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Theorem hspmbl 40606
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:   𝑥,𝑘   𝐾,𝑙,𝑥,𝑦   𝑋,𝑙,𝑥,𝑦   𝑌,𝑙,𝑥,𝑦   𝜑,𝑙   𝑘,𝑙,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 40550 . . 3 (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3 eqid 2620 . . 3 dom (voln*‘𝑋) = dom (voln*‘𝑋)
4 eqid 2620 . . 3 (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋))
5 ovex 6663 . . . . . . . . 9 (-∞(,)𝑌) ∈ V
6 reex 10012 . . . . . . . . 9 ℝ ∈ V
75, 6ifex 4147 . . . . . . . 8 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
87ixpssmap 7927 . . . . . . 7 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋)
9 iftrue 4083 . . . . . . . . . . . 12 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10 ioossre 12220 . . . . . . . . . . . . 13 (-∞(,)𝑌) ⊆ ℝ
1110a1i 11 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
129, 11eqsstrd 3631 . . . . . . . . . . 11 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13 iffalse 4086 . . . . . . . . . . . 12 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14 ssid 3616 . . . . . . . . . . . . 13 ℝ ⊆ ℝ
1514a1i 11 . . . . . . . . . . . 12 𝑝 = 𝐾 → ℝ ⊆ ℝ)
1613, 15eqsstrd 3631 . . . . . . . . . . 11 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
1712, 16pm2.61i 176 . . . . . . . . . 10 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
1817rgenw 2921 . . . . . . . . 9 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19 iunss 4552 . . . . . . . . 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 7885 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ) → ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋))
226, 20, 21mp2an 707 . . . . . . 7 ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋)
238, 22sstri 3604 . . . . . 6 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑𝑚 𝑋)
247rgenw 2921 . . . . . . . 8 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25 ixpexg 7917 . . . . . . . 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 4157 . . . . . . 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 1937 . . . . . . . . 9 𝑥 = 𝑥
33 eqid 2620 . . . . . . . . 9 ℝ = ℝ
34 equequ1 1950 . . . . . . . . . . 11 (𝑘 = 𝑝 → (𝑘 = 𝑙𝑝 = 𝑙))
3534ifbid 4099 . . . . . . . . . 10 (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3635cbvixpv 7911 . . . . . . . . 9 X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
3732, 33, 36mpt2eq123i 6703 . . . . . . . 8 (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)) = (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3837mpteq2i 4732 . . . . . . 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 2642 . . . . . 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 40586 . . . . 5 (𝜑 → (𝐾(𝐻𝑋)𝑌) = X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
431ovnf 40540 . . . . . . . . 9 (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑𝑚 𝑋)⟶(0[,]+∞))
44 fdm 6038 . . . . . . . . 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 4437 . . . . . . 7 (𝜑 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
47 unipw 4909 . . . . . . . 8 𝒫 (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 𝑋)
4847a1i 11 . . . . . . 7 (𝜑 𝒫 (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 𝑋))
4946, 48eqtrd 2654 . . . . . 6 (𝜑 dom (voln*‘𝑋) = (ℝ ↑𝑚 𝑋))
5049pweqd 4154 . . . . 5 (𝜑 → 𝒫 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
5142, 50eleq12d 2693 . . . 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 473 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝜑)
54 simpr 477 . . . . 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 2701 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
571adantr 481 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → 𝑋 ∈ Fin)
58 inss1 3825 . . . . . . . . . . . . 13 (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎
5958a1i 11 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎)
60 elpwi 4159 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
6159, 60sstrd 3605 . . . . . . . . . . 11 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6261adantl 482 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6357, 62ovnxrcl 40546 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6460adantl 482 . . . . . . . . . . 11 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
6564ssdifssd 3740 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (𝑎 ∖ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6657, 65ovnxrcl 40546 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6763, 66xaddcld 12116 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ∈ ℝ*)
68 pnfge 11949 . . . . . . . 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 481 . . . . . 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 2626 . . . . . . 7 (((voln*‘𝑋)‘𝑎) = +∞ → +∞ = ((voln*‘𝑋)‘𝑎))
7372adantl 482 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → +∞ = ((voln*‘𝑋)‘𝑎))
7470, 73breqtrd 4670 . . . . 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 473 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → (𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)))
7657, 64ovncl 40544 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
7776adantr 481 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞))
