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Theorem cvmliftlem1 34345
Description: Lemma for cvmlift 34359. In cvmliftlem15 34358, we picked an 𝑁 large enough so that the sections (𝐺 β€œ [(π‘˜ βˆ’ 1) / 𝑁, π‘˜ / 𝑁]) are all contained in an even covering, and the function 𝑇 enumerates these even coverings. So 1st β€˜(π‘‡β€˜π‘€) is a neighborhood of (𝐺 β€œ [(𝑀 βˆ’ 1) / 𝑁, 𝑀 / 𝑁]), and 2nd β€˜(π‘‡β€˜π‘€) is an even covering of 1st β€˜(π‘‡β€˜π‘€), which is to say a disjoint union of open sets in 𝐢 whose image is 1st β€˜(π‘‡β€˜π‘€). (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypotheses
Ref Expression
cvmliftlem.1 𝑆 = (π‘˜ ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐢 βˆ– {βˆ…}) ∣ (βˆͺ 𝑠 = (◑𝐹 β€œ π‘˜) ∧ βˆ€π‘’ ∈ 𝑠 (βˆ€π‘£ ∈ (𝑠 βˆ– {𝑒})(𝑒 ∩ 𝑣) = βˆ… ∧ (𝐹 β†Ύ 𝑒) ∈ ((𝐢 β†Ύt 𝑒)Homeo(𝐽 β†Ύt π‘˜))))})
cvmliftlem.b 𝐡 = βˆͺ 𝐢
cvmliftlem.x 𝑋 = βˆͺ 𝐽
cvmliftlem.f (πœ‘ β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
cvmliftlem.g (πœ‘ β†’ 𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p (πœ‘ β†’ 𝑃 ∈ 𝐡)
cvmliftlem.e (πœ‘ β†’ (πΉβ€˜π‘ƒ) = (πΊβ€˜0))
cvmliftlem.n (πœ‘ β†’ 𝑁 ∈ β„•)
cvmliftlem.t (πœ‘ β†’ 𝑇:(1...𝑁)⟢βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
cvmliftlem.a (πœ‘ β†’ βˆ€π‘˜ ∈ (1...𝑁)(𝐺 β€œ (((π‘˜ βˆ’ 1) / 𝑁)[,](π‘˜ / 𝑁))) βŠ† (1st β€˜(π‘‡β€˜π‘˜)))
cvmliftlem.l 𝐿 = (topGenβ€˜ran (,))
cvmliftlem1.m ((πœ‘ ∧ πœ“) β†’ 𝑀 ∈ (1...𝑁))
Assertion
Ref Expression
cvmliftlem1 ((πœ‘ ∧ πœ“) β†’ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
Distinct variable groups:   𝑣,𝐡   𝑗,π‘˜,𝑠,𝑒,𝑣,𝐹   𝑗,𝑀,π‘˜,𝑠,𝑒,𝑣   𝑃,π‘˜,𝑒,𝑣   𝐢,𝑗,π‘˜,𝑠,𝑒,𝑣   πœ‘,𝑗,𝑠   π‘˜,𝑁,𝑒,𝑣   𝑆,𝑗,π‘˜,𝑠,𝑒,𝑣   𝑗,𝑋   𝑗,𝐺,π‘˜,𝑠,𝑒,𝑣   𝑇,𝑗,π‘˜,𝑠,𝑒,𝑣   𝑗,𝐽,π‘˜,𝑠,𝑒,𝑣
Allowed substitution hints:   πœ‘(𝑣,𝑒,π‘˜)   πœ“(𝑣,𝑒,𝑗,π‘˜,𝑠)   𝐡(𝑒,𝑗,π‘˜,𝑠)   𝑃(𝑗,𝑠)   𝐿(𝑣,𝑒,𝑗,π‘˜,𝑠)   𝑁(𝑗,𝑠)   𝑋(𝑣,𝑒,π‘˜,𝑠)
Proof of Theorem cvmliftlem1
StepHypRef Expression
1 relxp 5694 . . . . . 6 Rel ({𝑗} Γ— (π‘†β€˜π‘—))
21rgenw 3065 . . . . 5 βˆ€π‘— ∈ 𝐽 Rel ({𝑗} Γ— (π‘†β€˜π‘—))
3 reliun 5816 . . . . 5 (Rel βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)) ↔ βˆ€π‘— ∈ 𝐽 Rel ({𝑗} Γ— (π‘†β€˜π‘—)))
42, 3mpbir 230 . . . 4 Rel βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—))
5 cvmliftlem.t . . . . . 6 (πœ‘ β†’ 𝑇:(1...𝑁)⟢βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
65adantr 481 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝑇:(1...𝑁)⟢βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
7 cvmliftlem1.m . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝑀 ∈ (1...𝑁))
86, 7ffvelcdmd 7087 . . . 4 ((πœ‘ ∧ πœ“) β†’ (π‘‡β€˜π‘€) ∈ βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
9 1st2nd 8027 . . . 4 ((Rel βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)) ∧ (π‘‡β€˜π‘€) ∈ βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—))) β†’ (π‘‡β€˜π‘€) = ⟨(1st β€˜(π‘‡β€˜π‘€)), (2nd β€˜(π‘‡β€˜π‘€))⟩)
104, 8, 9sylancr 587 . . 3 ((πœ‘ ∧ πœ“) β†’ (π‘‡β€˜π‘€) = ⟨(1st β€˜(π‘‡β€˜π‘€)), (2nd β€˜(π‘‡β€˜π‘€))⟩)
1110, 8eqeltrrd 2834 . 2 ((πœ‘ ∧ πœ“) β†’ ⟨(1st β€˜(π‘‡β€˜π‘€)), (2nd β€˜(π‘‡β€˜π‘€))⟩ ∈ βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
12 fveq2 6891 . . . 4 (𝑗 = (1st β€˜(π‘‡β€˜π‘€)) β†’ (π‘†β€˜π‘—) = (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
1312opeliunxp2 5838 . . 3 (⟨(1st β€˜(π‘‡β€˜π‘€)), (2nd β€˜(π‘‡β€˜π‘€))⟩ ∈ βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)) ↔ ((1st β€˜(π‘‡β€˜π‘€)) ∈ 𝐽 ∧ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€)))))
1413simprbi 497 . 2 (⟨(1st β€˜(π‘‡β€˜π‘€)), (2nd β€˜(π‘‡β€˜π‘€))⟩ ∈ βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)) β†’ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
1511, 14syl 17 1 ((πœ‘ ∧ πœ“) β†’ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908  βˆͺ ciun 4997   ↦ cmpt 5231   Γ— cxp 5674  β—‘ccnv 5675  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679  Rel wrel 5681  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  0cc0 11112  1c1 11113   βˆ’ cmin 11446   / cdiv 11873  β„•cn 12214  (,)cioo 13326  [,]cicc 13329  ...cfz 13486   β†Ύt crest 17368  topGenctg 17385   Cn ccn 22735  Homeochmeo 23264  IIcii 24398   CovMap ccvm 34315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-1st 7977  df-2nd 7978
This theorem is referenced by:  cvmliftlem6  34350  cvmliftlem8  34352  cvmliftlem9  34353
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