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Theorem cvmliftlem3 24966
Description: Lemma for cvmlift 24978. 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 6080 . . . . . . . . 9  |-  ( k  =  M  ->  (
k  -  1 )  =  ( M  - 
1 ) )
54oveq1d 6088 . . . . . . . 8  |-  ( k  =  M  ->  (
( k  -  1 )  /  N )  =  ( ( M  -  1 )  /  N ) )
6 oveq1 6080 . . . . . . . 8  |-  ( k  =  M  ->  (
k  /  N )  =  ( M  /  N ) )
75, 6oveq12d 6091 . . . . . . 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 2485 . . . . . 6  |-  ( k  =  M  ->  (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) )  =  W )
109imaeq2d 5195 . . . . 5  |-  ( k  =  M  ->  ( G " ( ( ( k  -  1 )  /  N ) [,] ( k  /  N
) ) )  =  ( G " W
) )
11 fveq2 5720 . . . . . 6  |-  ( k  =  M  ->  ( T `  k )  =  ( T `  M ) )
1211fveq2d 5724 . . . . 5  |-  ( k  =  M  ->  ( 1st `  ( T `  k ) )  =  ( 1st `  ( T `  M )
) )
1310, 12sseq12d 3369 . . . 4  |-  ( k  =  M  ->  (
( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) )  <->  ( G " W )  C_  ( 1st `  ( T `  M ) ) ) )
1413rspcv 3040 . . 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 18903 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
19 cvmliftlem.x . . . . . . . 8  |-  X  = 
U. J
2018, 19cnf 17302 . . . . . . 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 5585 . . . . 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 24965 . . . . 5  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
34 fdm 5587 . . . . . 6  |-  ( G : ( 0 [,] 1 ) --> X  ->  dom  G  =  ( 0 [,] 1 ) )
3522, 34syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  dom  G  =  ( 0 [,] 1 ) )
3633, 35sseqtr4d 3377 . . . 4  |-  ( (
ph  /\  ps )  ->  W  C_  dom  G )
37 funfvima2 5966 . . . 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 3341 1  |-  ( (
ph  /\  ps )  ->  ( G `  A
)  e.  ( 1st `  ( T `  M
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   U.cuni 4007   U_ciun 4085    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   0cc0 8982   1c1 8983    - cmin 9283    / cdiv 9669   NNcn 9992   (,)cioo 10908   [,]cicc 10911   ...cfz 11035   ↾t crest 13640   topGenctg 13657    Cn ccn 17280    Homeo chmeo 17777   IIcii 18897   CovMap ccvm 24934
This theorem is referenced by:  cvmliftlem6  24969  cvmliftlem8  24971  cvmliftlem9  24972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-icc 10915  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-cn 17283  df-ii 18899
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