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Theorem cvmliftlem1 31802
Description: Lemma for cvmlift 31816. In cvmliftlem15 31815, 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 5360 . . . . . 6 Rel ({𝑗} × (𝑆𝑗))
21rgenw 3133 . . . . 5 𝑗𝐽 Rel ({𝑗} × (𝑆𝑗))
3 reliun 5474 . . . . 5 (Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ ∀𝑗𝐽 Rel ({𝑗} × (𝑆𝑗)))
42, 3mpbir 223 . . . 4 Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗))
5 cvmliftlem.t . . . . . 6 (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
65adantr 474 . . . . 5 ((𝜑𝜓) → 𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
7 cvmliftlem1.m . . . . 5 ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))
86, 7ffvelrnd 6609 . . . 4 ((𝜑𝜓) → (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
9 1st2nd 7476 . . . 4 ((Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗))) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
104, 8, 9sylancr 581 . . 3 ((𝜑𝜓) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
1110, 8eqeltrrd 2907 . 2 ((𝜑𝜓) → ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
12 fveq2 6433 . . . 4 (𝑗 = (1st ‘(𝑇𝑀)) → (𝑆𝑗) = (𝑆‘(1st ‘(𝑇𝑀))))
1312opeliunxp2 5493 . . 3 (⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ↔ ((1st ‘(𝑇𝑀)) ∈ 𝐽 ∧ (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀)))))
1413simprbi 492 . 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 386   = wceq 1656  wcel 2164  wral 3117  {crab 3121  cdif 3795  cin 3797  wss 3798  c0 4144  𝒫 cpw 4378  {csn 4397  cop 4403   cuni 4658   ciun 4740  cmpt 4952   × cxp 5340  ccnv 5341  ran crn 5343  cres 5344  cima 5345  Rel wrel 5347  wf 6119  cfv 6123  (class class class)co 6905  1st c1st 7426  2nd c2nd 7427  0cc0 10252  1c1 10253  cmin 10585   / cdiv 11009  cn 11350  (,)cioo 12463  [,]cicc 12466  ...cfz 12619  t crest 16434  topGenctg 16451   Cn ccn 21399  Homeochmeo 21927  IIcii 23048   CovMap ccvm 31772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-1st 7428  df-2nd 7429
This theorem is referenced by:  cvmliftlem6  31807  cvmliftlem8  31809  cvmliftlem9  31810
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