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Theorem hoicvrrex 41091
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 11065 . . . . . . . . 9 (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
21renegcld 10495 . . . . . . . 8 (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
3 opelxpi 5182 . . . . . . . 8 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
42, 1, 3syl2anc 694 . . . . . . 7 (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
54ad2antlr 763 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
6 eqid 2651 . . . . . 6 (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
75, 6fmptd 6425 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
8 reex 10065 . . . . . . . . 9 ℝ ∈ V
98, 8xpex 7004 . . . . . . . 8 (ℝ × ℝ) ∈ V
109a1i 11 . . . . . . 7 (𝜑 → (ℝ × ℝ) ∈ V)
11 hoicvrrex.fi . . . . . . 7 (𝜑𝑋 ∈ Fin)
12 elmapg 7912 . . . . . . 7 (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
1310, 11, 12syl2anc 694 . . . . . 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 247 . . . 4 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
16 eqid 2651 . . . 4 (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
1715, 16fmptd 6425 . . 3 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
18 ovex 6718 . . . 4 ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V
19 nnex 11064 . . . 4 ℕ ∈ V
2018, 19elmap 7928 . . 3 ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)):ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
2117, 20sylibr 224 . 2 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
22 hoicvrrex.y . . . 4 (𝜑𝑌 ⊆ (ℝ ↑𝑚 𝑋))
23 eqid 2651 . . . . . 6 (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩)) = (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))
2423, 11hoicvr 41083 . . . . 5 (𝜑 → (ℝ ↑𝑚 𝑋) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
25 eqidd 2652 . . . . . . . . . . . . 13 (𝑙 = 𝑘 → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
2625cbvmptv 4783 . . . . . . . . . . . 12 (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
2726mpteq2i 4774 . . . . . . . . . . 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 6231 . . . . . . . . 9 (𝜑 → ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
3029coeq2d 5317 . . . . . . . 8 (𝜑 → ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
3130fveq1d 6231 . . . . . . 7 (𝜑 → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3231ixpeq2dv 7966 . . . . . 6 (𝜑X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3332iuneq2d 4579 . . . . 5 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3424, 33sseqtrd 3674 . . . 4 (𝜑 → (ℝ ↑𝑚 𝑋) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
3522, 34sstrd 3646 . . 3 (𝜑𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
36 simpr 476 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
3715elexd 3245 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
3816fvmpt2 6330 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
3936, 37, 38syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗) = (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
4039, 5fmpt3d 6426 . . . . . . . . . . . . 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 476 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
4341, 42fvovco 39695 . . . . . . . . . . 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 6231 . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → 𝑘𝑋)
47 opex 4962 . . . . . . . . . . . . . . . . . 18 ⟨-𝑗, 𝑗⟩ ∈ V
4847a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑋) → ⟨-𝑗, 𝑗⟩ ∈ V)
496fvmpt2 6330 . . . . . . . . . . . . . . . . 17 ((𝑘𝑋 ∧ ⟨-𝑗, 𝑗⟩ ∈ V) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5046, 48, 49syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝑋) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5150adantlr 751 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5245, 51eqtrd 2685 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘) = ⟨-𝑗, 𝑗⟩)
5352fveq2d 6233 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (1st ‘⟨-𝑗, 𝑗⟩))
54 negex 10317 . . . . . . . . . . . . . . 15 -𝑗 ∈ V
55 vex 3234 . . . . . . . . . . . . . . 15 𝑗 ∈ V
5654, 55op1st 7218 . . . . . . . . . . . . . 14 (1st ‘⟨-𝑗, 𝑗⟩) = -𝑗
