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Theorem cvmliftlem1 28997
Description: Lemma for cvmlift 29011. In cvmliftlem15 29010, 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 5098 . . . . . 6  |-  Rel  ( { j }  X.  ( S `  j ) )
21rgenw 2815 . . . . 5  |-  A. j  e.  J  Rel  ( { j }  X.  ( S `  j )
)
3 reliun 5111 . . . . 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 463 . . . . 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 6008 . . . 4  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  e.  U_ j  e.  J  ( {
j }  X.  ( S `  j )
) )
9 1st2nd 6819 . . . 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 661 . . 3  |-  ( (
ph  /\  ps )  ->  ( T `  M
)  =  <. ( 1st `  ( T `  M ) ) ,  ( 2nd `  ( T `  M )
) >. )
1110, 8eqeltrrd 2543 . 2  |-  ( (
ph  /\  ps )  -> 
<. ( 1st `  ( T `  M )
) ,  ( 2nd `  ( T `  M
) ) >.  e.  U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
12 fveq2 5848 . . . 4  |-  ( j  =  ( 1st `  ( T `  M )
)  ->  ( S `  j )  =  ( S `  ( 1st `  ( T `  M
) ) ) )
1312opeliunxp2 5130 . . 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 462 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   <.cop 4022   U.cuni 4235   U_ciun 4315    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   ran crn 4989    |` cres 4990   "cima 4991   Rel wrel 4993   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   0cc0 9481   1c1 9482    - cmin 9796    / cdiv 10202   NNcn 10531   (,)cioo 11532   [,]cicc 11535   ...cfz 11675   ↾t crest 14913   topGenctg 14930    Cn ccn 19895   Homeochmeo 20423   IIcii 21548   CovMap ccvm 28967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-1st 6773  df-2nd 6774
This theorem is referenced by:  cvmliftlem6  29002  cvmliftlem8  29004  cvmliftlem9  29005
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