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Theorem cvmliftlem1 30056
Description: Lemma for cvmlift 30070. In cvmliftlem15 30069, 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 4960 . . . . . 6  |-  Rel  ( { j }  X.  ( S `  j ) )
21rgenw 2760 . . . . 5  |-  A. j  e.  J  Rel  ( { j }  X.  ( S `  j )
)
3 reliun 4972 . . . . 5  |-  ( Rel  U_ j  e.  J  ( { j }  X.  ( S `  j ) )  <->  A. j  e.  J  Rel  ( { j }  X.  ( S `  j ) ) )
42, 3mpbir 214 . . . 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 471 . . . . 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 6045 . . . 4  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  e.  U_ j  e.  J  ( {
j }  X.  ( S `  j )
) )
9 1st2nd 6865 . . . 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 674 . . 3  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  =  <. ( 1st `  ( T `  M ) ) ,  ( 2nd `  ( T `  M )
) >. )
1110, 8eqeltrrd 2540 . 2  |-  ( (
ph  /\  ps )  -> 
<. ( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >.  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
12 fveq2 5887 . . . 4  |-  ( j  =  ( 1st `  ( T `  M )
)  ->  ( S `  j )  =  ( S `  ( 1st `  ( T `  M
) ) ) )
1312opeliunxp2 4991 . . 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 470 . 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 17 1  |-  ( (
ph  /\  ps )  ->  ( 2nd `  ( T `  M )
)  e.  ( S `
 ( 1st `  ( T `  M )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   {crab 2752    \ cdif 3412    i^i cin 3414    C_ wss 3415   (/)c0 3742   ~Pcpw 3962   {csn 3979   <.cop 3985   U.cuni 4211   U_ciun 4291    |-> cmpt 4474    X. cxp 4850   `'ccnv 4851   ran crn 4853    |` cres 4854   "cima 4855   Rel wrel 4857   -->wf 5596   ` cfv 5600  (class class class)co 6314   1stc1st 6817   2ndc2nd 6818   0cc0 9564   1c1 9565    - cmin 9885    / cdiv 10296   NNcn 10636   (,)cioo 11663   [,]cicc 11666   ...cfz 11812   ↾t crest 15367   topGenctg 15384    Cn ccn 20288   Homeochmeo 20816   IIcii 21955   CovMap ccvm 30026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fv 5608  df-1st 6819  df-2nd 6820
This theorem is referenced by:  cvmliftlem6  30061  cvmliftlem8  30063  cvmliftlem9  30064
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