5756a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘⟨-𝑗, 𝑗⟩) = -𝑗)
5853, 57eqtrd 2685 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = -𝑗)
5952fveq2d 6233 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = (2nd ‘⟨-𝑗, 𝑗⟩))
6054, 55op2nd 7219 . . . . . . . . . . . . . 14 (2nd ‘⟨-𝑗, 𝑗⟩) = 𝑗
6160a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘⟨-𝑗, 𝑗⟩) = 𝑗)
6259, 61eqtrd 2685 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘)) = 𝑗)
6358, 62oveq12d 6708 . . . . . . . . . . 11 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))[,)(2nd ‘(((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)‘𝑘))) = (-𝑗[,)𝑗))
6443, 63eqtrd 2685 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘) = (-𝑗[,)𝑗))
6564fveq2d 6233 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (vol‘(-𝑗[,)𝑗)))
66 volico 40518 . . . . . . . . . . . 12 ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
672, 1, 66syl2anc 694 . . . . . . . . . . 11 (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0))
68 nnrp 11880 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℝ+)
69 neglt 39810 . . . . . . . . . . . . 13 (𝑗 ∈ ℝ+ → -𝑗 < 𝑗)
7068, 69syl 17 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → -𝑗 < 𝑗)
7170iftrued 4127 . . . . . . . . . . 11 (𝑗 ∈ ℕ → if(-𝑗 < 𝑗, (𝑗 − -𝑗), 0) = (𝑗 − -𝑗))
721recnd 10106 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
7372, 72subnegd 10437 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (𝑗 + 𝑗))
74722timesd 11313 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (2 · 𝑗) = (𝑗 + 𝑗))
7573, 74eqtr4d 2688 . . . . . . . . . . 11 (𝑗 ∈ ℕ → (𝑗 − -𝑗) = (2 · 𝑗))
7667, 71, 753eqtrd 2689 . . . . . . . . . 10 (𝑗 ∈ ℕ → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
7776ad2antlr 763 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(-𝑗[,)𝑗)) = (2 · 𝑗))
7865, 77eqtrd 2685 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = (2 · 𝑗))
7978prodeq2dv 14697 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ∏𝑘𝑋 (2 · 𝑗))
8011adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
81 2cnd 11131 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℂ)
8272adantl 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
8381, 82mulcld 10098 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℂ)
84 fprodconst 14752 . . . . . . . 8 ((𝑋 ∈ Fin ∧ (2 · 𝑗) ∈ ℂ) → ∏𝑘𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(#‘𝑋)))
8580, 83, 84syl2anc 694 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (2 · 𝑗) = ((2 · 𝑗)↑(#‘𝑋)))
8679, 85eqtrd 2685 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)) = ((2 · 𝑗)↑(#‘𝑋)))
8786mpteq2dva 4777 . . . . 5 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(#‘𝑋))))
8887fveq2d 6233 . . . 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 3640 . . . . . . . . . 10 ℕ ⊆ ℝ+
91 ioorp 12289 . . . . . . . . . . 11 (0(,)+∞) = ℝ+
9291eqcomi 2660 . . . . . . . . . 10 + = (0(,)+∞)
9390, 92sseqtri 3670 . . . . . . . . 9 ℕ ⊆ (0(,)+∞)
94 ioossicc 12297 . . . . . . . . 9 (0(,)+∞) ⊆ (0[,]+∞)
9593, 94sstri 3645 . . . . . . . 8 ℕ ⊆ (0[,]+∞)
96 2nn 11223 . . . . . . . . . . 11 2 ∈ ℕ
9796a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 2 ∈ ℕ)
9897, 36nnmulcld 11106 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (2 · 𝑗) ∈ ℕ)
99 hashcl 13185 . . . . . . . . . . 11 (𝑋 ∈ Fin → (#‘𝑋) ∈ ℕ0)
10011, 99syl 17 . . . . . . . . . 10 (𝜑 → (#‘𝑋) ∈ ℕ0)
101100adantr 480 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (#‘𝑋) ∈ ℕ0)
102 nnexpcl 12913 . . . . . . . . 9 (((2 · 𝑗) ∈ ℕ ∧ (#‘𝑋) ∈ ℕ0) → ((2 · 𝑗)↑(#‘𝑋)) ∈ ℕ)
10398, 101, 102syl2anc 694 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((2 · 𝑗)↑(#‘𝑋)) ∈ ℕ)
10495, 103sseldi 3634 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ((2 · 𝑗)↑(#‘𝑋)) ∈ (0[,]+∞))
105 eqid 2651 . . . . . . 7 (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(#‘𝑋))) = (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(#‘𝑋)))
106104, 105fmptd 6425 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(#‘𝑋))):ℕ⟶(0[,]+∞))
10789, 106sge0xrcl 40920 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(#‘𝑋)))) ∈ ℝ*)
108 pnfxr 10130 . . . . . . 7 +∞ ∈ ℝ*
109108a1i 11 . . . . . 6 (𝜑 → +∞ ∈ ℝ*)
110 1nn 11069 . . . . . . . . . 10 1 ∈ ℕ
