
91 ioorp 13465 . . . . . . . . . . 11 (0(,)+∞) = ℝ+
9291eqcomi 2746 . . . . . . . . . 10 + = (0(,)+∞)
9390, 92sseqtri 4032 . . . . . . . . 9 ℕ ⊆ (0(,)+∞)
94 ioossicc 13473 . . . . . . . . 9 (0(,)+∞) ⊆ (0[,]+∞)
9593, 94sstri 3993 . . . . . . . 8 ℕ ⊆ (0[,]+∞)
96 2nn 12339 . . . . . . . . . . 11 2 ∈ ℕ
9796a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℕ)
9897, 36nnmulcld 12319 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℕ)
99 hashcl 14395 . . . . . . . . . . 11 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
10011, 99syl 17 . . . . . . . . . 10 (𝜑 → (♯‘𝑋) ∈ ℕ0)
101100adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (♯‘𝑋) ∈ ℕ0)
102 nnexpcl 14115 . . . . . . . . 9 (((2 · 𝑗) ∈ ℕ ∧ (♯‘𝑋) ∈ ℕ0) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
10398, 101, 102syl2anc 584 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ ℕ)
10495, 103sselid 3981 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((2 · 𝑗)↑(♯‘𝑋)) ∈ (0[,]+∞))
105 eqid 2737 . . . . . . 7 (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))
106104, 105fmptd 7134 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋))):ℕ⟶(0[,]+∞))
10789, 106sge0xrcl 46400 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))) ∈ ℝ*)
108 pnfxr 11315 . . . . . . 7 +∞ ∈ ℝ*
109108a1i 11 . . . . . 6 (𝜑 → +∞ ∈ ℝ*)
110 1nn 12277 . . . . . . . . . 10 1 ∈ ℕ
11195, 110sselii 3980 . . . . . . . . 9 1 ∈ (0[,]+∞)
112111a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
113 eqid 2737 . . . . . . . 8 (𝑗 ∈ ℕ ↦ 1) = (𝑗 ∈ ℕ ↦ 1)
114112, 113fmptd 7134 . . . . . . 7 (𝜑 → (𝑗 ∈ ℕ ↦ 1):ℕ⟶(0[,]+∞))
11589, 114sge0xrcl 46400 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) ∈ ℝ*)
116 nnnfi 14007 . . . . . . . . . 10 ¬ ℕ ∈ Fin
117116a1i 11 . . . . . . . . 9 (𝜑 → ¬ ℕ ∈ Fin)
118 1rp 13038 . . . . . . . . . 10 1 ∈ ℝ+
119118a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℝ+)
12089, 117, 119sge0rpcpnf 46436 . . . . . . . 8 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) = +∞)
121120eqcomd 2743 . . . . . . 7 (𝜑 → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ 1)))
122109, 121xreqled 45341 . . . . . 6 (𝜑 → +∞ ≤ (Σ^‘(𝑗 ∈ ℕ ↦ 1)))
123 nfv 1914 . . . . . . 7 𝑗𝜑
124114fvmptelcdm 7133 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
125103nnge1d 12314 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 1 ≤ ((2 · 𝑗)↑(♯‘𝑋)))
126123, 89, 124, 104, 125sge0lempt 46425 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
127109, 115, 107, 122, 126xrletrd 13204 . . . . 5 (𝜑 → +∞ ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(♯‘𝑋)))))
128107, 127xrgepnfd 45342 . . . 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 511 . 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 5250 . . . . . . 7 𝑗(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
134132, 133nfeq 2919 . . . . . 6 𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
135 nfcv 2905 . . . . . . . . 9 𝑘𝑖
136 nfcv 2905 . . . . . . . . . 10 𝑘
137 nfmpt1 5250 . . . . . . . . . 10 𝑘(𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
138136, 137nfmpt 5249 . . . . . . . . 9 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
139135, 138nfeq 2919 . . . . . . . 8 𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
140 fveq1 6905 . . . . . . . . . . 11 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑖𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
141140coeq2d 5873 . . . . . . . . . 10 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
142141fveq1d 6908 . . . . . . . . 9 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
143142adantr 480 . . . . . . . 8 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
144139, 143ixpeq2d 45073 . . . . . . 7 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
145144adantr 480 . . . . . 6 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
146134, 145iuneq2df 45052 . . . . 5 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
147146sseq2d 4016 . . . 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 6910 . . . . . . . . . . 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 3257 . . . . . . . . 9 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
151150adantr 480 . . . . . . . 8 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
152151prodeq2d 15957 . . . . . . 7 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
153134, 152mpteq2da 5240 . . . . . 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 6910 . . . . 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 3622 . 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 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  ifcif 4525  cop 4632   ciun 4991   class class class wbr 5143  cmpt 5225   × cxp 5683  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  m cmap 8866  Xcixp 8937  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  +∞cpnf 11292  *cxr 11294   < clt 11295  cmin 11492  -cneg 11493  cn 12266  2c2 12321  0cn0 12526  +crp 13034  (,)cioo 13387  [,)cico 13389  [,]cicc 13390  cexp 14102  chash 14369  cprod 15939  volcvol 25498  Σ^csumge0 46377
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-prod 15940  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500  df-sumge0 46378
This theorem is referenced by:  ovnpnfelsup  46574
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