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Theorem hspmbl 41357
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 41301 . . 3 (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3 eqid 2770 . . 3 dom (voln*‘𝑋) = dom (voln*‘𝑋)
4 eqid 2770 . . 3 (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋))
5 ovex 6822 . . . . . . . . 9 (-∞(,)𝑌) ∈ V
6 reex 10228 . . . . . . . . 9 ℝ ∈ V
75, 6ifex 4293 . . . . . . . 8 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
87ixpssmap 8095 . . . . . . 7 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋)
9 iftrue 4229 . . . . . . . . . . . 12 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10 ioossre 12439 . . . . . . . . . . . . 13 (-∞(,)𝑌) ⊆ ℝ
1110a1i 11 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
129, 11eqsstrd 3786 . . . . . . . . . . 11 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13 iffalse 4232 . . . . . . . . . . . 12 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14 ssid 3771 . . . . . . . . . . . . 13 ℝ ⊆ ℝ
1514a1i 11 . . . . . . . . . . . 12 𝑝 = 𝐾 → ℝ ⊆ ℝ)
1613, 15eqsstrd 3786 . . . . . . . . . . 11 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
1712, 16pm2.61i 176 . . . . . . . . . 10 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
1817rgenw 3072 . . . . . . . . 9 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19 iunss 4693 . . . . . . . . 9 ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ ↔ ∀𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
2018, 19mpbir 221 . . . . . . . 8 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
21 mapss 8053 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ) → ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋))
226, 20, 21mp2an 664 . . . . . . 7 ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋)
238, 22sstri 3759 . . . . . 6 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑𝑚 𝑋)
247rgenw 3072 . . . . . . . 8 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25 ixpexg 8085 . . . . . . . 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 4303 . . . . . . 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 221 . . . . 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 2096 . . . . . . . . 9 𝑥 = 𝑥
33 eqid 2770 . . . . . . . . 9 ℝ = ℝ
34 equequ1 2109 . . . . . . . . . . 11 (𝑘 = 𝑝 → (𝑘 = 𝑙𝑝 = 𝑙))
3534ifbid 4245 . . . . . . . . . 10 (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3635cbvixpv 8079 . . . . . . . . 9 X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
3732, 33, 36mpt2eq123i 6864 . . . . . . . 8 (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)) = (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3837mpteq2i 4873 . . . . . . 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 2792 . . . . . 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 41337 . . . . 5 (𝜑 → (𝐾(𝐻𝑋)𝑌) = X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
431ovnf 41291 . . . . . . . . 9 (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑𝑚 𝑋)⟶(0[,]+∞))
44 fdm 6191 . . . . . . . . 9 ((voln*‘𝑋):𝒫 (ℝ ↑𝑚 𝑋)⟶(0[,]+∞) → dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
4543, 44syl 17 . . . . . . . 8 (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
4645unieqd 4582 . . . . . . 7 (𝜑 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
47 unipw 5046 . . . . . . . 8 𝒫 (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 𝑋)
4847a1i 11 . . . . . . 7 (𝜑 𝒫 (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 𝑋))
4946, 48eqtrd 2804 . . . . . 6 (𝜑 dom (voln*‘𝑋) = (ℝ ↑𝑚 𝑋))
5049pweqd 4300 . . . . 5 (𝜑 → 𝒫 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
5142, 50eleq12d 2843 . . . 4 (𝜑 → ((𝐾(𝐻𝑋)𝑌) ∈ 𝒫 dom (voln*‘𝑋) ↔ X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑𝑚 𝑋)))
5230, 51mpbird 247 . . 3 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ 𝒫 dom (voln*‘𝑋))
53 simpl 468 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝜑)
54 simpr 471 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 dom (voln*‘𝑋))
5553, 50syl 17 . . . . 5 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝒫 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
5654, 55eleqtrd 2851 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
571adantr 466 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → 𝑋 ∈ Fin)
58 inss1 3979 . . . . . . . . . . . . 13 (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎
5958a1i 11 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎)
60 elpwi 4305 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
6159, 60sstrd 3760 . . . . . . . . . . 11 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6261adantl 467 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6357, 62ovnxrcl 41297 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6460adantl 467 . . . . . . . . . . 11 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
6564ssdifssd 3897 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (𝑎 ∖ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6657, 65ovnxrcl 41297 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6763, 66xaddcld 12335 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ∈ ℝ*)
68 pnfge 12168 . . . . . . . 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)))) ≤ +∞)
6967, 68syl 17 . . . . . . 7 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ +∞)
7069adantr 466 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ +∞)
71 id 22 . . . . . . . 8 (((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) = +∞)
7271eqcomd 2776 . . . . . . 7 (((voln*‘𝑋)‘𝑎) = +∞ → +∞ = ((voln*‘𝑋)‘𝑎))
7372adantl 467 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → +∞ = ((voln*‘𝑋)‘𝑎))
7470, 73breqtrd 4810 . . . . 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))
75 simpl 468 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)))
7657, 64ovncl 41295 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
