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Theorem cvmliftlem3 24755
Description: Lemma for cvmlift 24767. 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 452 . . 3  |-  ( (
ph  /\  ps )  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
4 oveq1 6029 . . . . . . . . 9  |-  ( k  =  M  ->  (
k  -  1 )  =  ( M  - 
1 ) )
54oveq1d 6037 . . . . . . . 8  |-  ( k  =  M  ->  (
( k  -  1 )  /  N )  =  ( ( M  -  1 )  /  N ) )
6 oveq1 6029 . . . . . . . 8  |-  ( k  =  M  ->  (
k  /  N )  =  ( M  /  N ) )
75, 6oveq12d 6040 . . . . . . 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 2439 . . . . . 6  |-  ( k  =  M  ->  (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) )  =  W )
109imaeq2d 5145 . . . . 5  |-  ( k  =  M  ->  ( G " ( ( ( k  -  1 )  /  N ) [,] ( k  /  N
) ) )  =  ( G " W
) )
11 fveq2 5670 . . . . . 6  |-  ( k  =  M  ->  ( T `  k )  =  ( T `  M ) )
1211fveq2d 5674 . . . . 5  |-  ( k  =  M  ->  ( 1st `  ( T `  k ) )  =  ( 1st `  ( T `  M )
) )
1310, 12sseq12d 3322 . . . 4  |-  ( k  =  M  ->  (
( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) )  <->  ( G " W )  C_  ( 1st `  ( T `  M ) ) ) )
1413rspcv 2993 . . 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 58 . 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 18784 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
19 cvmliftlem.x . . . . . . . 8  |-  X  = 
U. J
2018, 19cnf 17234 . . . . . . 7  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
2117, 20syl 16 . . . . . 6  |-  ( ph  ->  G : ( 0 [,] 1 ) --> X )
2221adantr 452 . . . . 5  |-  ( (
ph  /\  ps )  ->  G : ( 0 [,] 1 ) --> X )
23 ffun 5535 . . . . 5  |-  ( G : ( 0 [,] 1 ) --> X  ->  Fun  G )
2422, 23syl 16 . . . 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 24754 . . . . 5  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
34 fdm 5537 . . . . . 6  |-  ( G : ( 0 [,] 1 ) --> X  ->  dom  G  =  ( 0 [,] 1 ) )
3522, 34syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  dom  G  =  ( 0 [,] 1 ) )
3633, 35sseqtr4d 3330 . . . 4  |-  ( (
ph  /\  ps )  ->  W  C_  dom  G )
37 funfvima2 5915 . . . 4  |-  ( ( Fun  G  /\  W  C_ 
dom  G )  -> 
( A  e.  W  ->  ( G `  A
)  e.  ( G
" W ) ) )
3824, 36, 37syl2anc 643 . . 3  |-  ( (
ph  /\  ps )  ->  ( A  e.  W  ->  ( G `  A
)  e.  ( G
" W ) ) )
3916, 38mpd 15 . 2  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( G
" W ) )
4015, 39sseldd 3294 1  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( 1st `  ( T `  M
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655    \ cdif 3262    i^i cin 3264    C_ wss 3265   (/)c0 3573   ~Pcpw 3744   {csn 3759   U.cuni 3959   U_ciun 4037    e. cmpt 4209    X. cxp 4818   `'ccnv 4819   dom cdm 4820   ran crn 4821    |` cres 4822   "cima 4823   Fun wfun 5390   -->wf 5392   ` cfv 5396  (class class class)co 6022   1stc1st 6288   0cc0 8925   1c1 8926    - cmin 9225    / cdiv 9611   NNcn 9934   (,)cioo 10850   [,]cicc 10853   ...cfz 10977   ↾t crest 13577   topGenctg 13594    Cn ccn 17212    Homeo chmeo 17708   IIcii 18778   CovMap ccvm 24723
This theorem is referenced by:  cvmliftlem6  24758  cvmliftlem8  24760  cvmliftlem9  24761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-icc 10857  df-fz 10978  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-topgen 13596  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-top 16888  df-bases 16890  df-topon 16891  df-cn 17215  df-ii 18780
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