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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem1 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 34290. In cvmliftlem15 34289, we picked an π large enough so that the sections (πΊ β [(π β 1) / π, π / π]) are all contained in an even covering, and the function π enumerates these even coverings. So 1st β(πβπ) is a neighborhood of (πΊ β [(π β 1) / π, π / π]), and 2nd β(πβπ) is an even covering of 1st β(πβπ), which is to say a disjoint union of open sets in πΆ whose image is 1st β(πβπ). (Contributed by Mario Carneiro, 14-Feb-2015.) |
Ref | Expression |
---|---|
cvmliftlem.1 | β’ π = (π β π½ β¦ {π β (π« πΆ β {β }) β£ (βͺ π = (β‘πΉ β π) β§ βπ’ β π (βπ£ β (π β {π’})(π’ β© π£) = β β§ (πΉ βΎ π’) β ((πΆ βΎt π’)Homeo(π½ βΎt π))))}) |
cvmliftlem.b | β’ π΅ = βͺ πΆ |
cvmliftlem.x | β’ π = βͺ π½ |
cvmliftlem.f | β’ (π β πΉ β (πΆ CovMap π½)) |
cvmliftlem.g | β’ (π β πΊ β (II Cn π½)) |
cvmliftlem.p | β’ (π β π β π΅) |
cvmliftlem.e | β’ (π β (πΉβπ) = (πΊβ0)) |
cvmliftlem.n | β’ (π β π β β) |
cvmliftlem.t | β’ (π β π:(1...π)βΆβͺ π β π½ ({π} Γ (πβπ))) |
cvmliftlem.a | β’ (π β βπ β (1...π)(πΊ β (((π β 1) / π)[,](π / π))) β (1st β(πβπ))) |
cvmliftlem.l | β’ πΏ = (topGenβran (,)) |
cvmliftlem1.m | β’ ((π β§ π) β π β (1...π)) |
Ref | Expression |
---|---|
cvmliftlem1 | β’ ((π β§ π) β (2nd β(πβπ)) β (πβ(1st β(πβπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5695 | . . . . . 6 β’ Rel ({π} Γ (πβπ)) | |
2 | 1 | rgenw 3066 | . . . . 5 β’ βπ β π½ Rel ({π} Γ (πβπ)) |
3 | reliun 5817 | . . . . 5 β’ (Rel βͺ π β π½ ({π} Γ (πβπ)) β βπ β π½ Rel ({π} Γ (πβπ))) | |
4 | 2, 3 | mpbir 230 | . . . 4 β’ Rel βͺ π β π½ ({π} Γ (πβπ)) |
5 | cvmliftlem.t | . . . . . 6 β’ (π β π:(1...π)βΆβͺ π β π½ ({π} Γ (πβπ))) | |
6 | 5 | adantr 482 | . . . . 5 β’ ((π β§ π) β π:(1...π)βΆβͺ π β π½ ({π} Γ (πβπ))) |
7 | cvmliftlem1.m | . . . . 5 β’ ((π β§ π) β π β (1...π)) | |
8 | 6, 7 | ffvelcdmd 7088 | . . . 4 β’ ((π β§ π) β (πβπ) β βͺ π β π½ ({π} Γ (πβπ))) |
9 | 1st2nd 8025 | . . . 4 β’ ((Rel βͺ π β π½ ({π} Γ (πβπ)) β§ (πβπ) β βͺ π β π½ ({π} Γ (πβπ))) β (πβπ) = β¨(1st β(πβπ)), (2nd β(πβπ))β©) | |
10 | 4, 8, 9 | sylancr 588 | . . 3 β’ ((π β§ π) β (πβπ) = β¨(1st β(πβπ)), (2nd β(πβπ))β©) |
11 | 10, 8 | eqeltrrd 2835 | . 2 β’ ((π β§ π) β β¨(1st β(πβπ)), (2nd β(πβπ))β© β βͺ π β π½ ({π} Γ (πβπ))) |
12 | fveq2 6892 | . . . 4 β’ (π = (1st β(πβπ)) β (πβπ) = (πβ(1st β(πβπ)))) | |
13 | 12 | opeliunxp2 5839 | . . 3 β’ (β¨(1st β(πβπ)), (2nd β(πβπ))β© β βͺ π β π½ ({π} Γ (πβπ)) β ((1st β(πβπ)) β π½ β§ (2nd β(πβπ)) β (πβ(1st β(πβπ))))) |
14 | 13 | simprbi 498 | . 2 β’ (β¨(1st β(πβπ)), (2nd β(πβπ))β© β βͺ π β π½ ({π} Γ (πβπ)) β (2nd β(πβπ)) β (πβ(1st β(πβπ)))) |
15 | 11, 14 | syl 17 | 1 β’ ((π β§ π) β (2nd β(πβπ)) β (πβ(1st β(πβπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 {crab 3433 β cdif 3946 β© cin 3948 β wss 3949 β c0 4323 π« cpw 4603 {csn 4629 β¨cop 4635 βͺ cuni 4909 βͺ ciun 4998 β¦ cmpt 5232 Γ cxp 5675 β‘ccnv 5676 ran crn 5678 βΎ cres 5679 β cima 5680 Rel wrel 5682 βΆwf 6540 βcfv 6544 (class class class)co 7409 1st c1st 7973 2nd c2nd 7974 0cc0 11110 1c1 11111 β cmin 11444 / cdiv 11871 βcn 12212 (,)cioo 13324 [,]cicc 13327 ...cfz 13484 βΎt crest 17366 topGenctg 17383 Cn ccn 22728 Homeochmeo 23257 IIcii 24391 CovMap ccvm 34246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-1st 7975 df-2nd 7976 |
This theorem is referenced by: cvmliftlem6 34281 cvmliftlem8 34283 cvmliftlem9 34284 |
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