7776adantr 466 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
78 neqne 2950 . . . . . . . 8 (¬ ((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) ≠ +∞)
7978adantl 467 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ≠ +∞)
80 ge0xrre 40270 . . . . . . 7 ((((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln*‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8177, 79, 80syl2anc 565 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8257adantr 466 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑋 ∈ Fin)
8340ad2antrr 697 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝐾𝑋)
8441ad2antrr 697 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑌 ∈ ℝ)
85 simpr 471 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8664adantr 466 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
87 sseq1 3773 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝) ↔ 𝑏 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
8887rabbidv 3338 . . . . . . . 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)})
8988cbvmptv 4882 . . . . . . 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)})
90 simpl 468 . . . . . . . . . . . 12 ((𝑖 = 𝑝𝑋) → 𝑖 = )
9190coeq2d 5423 . . . . . . . . . . 11 ((𝑖 = 𝑝𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ))
9291fveq1d 6334 . . . . . . . . . 10 ((𝑖 = 𝑝𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ )‘𝑝))
9392fveq2d 6336 . . . . . . . . 9 ((𝑖 = 𝑝𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ )‘𝑝)))
9493prodeq2dv 14859 . . . . . . . 8 (𝑖 = → ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
9594cbvmptv 4882 . . . . . . 7 (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
96 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑝 → (([,) ∘ (𝑚𝑖))‘𝑛) = (([,) ∘ (𝑚𝑖))‘𝑝))
9796cbvixpv 8079 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝)
9897a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝))
99 fveq1 6331 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = → (𝑚𝑖) = (𝑖))
10099coeq2d 5423 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = → ([,) ∘ (𝑚𝑖)) = ([,) ∘ (𝑖)))
101100fveq1d 6334 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = → (([,) ∘ (𝑚𝑖))‘𝑝) = (([,) ∘ (𝑖))‘𝑝))
102101ixpeq2dv 8077 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
10398, 102eqtrd 2804 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
104103adantr 466 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖 ∈ ℕ) → X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
105104iuneq2dv 4674 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
106105sseq2d 3780 . . . . . . . . . . . . . . . . . 18 (𝑚 = → (𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) ↔ 𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝)))
107106cbvrabv 3348 . . . . . . . . . . . . . . . . 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)}
108 fveq1 6331 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑙 → (𝑖) = (𝑙𝑖))
109108coeq2d 5423 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = 𝑙 → ([,) ∘ (𝑖)) = ([,) ∘ (𝑙𝑖)))
110109fveq1d 6334 . . . . . . . . . . . . . . . . . . . . . . 23 ( = 𝑙 → (([,) ∘ (𝑖))‘𝑝) = (([,) ∘ (𝑙𝑖))‘𝑝))
111110ixpeq2dv 8077 . . . . . . . . . . . . . . . . . . . . . 22 ( = 𝑙X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
112111adantr 466 . . . . . . . . . . . . . . . . . . . . 21 (( = 𝑙𝑖 ∈ ℕ) → X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
113112iuneq2dv 4674 . . . . . . . . . . . . . . . . . . . 20 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
114 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗 → (𝑙𝑖) = (𝑙𝑗))
115114coeq2d 5423 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑗 → ([,) ∘ (𝑙𝑖)) = ([,) ∘ (𝑙𝑗)))
116115fveq1d 6334 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑗 → (([,) ∘ (𝑙𝑖))‘𝑝) = (([,) ∘ (𝑙𝑗))‘𝑝))
117116ixpeq2dv 8077 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑗X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
118117cbviunv 4691 . . . . . . . . . . . . . . . . . . . . 21 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)
119118a1i 11 . . . . . . . . . . . . . . . . . . . 20 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
120113, 119eqtrd 2804 . . . . . . . . . . . . . . . . . . 19 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
121120sseq2d 3780 . . . . . . . . . . . . . . . . . 18 ( = 𝑙 → (𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) ↔ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
122121cbvrabv 3348 . . . . . . . . . . . . . . . . 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)}
123107, 122eqtri 2792 . . . . . . . . . . . . . . . 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)}
124123mpteq2i 4873 . . . . . . . . . . . . . . 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)})
125124a1i 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)}))
126 id 22 . . . . . . . . . . . . . 14 (𝑐 = 𝑏𝑐 = 𝑏)
127125, 126fveq12d 6338 . . . . . . . . . . . . 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))
128127eleq2d 2835 . . . . . . . . . . . 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)))
129 fveq2 6332 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑝 → (([,) ∘ 𝑖)‘𝑚) = (([,) ∘ 𝑖)‘𝑝))
130129fveq2d 6336 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
131130cbvprodv 14852 . . . . . . . . . . . . . . . . . . 19 𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
132131mpteq2i 4873 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚))) = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))
133132a1i 11 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚))) = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))))
134 fveq2 6332 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑡𝑚) = (𝑡𝑗))
135133, 134fveq12d 6338 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)) = ((𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))
