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Theorem hspmbl 39519
 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 39463 . . 3 (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3 eqid 2610 . . 3 dom (voln*‘𝑋) = dom (voln*‘𝑋)
4 eqid 2610 . . 3 (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋))
5 ovex 6577 . . . . . . . . 9 (-∞(,)𝑌) ∈ V
6 reex 9906 . . . . . . . . 9 ℝ ∈ V
75, 6ifex 4106 . . . . . . . 8 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
87ixpssmap 7828 . . . . . . 7 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋)
9 iftrue 4042 . . . . . . . . . . . 12 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌))
10 ioossre 12106 . . . . . . . . . . . . 13 (-∞(,)𝑌) ⊆ ℝ
1110a1i 11 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ)
129, 11eqsstrd 3602 . . . . . . . . . . 11 (𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
13 iffalse 4045 . . . . . . . . . . . 12 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ)
14 ssid 3587 . . . . . . . . . . . . 13 ℝ ⊆ ℝ
1514a1i 11 . . . . . . . . . . . 12 𝑝 = 𝐾 → ℝ ⊆ ℝ)
1613, 15eqsstrd 3602 . . . . . . . . . . 11 𝑝 = 𝐾 → if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
1712, 16pm2.61i 175 . . . . . . . . . 10 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
1817rgenw 2908 . . . . . . . . 9 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
19 iunss 4497 . . . . . . . . 9 ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ ↔ ∀𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ)
2018, 19mpbir 220 . . . . . . . 8 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ
21 mapss 7786 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ) → ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋))
226, 20, 21mp2an 704 . . . . . . 7 ( 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋)
238, 22sstri 3577 . . . . . 6 X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ (ℝ ↑𝑚 𝑋)
247rgenw 2908 . . . . . . . 8 𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ V
25 ixpexg 7818 . . . . . . . 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 4116 . . . . . . 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 220 . . . . 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 1926 . . . . . . . . 9 𝑥 = 𝑥
33 eqid 2610 . . . . . . . . 9 ℝ = ℝ
34 equequ1 1939 . . . . . . . . . . 11 (𝑘 = 𝑝 → (𝑘 = 𝑙𝑝 = 𝑙))
3534ifbid 4058 . . . . . . . . . 10 (𝑘 = 𝑝 → if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3635cbvixpv 7812 . . . . . . . . 9 X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ) = X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ)
3732, 33, 36mpt2eq123i 6616 . . . . . . . 8 (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)) = (𝑙𝑥, 𝑦 ∈ ℝ ↦ X𝑝𝑥 if(𝑝 = 𝑙, (-∞(,)𝑦), ℝ))
3837mpteq2i 4669 . . . . . . 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 2632 . . . . . 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 39499 . . . . 5 (𝜑 → (𝐾(𝐻𝑋)𝑌) = X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ))
431ovnf 39453 . . . . . . . . 9 (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑𝑚 𝑋)⟶(0[,]+∞))
44 fdm 5964 . . . . . . . . 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 4382 . . . . . . 7 (𝜑 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
47 unipw 4845 . . . . . . . 8 𝒫 (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 𝑋)
4847a1i 11 . . . . . . 7 (𝜑 𝒫 (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 𝑋))
4946, 48eqtrd 2644 . . . . . 6 (𝜑 dom (voln*‘𝑋) = (ℝ ↑𝑚 𝑋))
5049pweqd 4113 . . . . 5 (𝜑 → 𝒫 dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
5142, 50eleq12d 2682 . . . 4 (𝜑 → ((𝐾(𝐻𝑋)𝑌) ∈ 𝒫 dom (voln*‘𝑋) ↔ X𝑝𝑋 if(𝑝 = 𝐾, (-∞(,)𝑌), ℝ) ∈ 𝒫 (ℝ ↑𝑚 𝑋)))
5230, 51mpbird 246 . . 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 2690 . . . 4 ((𝜑𝑎 ∈ 𝒫 dom (voln*‘𝑋)) → 𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
571adantr 480 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → 𝑋 ∈ Fin)
58 inss1 3795 . . . . . . . . . . . . 13 (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎
5958a1i 11 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝑎 ∩ (𝐾(𝐻𝑋)𝑌)) ⊆ 𝑎)
60 elpwi 4117 . . . . . . . . . . . 12 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) → 𝑎 ⊆ (ℝ ↑𝑚 𝑋))
6159, 60sstrd 3578 . . . . . . . . . . 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 39459 . . . . . . . . 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 3710 . . . . . . . . . 10 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (𝑎 ∖ (𝐾(𝐻𝑋)𝑌)) ⊆ (ℝ ↑𝑚 𝑋))
6657, 65ovnxrcl 39459 . . . . . . . . 9 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌))) ∈ ℝ*)
6763, 66xaddcld 12003 . . . . . . . 8 ((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) → (((voln*‘𝑋)‘(𝑎 ∩ (𝐾(𝐻𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ (𝐾(𝐻𝑋)𝑌)))) ∈ ℝ*)
