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Theorem hspmbl 44860
Description: Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
hspmbl.1 𝐻 = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))
hspmbl.x (𝜑𝑋 ∈ Fin)
hspmbl.i (𝜑𝐾𝑋)
hspmbl.y (𝜑𝑌 ∈ ℝ)
Assertion
Ref Expression
hspmbl (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ dom (voln‘𝑋))
Distinct variable groups:   𝐾,𝑙,𝑥,𝑦   𝑋,𝑙,𝑥,𝑦   𝑌,𝑙,𝑥,𝑦   𝜑,𝑙   𝑘,𝑙,𝑥,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑘)   𝐻(𝑥,𝑦,𝑘,𝑙)   𝐾(𝑘)   𝑋(𝑘)   𝑌(𝑘)

Proof of Theorem hspmbl
Dummy variables 𝑎 𝑗 𝑝 𝑡 𝑏 𝑐 𝑟 𝑠 𝑖 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hspmbl.x . . . 4 (𝜑𝑋 ∈ Fin)
21ovnome 44804 . . 3 (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3 eqid 2736 . . 3 dom (voln*‘𝑋) = dom (voln*‘𝑋)
4 eqid 2736 . . 3 (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋))
5 ovex 7390 . . . . . . . . 9 (-∞(,)𝑌) ∈ V
6 reex 11142 . . . . . . . . 9 ℝ ∈ V
75, 6ifex 4536 . . . . . . . 8 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
87ixpssmap 8870 . . . . . . 7 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋)
9 iftrue 4492 . . . . . . . . . . . 12 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10 ioossre 13325 . . . . . . . . . . . . 13 (-∞(,)𝑌) ⊆ ℝ
1110a1i 11 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
129, 11eqsstrd 3982 . . . . . . . . . . 11 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13 iffalse 4495 . . . . . . . . . . . 12 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14 ssid 3966 . . . . . . . . . . . . 13 ℝ ⊆ ℝ
1514a1i 11 . . . . . . . . . . . 12 𝑝 = 𝐾 → ℝ ⊆ ℝ)
1613, 15eqsstrd 3982 . . . . . . . . . . 11 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
1712, 16pm2.61i 182 . . . . . . . . . 10 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
1817rgenw 3068 . . . . . . . . 9 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19 iunss 5005 . . . . . . . . 9 ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ ↔ ∀𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
2018, 19mpbir 230 . . . . . . . 8 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
21 mapss 8827 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ) → ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋))
226, 20, 21mp2an 690 . . . . . . 7 ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋)
238, 22sstri 3953 . . . . . 6 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑m 𝑋)
247rgenw 3068 . . . . . . . 8 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25 ixpexg 8860 . . . . . . . 8 (∀𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V → X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V)
2624, 25ax-mp 5 . . . . . . 7 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
27 elpwg 4563 . . . . . . 7 (X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V → (X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋) ↔ X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑m 𝑋)))
2826, 27ax-mp 5 . . . . . 6 (X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋) ↔ X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑m 𝑋))
2923, 28mpbir 230 . . . . 5 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋)
3029a1i 11 . . . 4 (𝜑X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋))
31 hspmbl.1 . . . . . . 7 𝐻 = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))
32 equid 2015 . . . . . . . . 9 𝑥 = 𝑥
33 eqid 2736 . . . . . . . . 9 ℝ = ℝ
34 equequ1 2028 . . . . . . . . . . 11 (𝑘 = 𝑝 → (𝑘 = 𝑙𝑝 = 𝑙))
3534ifbid 4509 . . . . . . . . . 10 (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3635cbvixpv 8853 . . . . . . . . 9 X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
3732, 33, 36mpoeq123i 7433 . . . . . . . 8 (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)) = (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3837mpteq2i 5210 . . . . . . 7 (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ))) = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)))
3931, 38eqtri 2764 . . . . . 6 𝐻 = (𝑥 ∈ Fin ↦ (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)))
40 hspmbl.i . . . . . 6 (𝜑𝐾𝑋)
41 hspmbl.y . . . . . 6 (𝜑𝑌 ∈ ℝ)
4239, 1, 40, 41hspval 44840 . . . . 5 (𝜑 → (𝐾(𝐻𝑋)𝑌) = X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
431ovnf 44794 . . . . . . . . 9 (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑m 𝑋)⟶(0[,]+∞))
4443fdmd 6679 . . . . . . . 8 (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
4544unieqd 4879 . . . . . . 7 (𝜑 dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
46 unipw 5407 . . . . . . . 8 𝒫 (ℝ ↑m 𝑋) = (ℝ ↑m 𝑋)
4746a1i 11 . . . . . . 7 (𝜑 𝒫 (ℝ ↑m 𝑋) = (ℝ ↑m 𝑋))
4845, 47eqtrd 2776 . . . . . 6 (𝜑 dom (voln*‘𝑋) = (ℝ ↑m 𝑋))
4948pweqd 4577 . . . . 5 (𝜑 → 𝒫 dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
5042, 49eleq12d 2832 . . . 4 (𝜑 → ((𝐾(𝐻𝑋)𝑌) ∈ 𝒫 dom (voln*‘𝑋) ↔ X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑m 𝑋)))
5130, 50mpbird 256 . . 3 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ 𝒫 dom (voln*‘𝑋))
