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Theorem cvmliftlem1 27196
Description: Lemma for cvmlift 27210. In cvmliftlem15 27209, 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 4968 . . . . . 6  |-  Rel  ( { j }  X.  ( S `  j ) )
21rgenw 2804 . . . . 5  |-  A. j  e.  J  Rel  ( { j }  X.  ( S `  j )
)
3 reliun 4981 . . . . 5  |-  ( Rel  U_ j  e.  J  ( { j }  X.  ( S `  j ) )  <->  A. j  e.  J  Rel  ( { j }  X.  ( S `  j ) ) )
42, 3mpbir 209 . . . 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 465 . . . . 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 5865 . . . 4  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  e.  U_ j  e.  J  ( {
j }  X.  ( S `  j )
) )
9 1st2nd 6641 . . . 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 663 . . 3  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  =  <. ( 1st `  ( T `  M ) ) ,  ( 2nd `  ( T `  M )
) >. )
1110, 8eqeltrrd 2518 . 2  |-  ( (
ph  /\  ps )  -> 
<. ( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >.  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
12 fveq2 5712 . . . 4  |-  ( j  =  ( 1st `  ( T `  M )
)  ->  ( S `  j )  =  ( S `  ( 1st `  ( T `  M
) ) ) )
1312opeliunxp2 4999 . . 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 464 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   {crab 2740    \ cdif 3346    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   {csn 3898   <.cop 3904   U.cuni 4112   U_ciun 4192    e. cmpt 4371    X. cxp 4859   `'ccnv 4860   ran crn 4862    |` cres 4863   "cima 4864   Rel wrel 4866   -->wf 5435   ` cfv 5439  (class class class)co 6112   1stc1st 6596   2ndc2nd 6597   0cc0 9303   1c1 9304    - cmin 9616    / cdiv 10014   NNcn 10343   (,)cioo 11321   [,]cicc 11324   ...cfz 11458   ↾t crest 14380   topGenctg 14397    Cn ccn 18850   Homeochmeo 19348   IIcii 20473   CovMap ccvm 27166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-1st 6598  df-2nd 6599
This theorem is referenced by:  cvmliftlem6  27201  cvmliftlem8  27203  cvmliftlem9  27204
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