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Theorem cvmliftlem3 30010
Description: Lemma for cvmlift 30022. 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 467 . . 3  |-  ( (
ph  /\  ps )  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
4 oveq1 6297 . . . . . . . . 9  |-  ( k  =  M  ->  (
k  -  1 )  =  ( M  - 
1 ) )
54oveq1d 6305 . . . . . . . 8  |-  ( k  =  M  ->  (
( k  -  1 )  /  N )  =  ( ( M  -  1 )  /  N ) )
6 oveq1 6297 . . . . . . . 8  |-  ( k  =  M  ->  (
k  /  N )  =  ( M  /  N ) )
75, 6oveq12d 6308 . . . . . . 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 2503 . . . . . 6  |-  ( k  =  M  ->  (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) )  =  W )
109imaeq2d 5168 . . . . 5  |-  ( k  =  M  ->  ( G " ( ( ( k  -  1 )  /  N ) [,] ( k  /  N
) ) )  =  ( G " W
) )
11 fveq2 5865 . . . . . 6  |-  ( k  =  M  ->  ( T `  k )  =  ( T `  M ) )
1211fveq2d 5869 . . . . 5  |-  ( k  =  M  ->  ( 1st `  ( T `  k ) )  =  ( 1st `  ( T `  M )
) )
1310, 12sseq12d 3461 . . . 4  |-  ( k  =  M  ->  (
( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) )  <->  ( G " W )  C_  ( 1st `  ( T `  M ) ) ) )
1413rspcv 3146 . . 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 62 . 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 21913 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
19 cvmliftlem.x . . . . . . . 8  |-  X  = 
U. J
2018, 19cnf 20262 . . . . . . 7  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
2117, 20syl 17 . . . . . 6  |-  ( ph  ->  G : ( 0 [,] 1 ) --> X )
2221adantr 467 . . . . 5  |-  ( (
ph  /\  ps )  ->  G : ( 0 [,] 1 ) --> X )
23 ffun 5731 . . . . 5  |-  ( G : ( 0 [,] 1 ) --> X  ->  Fun  G )
2422, 23syl 17 . . . 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 30009 . . . . 5  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
34 fdm 5733 . . . . . 6  |-  ( G : ( 0 [,] 1 ) --> X  ->  dom  G  =  ( 0 [,] 1 ) )
3522, 34syl 17 . . . . 5  |-  ( (
ph  /\  ps )  ->  dom  G  =  ( 0 [,] 1 ) )
3633, 35sseqtr4d 3469 . . . 4  |-  ( (
ph  /\  ps )  ->  W  C_  dom  G )
37 funfvima2 6141 . . . 4  |-  ( ( Fun  G  /\  W  C_ 
dom  G )  -> 
( A  e.  W  ->  ( G `  A
)  e.  ( G
" W ) ) )
3824, 36, 37syl2anc 667 . . 3  |-  ( (
ph  /\  ps )  ->  ( A  e.  W  ->  ( G `  A
)  e.  ( G
" W ) ) )
3916, 38mpd 15 . 2  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( G
" W ) )
4015, 39sseldd 3433 1  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( 1st `  ( T `  M
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   {crab 2741    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {csn 3968   U.cuni 4198   U_ciun 4278    |-> cmpt 4461    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835    |` cres 4836   "cima 4837   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   0cc0 9539   1c1 9540    - cmin 9860    / cdiv 10269   NNcn 10609   (,)cioo 11635   [,]cicc 11638   ...cfz 11784   ↾t crest 15319   topGenctg 15336    Cn ccn 20240   Homeochmeo 20768   IIcii 21907   CovMap ccvm 29978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-top 19921  df-bases 19922  df-topon 19923  df-cn 20243  df-ii 21909
This theorem is referenced by:  cvmliftlem6  30013  cvmliftlem8  30015  cvmliftlem9  30016
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