11195, 110sselii 3633 . . . . . . . . 9 1 ∈ (0[,]+∞)
112111a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
113 eqid 2651 . . . . . . . 8 (𝑗 ∈ ℕ ↦ 1) = (𝑗 ∈ ℕ ↦ 1)
114112, 113fmptd 6425 . . . . . . 7 (𝜑 → (𝑗 ∈ ℕ ↦ 1):ℕ⟶(0[,]+∞))
11589, 114sge0xrcl 40920 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) ∈ ℝ*)
116 nnnfi 12805 . . . . . . . . . 10 ¬ ℕ ∈ Fin
117116a1i 11 . . . . . . . . 9 (𝜑 → ¬ ℕ ∈ Fin)
118 1rp 11874 . . . . . . . . . 10 1 ∈ ℝ+
119118a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℝ+)
12089, 117, 119sge0rpcpnf 40956 . . . . . . . 8 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) = +∞)
121120eqcomd 2657 . . . . . . 7 (𝜑 → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ 1)))
122109, 121xreqled 39859 . . . . . 6 (𝜑 → +∞ ≤ (Σ^‘(𝑗 ∈ ℕ ↦ 1)))
123 nfv 1883 . . . . . . 7 𝑗𝜑
124114mptex2 6424 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 1 ∈ (0[,]+∞))
125103nnge1d 11101 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 1 ≤ ((2 · 𝑗)↑(#‘𝑋)))
126123, 89, 124, 104, 125sge0lempt 40945 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 1)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(#‘𝑋)))))
127109, 115, 107, 122, 126xrletrd 12031 . . . . 5 (𝜑 → +∞ ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(#‘𝑋)))))
128107, 127xrgepnfd 39860 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((2 · 𝑗)↑(#‘𝑋)))) = +∞)
129 eqidd 2652 . . . 4 (𝜑 → +∞ = +∞)
13088, 128, 1293eqtrrd 2690 . . 3 (𝜑 → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))))
13135, 130jca 553 . 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 2793 . . . . . . 7 𝑗𝑖
133 nfmpt1 4780 . . . . . . 7 𝑗(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
134132, 133nfeq 2805 . . . . . 6 𝑗 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
135 nfcv 2793 . . . . . . . . 9 𝑘𝑖
136 nfcv 2793 . . . . . . . . . 10 𝑘
137 nfmpt1 4780 . . . . . . . . . 10 𝑘(𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)
138136, 137nfmpt 4779 . . . . . . . . 9 𝑘(𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
139135, 138nfeq 2805 . . . . . . . 8 𝑘 𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))
140 fveq1 6228 . . . . . . . . . . 11 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → (𝑖𝑗) = ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))
141140coeq2d 5317 . . . . . . . . . 10 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → ([,) ∘ (𝑖𝑗)) = ([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗)))
142141fveq1d 6231 . . . . . . . . 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 39551 . . . . . . 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 39526 . . . . 5 (𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) → 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘))
147146sseq2d 3666 . . . 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 6233 . . . . . . . . . . 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 2986 . . . . . . . . 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 14696 . . . . . . 7 ((𝑖 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩)) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ ((𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ ⟨-𝑗, 𝑗⟩))‘𝑗))‘𝑘)))
153134, 152mpteq2da 4776 . . . . . 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 6233 . . . . 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 2661 . . . 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 747 . . 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 3340 . 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 694 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607  ifcif 4119  cop 4216   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Xcixp 7950  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  *cxr 10111   < clt 10112  cmin 10304  -cneg 10305  cn 11058  2c2 11108  0cn0 11330  +crp 11870  (,)cioo 12213  [,)cico 12215  [,]cicc 12216  cexp 12900  #chash 13157  cprod 14679  volcvol 23278  Σ^csumge0 40897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-prod 14680  df-rest 16130  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-sumge0 40898
This theorem is referenced by:  ovnpnfelsup  41094
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