52 simpl 483 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝜑)
53 simpr 485 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 dom (voln*‘𝑋))
5452, 49syl 17 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝒫 dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋))
5553, 54eleqtrd 2840 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑m 𝑋))
561adantr 481 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → 𝑋 ∈ Fin)
57 inss1 4188 . . . . . . . . . . . . 13 (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎
5857a1i 11 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎)
59 elpwi 4567 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → 𝑎 ⊆ (ℝ ↑m 𝑋))
6058, 59sstrd 3954 . . . . . . . . . . 11 (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
6160adantl 482 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
6256, 61ovnxrcl 44800 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6359adantl 482 . . . . . . . . . . 11 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → 𝑎 ⊆ (ℝ ↑m 𝑋))
6463ssdifssd 4102 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (𝑎 ∖ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑m 𝑋))
6556, 64ovnxrcl 44800 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6662, 65xaddcld 13220 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ∈ ℝ*)
67 pnfge 13051 . . . . . . . 8 ((((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ∈ ℝ* → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ +∞)
6866, 67syl 17 . . . . . . 7 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ +∞)
6968adantr 481 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ +∞)
70 id 22 . . . . . . . 8 (((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) = +∞)
7170eqcomd 2742 . . . . . . 7 (((voln*‘𝑋)‘𝑎) = +∞ → +∞ = ((voln*‘𝑋)‘𝑎))
7271adantl 482 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → +∞ = ((voln*‘𝑋)‘𝑎))
7369, 72breqtrd 5131 . . . . 5 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
74 simpl 483 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)))
7556, 63ovncl 44798 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
7675adantr 481 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
77 neqne 2951 . . . . . . . 8 (¬ ((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) ≠ +∞)
7877adantl 482 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ≠ +∞)
79 ge0xrre 43759 . . . . . . 7 ((((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln*‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8076, 78, 79syl2anc 584 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8156adantr 481 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑋 ∈ Fin)
8240ad2antrr 724 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝐾𝑋)
8341ad2antrr 724 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑌 ∈ ℝ)
84 simpr 485 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8563adantr 481 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑎 ⊆ (ℝ ↑m 𝑋))
86 sseq1 3969 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝) ↔ 𝑏 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
8786rabbidv 3415 . . . . . . . 8 (𝑎 = 𝑏 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})
8887cbvmptv 5218 . . . . . . 7 (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)}) = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})
89 simpl 483 . . . . . . . . . . . 12 ((𝑖 = 𝑝𝑋) → 𝑖 = )
9089coeq2d 5818 . . . . . . . . . . 11 ((𝑖 = 𝑝𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ))
9190fveq1d 6844 . . . . . . . . . 10 ((𝑖 = 𝑝𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ )‘𝑝))
9291fveq2d 6846 . . . . . . . . 9 ((𝑖 = 𝑝𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ )‘𝑝)))
9392prodeq2dv 15806 . . . . . . . 8 (𝑖 = → ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
9493cbvmptv 5218 . . . . . . 7 (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = ( ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
95 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑝 → (([,) ∘ (𝑚𝑖))‘𝑛) = (([,) ∘ (𝑚𝑖))‘𝑝))
9695cbvixpv 8853 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝)
9796a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝))
98 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = → (𝑚𝑖) = (𝑖))
9998coeq2d 5818 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = → ([,) ∘ (𝑚𝑖)) = ([,) ∘ (𝑖)))
10099fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = → (([,) ∘ (𝑚𝑖))‘𝑝) = (([,) ∘ (𝑖))‘𝑝))
101100ixpeq2dv 8851 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
10297, 101eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
103102adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖 ∈ ℕ) → X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
104103iuneq2dv 4978 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
105104sseq2d 3976 . . . . . . . . . . . . . . . . . 18 (𝑚 = → (𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) ↔ 𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝)))
