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Theorem cvmliftlem1 31814
Description: Lemma for cvmlift 31828. In cvmliftlem15 31827, 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 5361 . . . . . 6 Rel ({𝑗} × (𝑆𝑗))
21rgenw 3134 . . . . 5 𝑗𝐽 Rel ({𝑗} × (𝑆𝑗))
3 reliun 5475 . . . . 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 6610 . . . 4 ((𝜑𝜓) → (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
9 1st2nd 7477 . . . 4 ((Rel 𝑗𝐽 ({𝑗} × (𝑆𝑗)) ∧ (𝑇𝑀) ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗))) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
104, 8, 9sylancr 583 . . 3 ((𝜑𝜓) → (𝑇𝑀) = ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩)
1110, 8eqeltrrd 2908 . 2 ((𝜑𝜓) → ⟨(1st ‘(𝑇𝑀)), (2nd ‘(𝑇𝑀))⟩ ∈ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
12 fveq2 6434 . . . 4 (𝑗 = (1st ‘(𝑇𝑀)) → (𝑆𝑗) = (𝑆‘(1st ‘(𝑇𝑀))))
1312opeliunxp2 5494 . . 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 1658  wcel 2166  wral 3118  {crab 3122  cdif 3796  cin 3798  wss 3799  c0 4145  𝒫 cpw 4379  {csn 4398  cop 4404   cuni 4659   ciun 4741  cmpt 4953   × cxp 5341  ccnv 5342  ran crn 5344  cres 5345  cima 5346  Rel wrel 5348  wf 6120  cfv 6124  (class class class)co 6906  1st c1st 7427  2nd c2nd 7428  0cc0 10253  1c1 10254  cmin 10586   / cdiv 11010  cn 11351  (,)cioo 12464  [,]cicc 12467  ...cfz 12620  t crest 16435  topGenctg 16452   Cn ccn 21400  Homeochmeo 21928  IIcii 23049   CovMap ccvm 31784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-fv 6132  df-1st 7429  df-2nd 7430
This theorem is referenced by:  cvmliftlem6  31819  cvmliftlem8  31821  cvmliftlem9  31822
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