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Theorem cvmliftlem1 30975
Description: Lemma for cvmlift 30989. In cvmliftlem15 30988, we picked an 𝑁 large enough so that the sections (𝐺 “ [(𝑘 − 1) / 𝑁, 𝑘 / 𝑁]) are all contained in an even covering, and the function 𝑇 enumerates these even coverings. So 1st ‘(𝑇𝑀) is a neighborhood of (𝐺 “ [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁]), and 2nd ‘(𝑇𝑀) is an even covering of 1st ‘(𝑇𝑀), which is to say a disjoint union of open sets in 𝐶 whose image is 1st ‘(𝑇𝑀). (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypotheses
Ref Expression
cvmliftlem.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
cvmliftlem.b 𝐵 = 𝐶
cvmliftlem.x 𝑋 = 𝐽
cvmliftlem.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftlem.g (𝜑𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p (𝜑𝑃𝐵)
cvmliftlem.e (𝜑 → (𝐹𝑃) = (𝐺‘0))
cvmliftlem.n (𝜑𝑁 ∈ ℕ)
cvmliftlem.t (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
cvmliftlem.a (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))
cvmliftlem.l 𝐿 = (topGen‘ran (,))
cvmliftlem1.m ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))
Assertion
Ref Expression
cvmliftlem1 ((𝜑𝜓) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
Distinct variable groups:   𝑣,𝐵   𝑗,𝑘,𝑠,𝑢,𝑣,𝐹   𝑗,𝑀,𝑘,𝑠,𝑢,𝑣   𝑃,𝑘,𝑢,𝑣   𝐶,𝑗,𝑘,𝑠,𝑢,𝑣   𝜑,𝑗,𝑠   𝑘,𝑁,𝑢,𝑣   𝑆,𝑗,𝑘,𝑠,𝑢,𝑣   𝑗,𝑋   𝑗,𝐺,𝑘,𝑠,𝑢,𝑣   𝑇,𝑗,𝑘,𝑠,𝑢,𝑣   𝑗,𝐽,𝑘,𝑠,𝑢,𝑣
Allowed substitution hints:   𝜑(𝑣,𝑢,𝑘)   𝜓(𝑣,𝑢,𝑗,𝑘,𝑠)   𝐵(𝑢,𝑗,𝑘,𝑠)   𝑃(𝑗,𝑠)   𝐿(𝑣,𝑢,𝑗,𝑘,𝑠)   𝑁(𝑗,𝑠)   𝑋(𝑣,𝑢,𝑘,𝑠)

Proof of Theorem cvmliftlem1
StepHypRef Expression
1 relxp 5188 . . . . . 6 Rel ({𝑗} × (𝑆𝑗))
21rgenw 2919 . . . . 5 𝑗𝐽 Rel ({𝑗} × (𝑆𝑗))
3 reliun 5200 . . . . 5 (Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ ∀𝑗𝐽 Rel ({𝑗} × (𝑆𝑗)))
42, 3mpbir 221 . . . 4 Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗))
5 cvmliftlem.t . . . . . 6 (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
65adantr 481 . . . . 5 ((𝜑𝜓) → 𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
7 cvmliftlem1.m . . . . 5 ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))
86, 7ffvelrnd 6316 . . . 4 ((𝜑𝜓) → (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
9 1st2nd 7159 . . . 4 ((Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗))) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
104, 8, 9sylancr 694 . . 3 ((𝜑𝜓) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
1110, 8eqeltrrd 2699 . 2 ((𝜑𝜓) → ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
12 fveq2 6148 . . . 4 (𝑗 = (1st ‘(𝑇𝑀)) → (𝑆𝑗) = (𝑆‘(1st ‘(𝑇𝑀))))
1312opeliunxp2 5220 . . 3 (⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ ((1st ‘(𝑇𝑀)) ∈ 𝐽 ∧ (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀)))))
1413simprbi 480 . 2 (⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
1511, 14syl 17 1 ((𝜑𝜓) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3552  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148  cop 4154   cuni 4402   ciun 4485  cmpt 4673   × cxp 5072  ccnv 5073  ran crn 5075  cres 5076  cima 5077  Rel wrel 5079  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  0cc0 9880  1c1 9881  cmin 10210   / cdiv 10628  cn 10964  (,)cioo 12117  [,]cicc 12120  ...cfz 12268  t crest 16002  topGenctg 16019   Cn ccn 20938  Homeochmeo 21466  IIcii 22586   CovMap ccvm 30945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-1st 7113  df-2nd 7114
This theorem is referenced by:  cvmliftlem6  30980  cvmliftlem8  30982  cvmliftlem9  30983
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