106105cbvrabv 3417 . . . . . . . . . . . . . . . . 17 {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)} = { ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝)}
107 fveq1 6841 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑙 → (𝑖) = (𝑙𝑖))
108107coeq2d 5818 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = 𝑙 → ([,) ∘ (𝑖)) = ([,) ∘ (𝑙𝑖)))
109108fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . 23 ( = 𝑙 → (([,) ∘ (𝑖))‘𝑝) = (([,) ∘ (𝑙𝑖))‘𝑝))
110109ixpeq2dv 8851 . . . . . . . . . . . . . . . . . . . . . 22 ( = 𝑙X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
111110adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (( = 𝑙𝑖 ∈ ℕ) → X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
112111iuneq2dv 4978 . . . . . . . . . . . . . . . . . . . 20 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
113 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗 → (𝑙𝑖) = (𝑙𝑗))
114113coeq2d 5818 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑗 → ([,) ∘ (𝑙𝑖)) = ([,) ∘ (𝑙𝑗)))
115114fveq1d 6844 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑗 → (([,) ∘ (𝑙𝑖))‘𝑝) = (([,) ∘ (𝑙𝑗))‘𝑝))
116115ixpeq2dv 8851 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑗X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
117116cbviunv 5000 . . . . . . . . . . . . . . . . . . . . 21 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)
118117a1i 11 . . . . . . . . . . . . . . . . . . . 20 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
119112, 118eqtrd 2776 . . . . . . . . . . . . . . . . . . 19 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
120119sseq2d 3976 . . . . . . . . . . . . . . . . . 18 ( = 𝑙 → (𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) ↔ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
121120cbvrabv 3417 . . . . . . . . . . . . . . . . 17 { ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)}
122106, 121eqtri 2764 . . . . . . . . . . . . . . . 16 {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)}
123122mpteq2i 5210 . . . . . . . . . . . . . . 15 (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)}) = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})
124123a1i 11 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)}) = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)}))
125 id 22 . . . . . . . . . . . . . 14 (𝑐 = 𝑏𝑐 = 𝑏)
126124, 125fveq12d 6849 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)})‘𝑐) = ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏))
127126eleq2d 2823 . . . . . . . . . . . 12 (𝑐 = 𝑏 → (𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)})‘𝑐) ↔ 𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏)))
128 2fveq3 6847 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
129128cbvprodv 15799 . . . . . . . . . . . . . . . . . . 19 𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
130129mpteq2i 5210 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚))) = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))
131130a1i 11 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚))) = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))))
132 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑡𝑚) = (𝑡𝑗))
133131, 132fveq12d 6849 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)) = ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))
134133cbvmptv 5218 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚))) = (𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))
135134a1i 11 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → (𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚))) = (𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗))))
136135fveq2d 6846 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))))
137 fveq2 6842 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → ((voln*‘𝑋)‘𝑐) = ((voln*‘𝑋)‘𝑏))
138137oveq1d 7372 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))
139136, 138breq12d 5118 . . . . . . . . . . . 12 (𝑐 = 𝑏 → ((Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)))
140127, 139anbi12d 631 . . . . . . . . . . 11 (𝑐 = 𝑏 → ((𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)})‘𝑐) ∧ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)) ↔ (𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))))
141140rabbidva2 3409 . . . . . . . . . 10 (𝑐 = 𝑏 → {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)})‘𝑐) ∣ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)} = {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)})
142141mpteq2dv 5207 . . . . . . . . 9 (𝑐 = 𝑏 → (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)})‘𝑐) ∣ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)}) = (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)}))
143 eqidd 2737 . . . . . . . . . . . . . 14 (𝑠 = 𝑟 → ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) = ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏))