78 neqne 2799 . . . . . . . 8 (¬ ((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) ≠ +∞)
7978adantl 482 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ≠ +∞)
80 ge0xrre 39561 . . . . . . 7 ((((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln*‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8177, 79, 80syl2anc 692 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8257adantr 481 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑋 ∈ Fin)
8340ad2antrr 761 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝐾𝑋)
8441ad2antrr 761 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑌 ∈ ℝ)
85 simpr 477 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8664adantr 481 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
87 sseq1 3618 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝) ↔ 𝑏 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
8887rabbidv 3184 . . . . . . . 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 4741 . . . . . . 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 473 . . . . . . . . . . . 12 ((𝑖 = 𝑝𝑋) → 𝑖 = )
9190coeq2d 5273 . . . . . . . . . . 11 ((𝑖 = 𝑝𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ))
9291fveq1d 6180 . . . . . . . . . 10 ((𝑖 = 𝑝𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ )‘𝑝))
9392fveq2d 6182 . . . . . . . . 9 ((𝑖 = 𝑝𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ )‘𝑝)))
9493prodeq2dv 14634 . . . . . . . 8 (𝑖 = → ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
9594cbvmptv 4741 . . . . . . 7 (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
96 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑝 → (([,) ∘ (𝑚𝑖))‘𝑛) = (([,) ∘ (𝑚𝑖))‘𝑝))
9796cbvixpv 7911 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝)
9897a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝))
99 fveq1 6177 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = → (𝑚𝑖) = (𝑖))
10099coeq2d 5273 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = → ([,) ∘ (𝑚𝑖)) = ([,) ∘ (𝑖)))
101100fveq1d 6180 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = → (([,) ∘ (𝑚𝑖))‘𝑝) = (([,) ∘ (𝑖))‘𝑝))
102101ixpeq2dv 7909 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
10398, 102eqtrd 2654 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
104103adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖 ∈ ℕ) → X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
105104iuneq2dv 4533 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
106105sseq2d 3625 . . . . . . . . . . . . . . . . . 18 (𝑚 = → (𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) ↔ 𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝)))
107106cbvrabv 3194 . . . . . . . . . . . . . . . . 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 6177 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑙 → (𝑖) = (𝑙𝑖))
109108coeq2d 5273 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = 𝑙 → ([,) ∘ (𝑖)) = ([,) ∘ (𝑙𝑖)))
110109fveq1d 6180 . . . . . . . . . . . . . . . . . . . . . . 23 ( = 𝑙 → (([,) ∘ (𝑖))‘𝑝) = (([,) ∘ (𝑙𝑖))‘𝑝))
111110ixpeq2dv 7909 . . . . . . . . . . . . . . . . . . . . . 22 ( = 𝑙X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
112111adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (( = 𝑙𝑖 ∈ ℕ) → X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
113112iuneq2dv 4533 . . . . . . . . . . . . . . . . . . . 20 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
114 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗 → (𝑙𝑖) = (𝑙𝑗))
115114coeq2d 5273 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑗 → ([,) ∘ (𝑙𝑖)) = ([,) ∘ (𝑙𝑗)))
116115fveq1d 6180 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑗 → (([,) ∘ (𝑙𝑖))‘𝑝) = (([,) ∘ (𝑙𝑗))‘𝑝))
117116ixpeq2dv 7909 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑗X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
118117cbviunv 4550 . . . . . . . . . . . . . . . . . . . . 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 2654 . . . . . . . . . . . . . . . . . . 19 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
121120sseq2d 3625 . . . . . . . . . . . . . . . . . 18 ( = 𝑙 → (𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) ↔ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
122121cbvrabv 3194 . . . . . . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . . . 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 4732 . . . . . . . . . . . . . . 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 6184 . . . . . . . . . . . . 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 2685 . . . . . . . . . . . 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 6178 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑝 → (([,) ∘ 𝑖)‘𝑚) = (([,) ∘ 𝑖)‘𝑝))
130129fveq2d 6182 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
131130cbvprodv 14627 . . . . . . . . . . . . . . . . . . 19 𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
132131mpteq2i 4732 . . . . . . . . . . . . . . . . . 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 6178 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑡𝑚) = (𝑡𝑗))
135133, 134fveq12d 6184 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)) = ((𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))
