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Theorem hspmbl 41164
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 41108 . . 3 (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3 eqid 2651 . . 3 dom (voln*‘𝑋) = dom (voln*‘𝑋)
4 eqid 2651 . . 3 (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋))
5 ovex 6718 . . . . . . . . 9 (-∞(,)𝑌) ∈ V
6 reex 10065 . . . . . . . . 9 ℝ ∈ V
75, 6ifex 4189 . . . . . . . 8 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
87ixpssmap 7984 . . . . . . 7 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋)
9 iftrue 4125 . . . . . . . . . . . 12 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10 ioossre 12273 . . . . . . . . . . . . 13 (-∞(,)𝑌) ⊆ ℝ
1110a1i 11 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
129, 11eqsstrd 3672 . . . . . . . . . . 11 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13 iffalse 4128 . . . . . . . . . . . 12 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14 ssid 3657 . . . . . . . . . . . . 13 ℝ ⊆ ℝ
1514a1i 11 . . . . . . . . . . . 12 𝑝 = 𝐾 → ℝ ⊆ ℝ)
1613, 15eqsstrd 3672 . . . . . . . . . . 11 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
1712, 16pm2.61i 176 . . . . . . . . . 10 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
1817rgenw 2953 . . . . . . . . 9 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19 iunss 4593 . . . . . . . . 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 7942 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ) → ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋))
226, 20, 21mp2an 708 . . . . . . 7 ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋)
238, 22sstri 3645 . . . . . 6 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑𝑚 𝑋)
247rgenw 2953 . . . . . . . 8 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25 ixpexg 7974 . . . . . . . 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 4199 . . . . . . 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 1985 . . . . . . . . 9 𝑥 = 𝑥
33 eqid 2651 . . . . . . . . 9 ℝ = ℝ
34 equequ1 1998 . . . . . . . . . . 11 (𝑘 = 𝑝 → (𝑘 = 𝑙𝑝 = 𝑙))
3534ifbid 4141 . . . . . . . . . 10 (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3635cbvixpv 7968 . . . . . . . . 9 X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
3732, 33, 36mpt2eq123i 6760 . . . . . . . 8 (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)) = (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3837mpteq2i 4774 . . . . . . 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 2673 . . . . . 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 41144 . . . . 5 (𝜑 → (𝐾(𝐻𝑋)𝑌) = X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
431ovnf 41098 . . . . . . . . 9 (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑𝑚 𝑋)⟶(0[,]+∞))
44 fdm 6089 . . . . . . . . 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 4478 . . . . . . 7 (𝜑 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
47 unipw 4948 . . . . . . . 8 𝒫 (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 𝑋)
4847a1i 11 . . . . . . 7 (𝜑 𝒫 (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 𝑋))
4946, 48eqtrd 2685 . . . . . 6 (𝜑 dom (voln*‘𝑋) = (ℝ ↑𝑚 𝑋))
5049pweqd 4196 . . . . 5 (𝜑 → 𝒫 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
5142, 50eleq12d 2724 . . . 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 472 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝜑)
54 simpr 476 . . . . 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 2732 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
571adantr 480 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → 𝑋 ∈ Fin)
58 inss1 3866 . . . . . . . . . . . . 13 (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎
5958a1i 11 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎)
60 elpwi 4201 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
6159, 60sstrd 3646 . . . . . . . . . . 11 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6261adantl 481 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6357, 62ovnxrcl 41104 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6460adantl 481 . . . . . . . . . . 11 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
6564ssdifssd 3781 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (𝑎 ∖ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6657, 65ovnxrcl 41104 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6763, 66xaddcld 12169 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ∈ ℝ*)
68 pnfge 12002 . . . . . . . 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 480 . . . . . 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 2657 . . . . . . 7 (((voln*‘𝑋)‘𝑎) = +∞ → +∞ = ((voln*‘𝑋)‘𝑎))
7372adantl 481 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → +∞ = ((voln*‘𝑋)‘𝑎))
7470, 73breqtrd 4711 . . . . 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 472 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)))
7657, 64ovncl 41102 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
7776adantr 480 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
78 neqne 2831 . . . . . . . 8 (¬ ((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) ≠ +∞)
7978adantl 481 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ≠ +∞)
80 ge0xrre 40076 . . . . . . 7 ((((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln*‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8177, 79, 80syl2anc 694 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8257adantr 480 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑋 ∈ Fin)
8340ad2antrr 762 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝐾𝑋)
8441ad2antrr 762 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑌 ∈ ℝ)
85 simpr 476 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8664adantr 480 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
87 sseq1 3659 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝) ↔ 𝑏 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
8887rabbidv 3220 . . . . . . . 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 4783 . . . . . . 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 472 . . . . . . . . . . . 12 ((𝑖 = 𝑝𝑋) → 𝑖 = )
9190coeq2d 5317 . . . . . . . . . . 11 ((𝑖 = 𝑝𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ))
9291fveq1d 6231 . . . . . . . . . 10 ((𝑖 = 𝑝𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ )‘𝑝))
9392fveq2d 6233 . . . . . . . . 9 ((𝑖 = 𝑝𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ )‘𝑝)))
9493prodeq2dv 14697 . . . . . . . 8 (𝑖 = → ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
9594cbvmptv 4783 . . . . . . 7 (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
96 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑝 → (([,) ∘ (𝑚𝑖))‘𝑛) = (([,) ∘ (𝑚𝑖))‘𝑝))
9796cbvixpv 7968 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝)
9897a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝))
99 fveq1 6228 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = → (𝑚𝑖) = (𝑖))
10099coeq2d 5317 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = → ([,) ∘ (𝑚𝑖)) = ([,) ∘ (𝑖)))
101100fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = → (([,) ∘ (𝑚𝑖))‘𝑝) = (([,) ∘ (𝑖))‘𝑝))
102101ixpeq2dv 7966 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