68 pnfge 11840 . . . . . . . 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 2616 . . . . . . 7 (((voln*‘𝑋)‘𝑎) = +∞ → +∞ = ((voln*‘𝑋)‘𝑎))
7372adantl 481 . . . . . 6 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) = +∞) → +∞ = ((voln*‘𝑋)‘𝑎))
7470, 73breqtrd 4609 . . . . 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 39457 . . . . . . . 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 2790 . . . . . . . 8 (¬ ((voln*‘𝑋)‘𝑎) = +∞ → ((voln*‘𝑋)‘𝑎) ≠ +∞)
7978adantl 481 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ¬ ((voln*‘𝑋)‘𝑎) = +∞) → ((voln*‘𝑋)‘𝑎) ≠ +∞)
80 ge0xrre 38605 . . . . . . 7 ((((voln*‘𝑋)‘𝑎) ∈ (0[,]+∞) ∧ ((voln*‘𝑋)‘𝑎) ≠ +∞) → ((voln*‘𝑋)‘𝑎) ∈ ℝ)
8177, 79, 80syl2anc 691 . . . . . 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 758 . . . . . . 7 (((𝜑𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)) ∧ ((voln*‘𝑋)‘𝑎) ∈ ℝ) → 𝐾𝑋)
8441ad2antrr 758 . . . . . . 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 3589 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝) ↔ 𝑏 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
8887rabbidv 3164 . . . . . . . 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 4678 . . . . . . 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 5206 . . . . . . . . . . 11 ((𝑖 = 𝑝𝑋) → ([,) ∘ 𝑖) = ([,) ∘ ))
9291fveq1d 6105 . . . . . . . . . 10 ((𝑖 = 𝑝𝑋) → (([,) ∘ 𝑖)‘𝑝) = (([,) ∘ )‘𝑝))
9392fveq2d 6107 . . . . . . . . 9 ((𝑖 = 𝑝𝑋) → (vol‘(([,) ∘ 𝑖)‘𝑝)) = (vol‘(([,) ∘ )‘𝑝)))
9493prodeq2dv 14492 . . . . . . . 8 (𝑖 = → ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)) = ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
9594cbvmptv 4678 . . . . . . 7 (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))) = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ )‘𝑝)))
96 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑝 → (([,) ∘ (𝑚𝑖))‘𝑛) = (([,) ∘ (𝑚𝑖))‘𝑝))
9796cbvixpv 7812 . . . . . . . . . . . . . . . . . . . . . . 23 X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝)
9897a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝))
99 fveq1 6102 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = → (𝑚𝑖) = (𝑖))
10099coeq2d 5206 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = → ([,) ∘ (𝑚𝑖)) = ([,) ∘ (𝑖)))
101100fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = → (([,) ∘ (𝑚𝑖))‘𝑝) = (([,) ∘ (𝑖))‘𝑝))
102101ixpeq2dv 7810 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = X𝑝𝑋 (([,) ∘ (𝑚𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
10398, 102eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
104103adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑖 ∈ ℕ) → X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
105104iuneq2dv 4478 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝))
106105sseq2d 3596 . . . . . . . . . . . . . . . . . 18 (𝑚 = → (𝑎 𝑖 ∈ ℕ X𝑛𝑋 (([,) ∘ (𝑚𝑖))‘𝑛) ↔ 𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝)))
107106cbvrabv 3172 . . . . . . . . . . . . . . . . 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 6102 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( = 𝑙 → (𝑖) = (𝑙𝑖))
109108coeq2d 5206 . . . . . . . . . . . . . . . . . . . . . . . 24 ( = 𝑙 → ([,) ∘ (𝑖)) = ([,) ∘ (𝑙𝑖)))
110109fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . 23 ( = 𝑙 → (([,) ∘ (𝑖))‘𝑝) = (([,) ∘ (𝑙𝑖))‘𝑝))
111110ixpeq2dv 7810 . . . . . . . . . . . . . . . . . . . . . 22 ( = 𝑙X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
112111adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (( = 𝑙𝑖 ∈ ℕ) → X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
113112iuneq2dv 4478 . . . . . . . . . . . . . . . . . . . 20 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝))
114 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑗 → (𝑙𝑖) = (𝑙𝑗))
115114coeq2d 5206 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑗 → ([,) ∘ (𝑙𝑖)) = ([,) ∘ (𝑙𝑗)))
116115fveq1d 6105 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑗 → (([,) ∘ (𝑙𝑖))‘𝑝) = (([,) ∘ (𝑙𝑗))‘𝑝))
117116ixpeq2dv 7810 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑗X𝑝𝑋 (([,) ∘ (𝑙𝑖))‘𝑝) = X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
118117cbviunv 4495 . . . . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . . . . 19 ( = 𝑙 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) = 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝))
121120sseq2d 3596 . . . . . . . . . . . . . . . . . 18 ( = 𝑙 → (𝑎 𝑖 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑖))‘𝑝) ↔ 𝑎 𝑗 ∈ ℕ X𝑝𝑋 (([,) ∘ (𝑙𝑗))‘𝑝)))