136135cbvmptv 4741 . . . . . . . . . . . . . . 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 6182 . . . . . . . . . . . . 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 6178 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → ((voln*‘𝑋)‘𝑐) = ((voln*‘𝑋)‘𝑏))
140139oveq1d 6650 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))
141138, 140breq12d 4657 . . . . . . . . . . . 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 746 . . . . . . . . . . 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 3181 . . . . . . . . . 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 4736 . . . . . . . . 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 2621 . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . 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 6643 . . . . . . . . . . . . . 14 (𝑠 = 𝑟 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))
148147breq2d 4656 . . . . . . . . . . . . 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 746 . . . . . . . . . . . 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 3181 . . . . . . . . . . 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 4741 . . . . . . . . . 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 2654 . . . . . . . 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 4741 . . . . . . 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 6178 . . . . . . . . . 10 (𝑚 = 𝑝 → ((𝑡𝑗)‘𝑚) = ((𝑡𝑗)‘𝑝))
156155fveq2d 6182 . . . . . . . . 9 (𝑚 = 𝑝 → (1st ‘((𝑡𝑗)‘𝑚)) = (1st ‘((𝑡𝑗)‘𝑝)))
157156cbvmptv 4741 . . . . . . . 8 (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝)))
158157mpteq2i 4732 . . . . . . 7 (𝑗 ∈ ℕ ↦ (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝))))
159 fveq2 6178 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑡𝑖) = (𝑡𝑗))
160159fveq1d 6180 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑡𝑖)‘𝑚) = ((𝑡𝑗)‘𝑚))
161160fveq2d 6182 . . . . . . . . . 10 (𝑖 = 𝑗 → (2nd ‘((𝑡𝑖)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑚)))
162161mpteq2dv 4736 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))))
163155fveq2d 6182 . . . . . . . . . . 11 (𝑚 = 𝑝 → (2nd ‘((𝑡𝑗)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑝)))
164163cbvmptv 4741 . . . . . . . . . 10 (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝)))
165164a1i 11 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
166162, 165eqtrd 2654 . . . . . . . 8 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
167166cbvmptv 4741 . . . . . . 7 (𝑖 ∈ ℕ ↦ (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
16839, 82, 83, 84, 85, 86, 89, 95, 154, 158, 167hspmbllem3 40605 . . . . . 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 692 . . . . 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 831 . . . 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 692 . . 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 40509 . 2 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
1731dmvon 40583 . . 3 (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
174173eqcomd 2626 . 2 (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
175172, 174eleqtrd 2701 1 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ dom (voln‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  {crab 2913  Vcvv 3195  cdif 3564  cin 3566  wss 3567  ifcif 4077  𝒫 cpw 4149   cuni 4427   ciun 4511   class class class wbr 4644  cmpt 4720   × cxp 5102  dom cdm 5104  ccom 5108  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  𝑚 cmap 7842  Xcixp 7893  Fincfn 7940  cr 9920  0cc0 9921  +∞cpnf 10056  -∞cmnf 10057  *cxr 10058  cle 10060  cn 11005  +crp 11817   +𝑒 cxad 11929  (,)cioo 12160  [,)cico 12162  [,]cicc 12163  cprod 14616  volcvol 23213  Σ^csumge0 40342  CaraGenccaragen 40468  voln*covoln 40513  volncvoln 40515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cc 9242  ax-ac2 9270  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999  ax-addf 10000  ax-mulf 10001
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-disj 4612  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-tpos 7337  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fi 8302  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-acn 8753  df-ac 8924  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-ioo 12164  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-fl 12576  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-rlim 14201  df-sum 14398  df-prod 14617  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-starv 15937  df-tset 15941  df-ple 15942  df-ds 15945  df-unif 15946  df-rest 16064  df-0g 16083  df-topgen 16085  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-subg 17572  df-cmn 18176  df-abl 18177  df-mgp 18471  df-ur 18483  df-ring 18530  df-cring 18531  df-oppr 18604  df-dvdsr 18622  df-unit 18623  df-invr 18653  df-dvr 18664  df-drng 18730  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-cnfld 19728  df-top 20680  df-topon 20697  df-bases 20731  df-cmp 21171  df-ovol 23214  df-vol 23215  df-sumge0 40343  df-ome 40467  df-caragen 40469  df-ovoln 40514  df-voln 40516
This theorem is referenced by:  hoimbllem  40607
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