10398, 102eqtrd 2685 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
104103adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖 ∈ ℕ) → X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
105104iuneq2dv 4574 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
106105sseq2d 3666 . . . . . . . . . . . . . . . . . 18 (𝑚 = → (𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) ↔ 𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝)))
107106cbvrabv 3230 . . . . . . . . . . . . . . . . 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 6228 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑙 → (𝑖) = (𝑙𝑖))
109108coeq2d 5317 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = 𝑙 → ([,) ∘ (𝑖)) = ([,) ∘ (𝑙𝑖)))
110109fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . 23 ( = 𝑙 → (([,) ∘ (𝑖))‘𝑝) = (([,) ∘ (𝑙𝑖))‘𝑝))
111110ixpeq2dv 7966 . . . . . . . . . . . . . . . . . . . . . 22 ( = 𝑙X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
112111adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (( = 𝑙𝑖 ∈ ℕ) → X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
113112iuneq2dv 4574 . . . . . . . . . . . . . . . . . . . 20 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
114 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗 → (𝑙𝑖) = (𝑙𝑗))
115114coeq2d 5317 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑗 → ([,) ∘ (𝑙𝑖)) = ([,) ∘ (𝑙𝑗)))
116115fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑗 → (([,) ∘ (𝑙𝑖))‘𝑝) = (([,) ∘ (𝑙𝑗))‘𝑝))
117116ixpeq2dv 7966 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑗X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
118117cbviunv 4591 . . . . . . . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . . . . . . 19 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
121120sseq2d 3666 . . . . . . . . . . . . . . . . . 18 ( = 𝑙 → (𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) ↔ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
122121cbvrabv 3230 . . . . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . . . 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 4774 . . . . . . . . . . . . . . 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 6235 . . . . . . . . . . . . 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 2716 . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑝 → (([,) ∘ 𝑖)‘𝑚) = (([,) ∘ 𝑖)‘𝑝))
130129fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
131130cbvprodv 14690 . . . . . . . . . . . . . . . . . . 19 𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
132131mpteq2i 4774 . . . . . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑡𝑚) = (𝑡𝑗))
135133, 134fveq12d 6235 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)) = ((𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))
136135cbvmptv 4783 . . . . . . . . . . . . . . 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 6233 . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → ((voln*‘𝑋)‘𝑐) = ((voln*‘𝑋)‘𝑏))
140139oveq1d 6705 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))
141138, 140breq12d 4698 . . . . . . . . . . . 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 747 . . . . . . . . . . 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 3217 . . . . . . . . . 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 4778 . . . . . . . . 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 2652 . . . . . . . . . . . . . 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 2716 . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . 14 (𝑠 = 𝑟 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))
148147breq2d 4697 . . . . . . . . . . . . 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 747 . . . . . . . . . . . 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 3217 . . . . . . . . . . 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 4783 . . . . . . . . . 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 2685 . . . . . . . 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 4783 . . . . . . 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 6229 . . . . . . . . . 10 (𝑚 = 𝑝 → ((𝑡𝑗)‘𝑚) = ((𝑡𝑗)‘𝑝))
156155fveq2d 6233 . . . . . . . . 9 (𝑚 = 𝑝 → (1st ‘((𝑡𝑗)‘𝑚)) = (1st ‘((𝑡𝑗)‘𝑝)))
157156cbvmptv 4783 . . . . . . . 8 (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝)))
158157mpteq2i 4774 . . . . . . 7 (𝑗 ∈ ℕ ↦ (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝))))
159 fveq2 6229 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑡𝑖) = (𝑡𝑗))
160159fveq1d 6231 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑡𝑖)‘𝑚) = ((𝑡𝑗)‘𝑚))
161160fveq2d 6233 . . . . . . . . . 10 (𝑖 = 𝑗 → (2nd ‘((𝑡𝑖)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑚)))
162161mpteq2dv 4778 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))))
163155fveq2d 6233 . . . . . . . . . . 11 (𝑚 = 𝑝 → (2nd ‘((𝑡𝑗)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑝)))
164163cbvmptv 4783 . . . . . . . . . 10 (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝)))
165164a1i 11 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
166162, 165eqtrd 2685 . . . . . . . 8 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
167166cbvmptv 4783 . . . . . . 7 (𝑖 ∈ ℕ ↦ (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
16839, 82, 83, 84, 85, 86, 89, 95, 154, 158, 167hspmbllem3 41163 . . . . . 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 694 . . . . 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 849 . . . 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 694 . . 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 41067 . 2 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
1731dmvon 41141 . . 3 (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
174173eqcomd 2657 . 2 (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
175172, 174eleqtrd 2732 1 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ dom (voln‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607  ifcif 4119  𝒫 cpw 4191   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Xcixp 7950  Fincfn 7997  cr 9973  0cc0 9974  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111  cle 10113  cn 11058  +crp 11870   +𝑒 cxad 11982  (,)cioo 12213  [,)cico 12215  [,]cicc 12216  cprod 14679  volcvol 23278  Σ^csumge0 40897  CaraGenccaragen 41026  voln*covoln 41071  volncvoln 41073
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-cc 9295  ax-ac2 9323  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  ax-addf 10053  ax-mulf 10054
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-disj 4653  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-tpos 7397  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-acn 8806  df-ac 8977  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-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  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-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-rest 16130  df-0g 16149  df-topgen 16151  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-subg 17638  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-bases 20798  df-cmp 21238  df-ovol 23279  df-vol 23280  df-sumge0 40898  df-ome 41025  df-caragen 41027  df-ovoln 41072  df-voln 41074
This theorem is referenced by:  hoimbllem  41165
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