136135cbvmptv 4882 . . . . . . . . . . . . . . 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)))
137136a1i 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))))
138137fveq2d 6336 . . . . . . . . . . . . 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)))))
139 fveq2 6332 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → ((voln*‘𝑋)‘𝑐) = ((voln*‘𝑋)‘𝑏))
140139oveq1d 6807 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))
141138, 140breq12d 4797 . . . . . . . . . . . 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)))
142128, 141anbi12d 608 . . . . . . . . . . 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))))
143142rabbidva2 3335 . . . . . . . . . 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)})
144143mpteq2dv 4877 . . . . . . . . 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)}))
145 eqidd 2771 . . . . . . . . . . . . . 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))
146145eleq2d 2835 . . . . . . . . . . . . 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)))
147 oveq2 6800 . . . . . . . . . . . . . 14 (𝑠 = 𝑟 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))
148147breq2d 4796 . . . . . . . . . . . . 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)))
149146, 148anbi12d 608 . . . . . . . . . . . 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))))
150149rabbidva2 3335 . . . . . . . . . . 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)})
151150cbvmptv 4882 . . . . . . . . . 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)})
152151a1i 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)}))
153144, 152eqtrd 2804 . . . . . . . 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)}))
154153cbvmptv 4882 . . . . . . 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)}))
155 fveq2 6332 . . . . . . . . . 10 (𝑚 = 𝑝 → ((𝑡𝑗)‘𝑚) = ((𝑡𝑗)‘𝑝))
156155fveq2d 6336 . . . . . . . . 9 (𝑚 = 𝑝 → (1st ‘((𝑡𝑗)‘𝑚)) = (1st ‘((𝑡𝑗)‘𝑝)))
157156cbvmptv 4882 . . . . . . . 8 (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝)))
158157mpteq2i 4873 . . . . . . 7 (𝑗 ∈ ℕ ↦ (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝))))
159 fveq2 6332 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑡𝑖) = (𝑡𝑗))
160159fveq1d 6334 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑡𝑖)‘𝑚) = ((𝑡𝑗)‘𝑚))
161160fveq2d 6336 . . . . . . . . . 10 (𝑖 = 𝑗 → (2nd ‘((𝑡𝑖)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑚)))
162161mpteq2dv 4877 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))))
163155fveq2d 6336 . . . . . . . . . . 11 (𝑚 = 𝑝 → (2nd ‘((𝑡𝑗)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑝)))
164163cbvmptv 4882 . . . . . . . . . 10 (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝)))
165164a1i 11 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
166162, 165eqtrd 2804 . . . . . . . 8 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
167166cbvmptv 4882 . . . . . . 7 (𝑖 ∈ ℕ ↦ (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
16839, 82, 83, 84, 85, 86, 89, 95, 154, 158, 167hspmbllem3 41356 . . . . . 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))
16975, 81, 168syl2anc 565 . . . . 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))
17074, 169pm2.61dan 796 . . . 4 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
17153, 56, 170syl2anc 565 . . 3 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝑎))
1722, 3, 4, 52, 171caragenel2d 41260 . 2 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
1731dmvon 41334 . . 3 (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
174173eqcomd 2776 . 2 (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
175172, 174eleqtrd 2851 1 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ dom (voln‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  {crab 3064  Vcvv 3349  cdif 3718  cin 3720  wss 3721  ifcif 4223  𝒫 cpw 4295   cuni 4572   ciun 4652   class class class wbr 4784  cmpt 4861   × cxp 5247  dom cdm 5249  ccom 5253  wf 6027  cfv 6031  (class class class)co 6792  cmpt2 6794  1st c1st 7312  2nd c2nd 7313  𝑚 cmap 8008  Xcixp 8061  Fincfn 8108  cr 10136  0cc0 10137  +∞cpnf 10272  -∞cmnf 10273  *cxr 10274  cle 10276  cn 11221  +crp 12034   +𝑒 cxad 12148  (,)cioo 12379  [,)cico 12381  [,]cicc 12382  cprod 14841  volcvol 23450  Σ^csumge0 41090  CaraGenccaragen 41219  voln*covoln 41264  volncvoln 41266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cc 9458  ax-ac2 9486  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215  ax-addf 10216  ax-mulf 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-disj 4753  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-tpos 7503  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fi 8472  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-acn 8967  df-ac 9138  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-q 11991  df-rp 12035  df-xneg 12150  df-xadd 12151  df-xmul 12152  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-fl 12800  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-rlim 14427  df-sum 14624  df-prod 14842  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-starv 16163  df-tset 16167  df-ple 16168  df-ds 16171  df-unif 16172  df-rest 16290  df-0g 16309  df-topgen 16311  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-subg 17798  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-cring 18757  df-oppr 18830  df-dvdsr 18848  df-unit 18849  df-invr 18879  df-dvr 18890  df-drng 18958  df-psmet 19952  df-xmet 19953  df-met 19954  df-bl 19955  df-mopn 19956  df-cnfld 19961  df-top 20918  df-topon 20935  df-bases 20970  df-cmp 21410  df-ovol 23451  df-vol 23452  df-sumge0 41091  df-ome 41218  df-caragen 41220  df-ovoln 41265  df-voln 41267
This theorem is referenced by:  hoimbllem  41358
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