144143eleq2d 2823 . . . . . . . . . . . . 13 (𝑠 = 𝑟 → (𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ↔ 𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏)))
145 oveq2 7365 . . . . . . . . . . . . . 14 (𝑠 = 𝑟 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))
146145breq2d 5117 . . . . . . . . . . . . 13 (𝑠 = 𝑟 → ((Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)))
147144, 146anbi12d 631 . . . . . . . . . . . 12 (𝑠 = 𝑟 → ((𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)) ↔ (𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))))
148147rabbidva2 3409 . . . . . . . . . . 11 (𝑠 = 𝑟 → {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)} = {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)})
149148cbvmptv 5218 . . . . . . . . . 10 (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)}) = (𝑟 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)})
150149a1i 11 . . . . . . . . 9 (𝑐 = 𝑏 → (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑠)}) = (𝑟 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)}))
151142, 150eqtrd 2776 . . . . . . . 8 (𝑐 = 𝑏 → (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)})‘𝑐) ∣ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)}) = (𝑟 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)}))
152151cbvmptv 5218 . . . . . . 7 (𝑐 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑠 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑚 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛)})‘𝑐) ∣ (Σ^‘(𝑚 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)))) ≤ (((voln*‘𝑋)‘𝑐) +𝑒 𝑠)})) = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑡 ∈ ((𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)})‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑟)}))
153 2fveq3 6847 . . . . . . . . 9 (𝑚 = 𝑝 → (1st ‘((𝑡𝑗)‘𝑚)) = (1st ‘((𝑡𝑗)‘𝑝)))
154153cbvmptv 5218 . . . . . . . 8 (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝)))
155154mpteq2i 5210 . . . . . . 7 (𝑗 ∈ ℕ ↦ (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝))))
156 fveq2 6842 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑡𝑖) = (𝑡𝑗))
157156fveq1d 6844 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑡𝑖)‘𝑚) = ((𝑡𝑗)‘𝑚))
158157fveq2d 6846 . . . . . . . . . 10 (𝑖 = 𝑗 → (2nd ‘((𝑡𝑖)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑚)))
159158mpteq2dv 5207 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))))
160 2fveq3 6847 . . . . . . . . . . 11 (𝑚 = 𝑝 → (2nd ‘((𝑡𝑗)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑝)))
161160cbvmptv 5218 . . . . . . . . . 10 (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝)))
162161a1i 11 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
163159, 162eqtrd 2776 . . . . . . . 8 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
164163cbvmptv 5218 . . . . . . 7 (𝑖 ∈ ℕ ↦ (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
16539, 81, 82, 83, 84, 85, 88, 94, 152, 155, 164hspmbllem3 44859 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
16674, 80, 165syl2anc 584 . . . . 5 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
16773, 166pm2.61dan 811 . . . 4 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
16852, 55, 167syl2anc 584 . . 3 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
1692, 3, 4, 51, 168caragenel2d 44763 . 2 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
1701dmvon 44837 . . 3 (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
171170eqcomd 2742 . 2 (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
172169, 171eleqtrd 2840 1 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ dom (voln‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  cin 3909  wss 3910  ifcif 4486  𝒫 cpw 4560   cuni 4865   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ccom 5637  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  m cmap 8765  Xcixp 8835  Fincfn 8883  cr 11050  0cc0 11051  +∞cpnf 11186  -∞cmnf 11187  *cxr 11188  cle 11190  cn 12153  +crp 12915   +𝑒 cxad 13031  (,)cioo 13264  [,)cico 13266  [,]cicc 13267  cprod 15788  volcvol 24827  Σ^csumge0 44593  CaraGenccaragen 44722  voln*covoln 44767  volncvoln 44769
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-cc 10371  ax-ac2 10399  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  ax-addf 11130  ax-mulf 11131
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-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  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-tpos 8157  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-acn 9878  df-ac 10052  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-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  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-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-rest 17304  df-0g 17323  df-topgen 17325  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828  df-vol 24829  df-sumge0 44594  df-ome 44721  df-caragen 44723  df-ovoln 44768  df-voln 44770
This theorem is referenced by:  hoimbllem  44861
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