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Theorem cvmliftlem3 27188
Description: Lemma for cvmlift 27200. 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 465 . . 3  |-  ( (
ph  /\  ps )  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
4 oveq1 6110 . . . . . . . . 9  |-  ( k  =  M  ->  (
k  -  1 )  =  ( M  - 
1 ) )
54oveq1d 6118 . . . . . . . 8  |-  ( k  =  M  ->  (
( k  -  1 )  /  N )  =  ( ( M  -  1 )  /  N ) )
6 oveq1 6110 . . . . . . . 8  |-  ( k  =  M  ->  (
k  /  N )  =  ( M  /  N ) )
75, 6oveq12d 6121 . . . . . . 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 2493 . . . . . 6  |-  ( k  =  M  ->  (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) )  =  W )
109imaeq2d 5181 . . . . 5  |-  ( k  =  M  ->  ( G " ( ( ( k  -  1 )  /  N ) [,] ( k  /  N
) ) )  =  ( G " W
) )
11 fveq2 5703 . . . . . 6  |-  ( k  =  M  ->  ( T `  k )  =  ( T `  M ) )
1211fveq2d 5707 . . . . 5  |-  ( k  =  M  ->  ( 1st `  ( T `  k ) )  =  ( 1st `  ( T `  M )
) )
1310, 12sseq12d 3397 . . . 4  |-  ( k  =  M  ->  (
( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) )  <->  ( G " W )  C_  ( 1st `  ( T `  M ) ) ) )
1413rspcv 3081 . . 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 60 . 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 20469 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
19 cvmliftlem.x . . . . . . . 8  |-  X  = 
U. J
2018, 19cnf 18862 . . . . . . 7  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
2117, 20syl 16 . . . . . 6  |-  ( ph  ->  G : ( 0 [,] 1 ) --> X )
2221adantr 465 . . . . 5  |-  ( (
ph  /\  ps )  ->  G : ( 0 [,] 1 ) --> X )
23 ffun 5573 . . . . 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 27187 . . . . 5  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
34 fdm 5575 . . . . . 6  |-  ( G : ( 0 [,] 1 ) --> X  ->  dom  G  =  ( 0 [,] 1 ) )
3522, 34syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  dom  G  =  ( 0 [,] 1 ) )
3633, 35sseqtr4d 3405 . . . 4  |-  ( (
ph  /\  ps )  ->  W  C_  dom  G )
37 funfvima2 5965 . . . 4  |-  ( ( Fun  G  /\  W  C_ 
dom  G )  -> 
( A  e.  W  ->  ( G `  A
)  e.  ( G
" W ) ) )
3824, 36, 37syl2anc 661 . . 3  |-  ( (
ph  /\  ps )  ->  ( A  e.  W  ->  ( G `  A
)  e.  ( G
" W ) ) )
3916, 38mpd 15 . 2  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( G
" W ) )
4015, 39sseldd 3369 1  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( 1st `  ( T `  M
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727   {crab 2731    \ cdif 3337    i^i cin 3339    C_ wss 3340   (/)c0 3649   ~Pcpw 3872   {csn 3889   U.cuni 4103   U_ciun 4183    e. cmpt 4362    X. cxp 4850   `'ccnv 4851   dom cdm 4852   ran crn 4853    |` cres 4854   "cima 4855   Fun wfun 5424   -->wf 5426   ` cfv 5430  (class class class)co 6103   1stc1st 6587   0cc0 9294   1c1 9295    - cmin 9607    / cdiv 10005   NNcn 10334   (,)cioo 11312   [,]cicc 11315   ...cfz 11449   ↾t crest 14371   topGenctg 14388    Cn ccn 18840   Homeochmeo 19338   IIcii 20463   CovMap ccvm 27156
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-icc 11319  df-fz 11450  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-topgen 14394  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-top 18515  df-bases 18517  df-topon 18518  df-cn 18843  df-ii 20465
This theorem is referenced by:  cvmliftlem6  27191  cvmliftlem8  27193  cvmliftlem9  27194
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