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Theorem cvmliftlem3 23818
Description: Lemma for cvmlift 23830. Since  1st `  ( T `  M
) is a neighborhood of  ( G " W ), every element  A  e.  W satisfies  ( G `  A )  e.  ( 1st `  ( T `
 M ) ). (Contributed by Mario Carneiro, 16-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 ) )
cvmliftlem3.3  |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
cvmliftlem3.m  |-  ( (
ph  /\  ps )  ->  A  e.  W )
Assertion
Ref Expression
cvmliftlem3  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( 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    k, W
Allowed substitution hints:    ph( v, u, k)    ps( v, u, j, k, s)    A( v, u, j, k, s)    B( u, j, k, s)    P( j, s)    L( v, u, j, k, s)    N( j, s)    W( v, u, j, s)    X( v, u, k, s)

Proof of Theorem cvmliftlem3
StepHypRef Expression
1 cvmliftlem1.m . . 3  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
2 cvmliftlem.a . . . 4  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
32adantr 451 . . 3  |-  ( (
ph  /\  ps )  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
4 oveq1 5865 . . . . . . . . 9  |-  ( k  =  M  ->  (
k  -  1 )  =  ( M  - 
1 ) )
54oveq1d 5873 . . . . . . . 8  |-  ( k  =  M  ->  (
( k  -  1 )  /  N )  =  ( ( M  -  1 )  /  N ) )
6 oveq1 5865 . . . . . . . 8  |-  ( k  =  M  ->  (
k  /  N )  =  ( M  /  N ) )
75, 6oveq12d 5876 . . . . . . 7  |-  ( k  =  M  ->  (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) )  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N
) ) )
8 cvmliftlem3.3 . . . . . . 7  |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
97, 8syl6eqr 2333 . . . . . 6  |-  ( k  =  M  ->  (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) )  =  W )
109imaeq2d 5012 . . . . 5  |-  ( k  =  M  ->  ( G " ( ( ( k  -  1 )  /  N ) [,] ( k  /  N
) ) )  =  ( G " W
) )
11 fveq2 5525 . . . . . 6  |-  ( k  =  M  ->  ( T `  k )  =  ( T `  M ) )
1211fveq2d 5529 . . . . 5  |-  ( k  =  M  ->  ( 1st `  ( T `  k ) )  =  ( 1st `  ( T `  M )
) )
1310, 12sseq12d 3207 . . . 4  |-  ( k  =  M  ->  (
( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) )  <->  ( G " W )  C_  ( 1st `  ( T `  M ) ) ) )
1413rspcv 2880 . . 3  |-  ( M  e.  ( 1 ... N )  ->  ( A. k  e.  (
1 ... N ) ( G " ( ( ( k  -  1 )  /  N ) [,] ( k  /  N ) ) ) 
C_  ( 1st `  ( T `  k )
)  ->  ( G " W )  C_  ( 1st `  ( T `  M ) ) ) )
151, 3, 14sylc 56 . 2  |-  ( (
ph  /\  ps )  ->  ( G " W
)  C_  ( 1st `  ( T `  M
) ) )
16 cvmliftlem3.m . . 3  |-  ( (
ph  /\  ps )  ->  A  e.  W )
17 cvmliftlem.g . . . . . . 7  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
18 iiuni 18385 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
19 cvmliftlem.x . . . . . . . 8  |-  X  = 
U. J
2018, 19cnf 16976 . . . . . . 7  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
2117, 20syl 15 . . . . . 6  |-  ( ph  ->  G : ( 0 [,] 1 ) --> X )
2221adantr 451 . . . . 5  |-  ( (
ph  /\  ps )  ->  G : ( 0 [,] 1 ) --> X )
23 ffun 5391 . . . . 5  |-  ( G : ( 0 [,] 1 ) --> X  ->  Fun  G )
2422, 23syl 15 . . . 4  |-  ( (
ph  /\  ps )  ->  Fun  G )
25 cvmliftlem.1 . . . . . 6  |-  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 ) ) ) ) } )
26 cvmliftlem.b . . . . . 6  |-  B  = 
U. C
27 cvmliftlem.f . . . . . 6  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
28 cvmliftlem.p . . . . . 6  |-  ( ph  ->  P  e.  B )
29 cvmliftlem.e . . . . . 6  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
30 cvmliftlem.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
31 cvmliftlem.t . . . . . 6  |-  ( ph  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
32 cvmliftlem.l . . . . . 6  |-  L  =  ( topGen `  ran  (,) )
3325, 26, 19, 27, 17, 28, 29, 30, 31, 2, 32, 1, 8cvmliftlem2 23817 . . . . 5  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
34 fdm 5393 . . . . . 6  |-  ( G : ( 0 [,] 1 ) --> X  ->  dom  G  =  ( 0 [,] 1 ) )
3522, 34syl 15 . . . . 5  |-  ( (
ph  /\  ps )  ->  dom  G  =  ( 0 [,] 1 ) )
3633, 35sseqtr4d 3215 . . . 4  |-  ( (
ph  /\  ps )  ->  W  C_  dom  G )
37 funfvima2 5754 . . . 4  |-  ( ( Fun  G  /\  W  C_ 
dom  G )  -> 
( A  e.  W  ->  ( G `  A
)  e.  ( G
" W ) ) )
3824, 36, 37syl2anc 642 . . 3  |-  ( (
ph  /\  ps )  ->  ( A  e.  W  ->  ( G `  A
)  e.  ( G
" W ) ) )
3916, 38mpd 14 . 2  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( G
" W ) )
4015, 39sseldd 3181 1  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( 1st `  ( T `  M
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827   U_ciun 3905    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   0cc0 8737   1c1 8738    - cmin 9037    / cdiv 9423   NNcn 9746   (,)cioo 10656   [,]cicc 10659   ...cfz 10782   ↾t crest 13325   topGenctg 13342    Cn ccn 16954    Homeo chmeo 17444   IIcii 18379   CovMap ccvm 23786
This theorem is referenced by:  cvmliftlem6  23821  cvmliftlem8  23823  cvmliftlem9  23824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-ii 18381
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