Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmliftlem1 Structured version   Unicode version

Theorem cvmliftlem1 24972
Description: Lemma for cvmlift 24986. In cvmliftlem15 24985, 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 `  ( T `  M
) is a neighborhood of  ( G " [
( M  -  1 )  /  N ,  M  /  N ] ), and  2nd `  ( T `  M ) is an even covering of  1st `  ( T `  M ), which is to say a disjoint union of open sets in  C whose image is  1st `  ( T `
 M ). (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypotheses
Ref Expression
cvmliftlem.1  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. u  e.  s  ( A. v  e.  ( s  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u ) 
Homeo  ( Jt  k ) ) ) ) } )
cvmliftlem.b  |-  B  = 
U. C
cvmliftlem.x  |-  X  = 
U. J
cvmliftlem.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmliftlem.g  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
cvmliftlem.p  |-  ( ph  ->  P  e.  B )
cvmliftlem.e  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
cvmliftlem.n  |-  ( ph  ->  N  e.  NN )
cvmliftlem.t  |-  ( ph  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
cvmliftlem.a  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
cvmliftlem.l  |-  L  =  ( topGen `  ran  (,) )
cvmliftlem1.m  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
Assertion
Ref Expression
cvmliftlem1  |-  ( (
ph  /\  ps )  ->  ( 2nd `  ( T `  M )
)  e.  ( S `
 ( 1st `  ( T `  M )
) ) )
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    ph, 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:    ph( v, u, k)    ps( 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 4983 . . . . . 6  |-  Rel  ( { j }  X.  ( S `  j ) )
21rgenw 2773 . . . . 5  |-  A. j  e.  J  Rel  ( { j }  X.  ( S `  j )
)
3 reliun 4995 . . . . 5  |-  ( Rel  U_ j  e.  J  ( { j }  X.  ( S `  j ) )  <->  A. j  e.  J  Rel  ( { j }  X.  ( S `  j ) ) )
42, 3mpbir 201 . . . 4  |-  Rel  U_ j  e.  J  ( {
j }  X.  ( S `  j )
)
5 cvmliftlem.t . . . . . 6  |-  ( ph  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
65adantr 452 . . . . 5  |-  ( (
ph  /\  ps )  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
7 cvmliftlem1.m . . . . 5  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
86, 7ffvelrnd 5871 . . . 4  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  e.  U_ j  e.  J  ( {
j }  X.  ( S `  j )
) )
9 1st2nd 6393 . . . 4  |-  ( ( Rel  U_ j  e.  J  ( { j }  X.  ( S `  j ) )  /\  ( T `
 M )  e. 
U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )  ->  ( T `  M )  =  <. ( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >. )
104, 8, 9sylancr 645 . . 3  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  =  <. ( 1st `  ( T `  M ) ) ,  ( 2nd `  ( T `  M )
) >. )
1110, 8eqeltrrd 2511 . 2  |-  ( (
ph  /\  ps )  -> 
<. ( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >.  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
12 fveq2 5728 . . . 4  |-  ( j  =  ( 1st `  ( T `  M )
)  ->  ( S `  j )  =  ( S `  ( 1st `  ( T `  M
) ) ) )
1312opeliunxp2 5013 . . 3  |-  ( <.
( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >.  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) )  <->  ( ( 1st `  ( T `  M
) )  e.  J  /\  ( 2nd `  ( T `  M )
)  e.  ( S `
 ( 1st `  ( T `  M )
) ) ) )
1413simprbi 451 . 2  |-  ( <.
( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >.  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) )  ->  ( 2nd `  ( T `  M
) )  e.  ( S `  ( 1st `  ( T `  M
) ) ) )
1511, 14syl 16 1  |-  ( (
ph  /\  ps )  ->  ( 2nd `  ( T `  M )
)  e.  ( S `
 ( 1st `  ( T `  M )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   <.cop 3817   U.cuni 4015   U_ciun 4093    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   ran crn 4879    |` cres 4880   "cima 4881   Rel wrel 4883   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   0cc0 8990   1c1 8991    - cmin 9291    / cdiv 9677   NNcn 10000   (,)cioo 10916   [,]cicc 10919   ...cfz 11043   ↾t crest 13648   topGenctg 13665    Cn ccn 17288    Homeo chmeo 17785   IIcii 18905   CovMap ccvm 24942
This theorem is referenced by:  cvmliftlem6  24977  cvmliftlem8  24979  cvmliftlem9  24980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-1st 6349  df-2nd 6350
  Copyright terms: Public domain W3C validator