122121cbvrabv 3172 . . . . . . . . . . . . . . . . 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 2632 . . . . . . . . . . . . . . . 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 4669 . . . . . . . . . . . . . . 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 6109 . . . . . . . . . . . . 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 2673 . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑝 → (([,) ∘ 𝑖)‘𝑚) = (([,) ∘ 𝑖)‘𝑝))
130129fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑝 → (vol‘(([,) ∘ 𝑖)‘𝑚)) = (vol‘(([,) ∘ 𝑖)‘𝑝)))
131130cbvprodv 14485 . . . . . . . . . . . . . . . . . . 19 𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)) = ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝))
132131mpteq2i 4669 . . . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑗 → (𝑡𝑚) = (𝑡𝑗))
135133, 134fveq12d 6109 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑗 → ((𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑚𝑋 (vol‘(([,) ∘ 𝑖)‘𝑚)))‘(𝑡𝑚)) = ((𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑝𝑋 (vol‘(([,) ∘ 𝑖)‘𝑝)))‘(𝑡𝑗)))
136135cbvmptv 4678 . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . 14 (𝑐 = 𝑏 → ((voln*‘𝑋)‘𝑐) = ((voln*‘𝑋)‘𝑏))
140139oveq1d 6564 . . . . . . . . . . . . 13 (𝑐 = 𝑏 → (((voln*‘𝑋)‘𝑐) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑠))
141138, 140breq12d 4596 . . . . . . . . . . . 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 743 . . . . . . . . . . 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 3162 . . . . . . . . . 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 4673 . . . . . . . . 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 2611 . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . 14 (𝑠 = 𝑟 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑠) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑟))
148147breq2d 4595 . . . . . . . . . . . . 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 743 . . . . . . . . . . . 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 3162 . . . . . . . . . . 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 4678 . . . . . . . . . 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 2644 . . . . . . . 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 4678 . . . . . . 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 6103 . . . . . . . . . 10 (𝑚 = 𝑝 → ((𝑡𝑗)‘𝑚) = ((𝑡𝑗)‘𝑝))
156155fveq2d 6107 . . . . . . . . 9 (𝑚 = 𝑝 → (1st ‘((𝑡𝑗)‘𝑚)) = (1st ‘((𝑡𝑗)‘𝑝)))
157156cbvmptv 4678 . . . . . . . 8 (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝)))
158157mpteq2i 4669 . . . . . . 7 (𝑗 ∈ ℕ ↦ (𝑚𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (1st ‘((𝑡𝑗)‘𝑝))))
159 fveq2 6103 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑡𝑖) = (𝑡𝑗))
160159fveq1d 6105 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑡𝑖)‘𝑚) = ((𝑡𝑗)‘𝑚))
161160fveq2d 6107 . . . . . . . . . 10 (𝑖 = 𝑗 → (2nd ‘((𝑡𝑖)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑚)))
162161mpteq2dv 4673 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))))
163155fveq2d 6107 . . . . . . . . . . 11 (𝑚 = 𝑝 → (2nd ‘((𝑡𝑗)‘𝑚)) = (2nd ‘((𝑡𝑗)‘𝑝)))
164163cbvmptv 4678 . . . . . . . . . 10 (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝)))
165164a1i 11 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
166162, 165eqtrd 2644 . . . . . . . 8 (𝑖 = 𝑗 → (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚))) = (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
167166cbvmptv 4678 . . . . . . 7 (𝑖 ∈ ℕ ↦ (𝑚𝑋 ↦ (2nd ‘((𝑡𝑖)‘𝑚)))) = (𝑗 ∈ ℕ ↦ (𝑝𝑋 ↦ (2nd ‘((𝑡𝑗)‘𝑝))))
16839, 82, 83, 84, 85, 86, 89, 95, 154, 158, 167hspmbllem3 39518 . . . . . 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 691 . . . . 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 828 . . . 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 691 . . 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 39422 . 2 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ (CaraGen‘(voln*‘𝑋)))
1731dmvon 39496 . . 3 (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
174173eqcomd 2616 . 2 (𝜑 → (CaraGen‘(voln*‘𝑋)) = dom (voln‘𝑋))
175172, 174eleqtrd 2690 1 (𝜑 → (𝐾(𝐻𝑋)𝑌) ∈ dom (voln‘𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ifcif 4036  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  Xcixp 7794  Fincfn 7841  ℝcr 9814  0cc0 9815  +∞cpnf 9950  -∞cmnf 9951  ℝ*cxr 9952   ≤ cle 9954  ℕcn 10897  ℝ+crp 11708   +𝑒 cxad 11820  (,)cioo 12046  [,)cico 12048  [,]cicc 12049  ∏cprod 14474  volcvol 23039  Σ^csumge0 39255  CaraGenccaragen 39381  voln*covoln 39426  volncvoln 39428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-0g 15925  df-topgen 15927  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256  df-ome 39380  df-caragen 39382  df-ovoln 39427  df-voln 39429 This theorem is referenced by:  hoimbllem  39520
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