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Theorem cvmliftlem1 24100
Description: Lemma for cvmlift 24114. In cvmliftlem15 24113, 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 4831 . . . . . 6  |-  Rel  ( { j }  X.  ( S `  j ) )
21rgenw 2644 . . . . 5  |-  A. j  e.  J  Rel  ( { j }  X.  ( S `  j )
)
3 reliun 4843 . . . . 5  |-  ( Rel  U_ j  e.  J  ( { j }  X.  ( S `  j ) )  <->  A. j  e.  J  Rel  ( { j }  X.  ( S `  j ) ) )
42, 3mpbir 200 . . . 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 451 . . . . 5  |-  ( (
ph  /\  ps )  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
7 cvmliftlem1.m . . . . 5  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
8 ffvelrn 5701 . . . . 5  |-  ( ( T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) )  /\  M  e.  ( 1 ... N
) )  ->  ( T `  M )  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
96, 7, 8syl2anc 642 . . . 4  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  e.  U_ j  e.  J  ( {
j }  X.  ( S `  j )
) )
10 1st2nd 6208 . . . 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
) ) >. )
114, 9, 10sylancr 644 . . 3  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  =  <. ( 1st `  ( T `  M ) ) ,  ( 2nd `  ( T `  M )
) >. )
1211, 9eqeltrrd 2391 . 2  |-  ( (
ph  /\  ps )  -> 
<. ( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >.  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
13 fveq2 5563 . . . 4  |-  ( j  =  ( 1st `  ( T `  M )
)  ->  ( S `  j )  =  ( S `  ( 1st `  ( T `  M
) ) ) )
1413opeliunxp2 4861 . . 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 )
) ) ) )
1514simprbi 450 . 2  |-  ( <.
( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >.  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) )  ->  ( 2nd `  ( T `  M
) )  e.  ( S `  ( 1st `  ( T `  M
) ) ) )
1612, 15syl 15 1  |-  ( (
ph  /\  ps )  ->  ( 2nd `  ( T `  M )
)  e.  ( S `
 ( 1st `  ( T `  M )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   {crab 2581    \ cdif 3183    i^i cin 3185    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   {csn 3674   <.cop 3677   U.cuni 3864   U_ciun 3942    e. cmpt 4114    X. cxp 4724   `'ccnv 4725   ran crn 4727    |` cres 4728   "cima 4729   Rel wrel 4731   -->wf 5288   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   0cc0 8782   1c1 8783    - cmin 9082    / cdiv 9468   NNcn 9791   (,)cioo 10703   [,]cicc 10706   ...cfz 10829   ↾t crest 13374   topGenctg 13391    Cn ccn 17010    Homeo chmeo 17500   IIcii 18431   CovMap ccvm 24070
This theorem is referenced by:  cvmliftlem6  24105  cvmliftlem8  24107  cvmliftlem9  24108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-1st 6164  df-2nd 6165
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