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Theorem cvmlift2 30111
Description: A two-dimensional version of cvmlift 30094. There is a unique lift of functions on the unit square 
II  tX  II which commutes with the covering map. (Contributed by Mario Carneiro, 1-Jun-2015.)
Hypotheses
Ref Expression
cvmlift2.b  |-  B  = 
U. C
cvmlift2.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift2.g  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
cvmlift2.p  |-  ( ph  ->  P  e.  B )
cvmlift2.i  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
Assertion
Ref Expression
cvmlift2  |-  ( ph  ->  E! f  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  f
)  =  G  /\  ( 0 f 0 )  =  P ) )
Distinct variable groups:    f, F    ph, f    f, J    f, G    C, f    P, f
Allowed substitution hint:    B( f)

Proof of Theorem cvmlift2
Dummy variables  g  h  k  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift2.b . 2  |-  B  = 
U. C
2 cvmlift2.f . 2  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
3 cvmlift2.g . 2  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
4 cvmlift2.p . 2  |-  ( ph  ->  P  e.  B )
5 cvmlift2.i . 2  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
6 coeq2 4998 . . . . 5  |-  ( h  =  g  ->  ( F  o.  h )  =  ( F  o.  g ) )
7 oveq1 6315 . . . . . . 7  |-  ( w  =  z  ->  (
w G 0 )  =  ( z G 0 ) )
87cbvmptv 4488 . . . . . 6  |-  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( z G 0 ) )
98a1i 11 . . . . 5  |-  ( h  =  g  ->  (
w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) )
106, 9eqeq12d 2486 . . . 4  |-  ( h  =  g  ->  (
( F  o.  h
)  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  <-> 
( F  o.  g
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) ) )
11 fveq1 5878 . . . . 5  |-  ( h  =  g  ->  (
h `  0 )  =  ( g ` 
0 ) )
1211eqeq1d 2473 . . . 4  |-  ( h  =  g  ->  (
( h `  0
)  =  P  <->  ( g `  0 )  =  P ) )
1310, 12anbi12d 725 . . 3  |-  ( h  =  g  ->  (
( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P )  <->  ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1
)  |->  ( z G 0 ) )  /\  ( g `  0
)  =  P ) ) )
1413cbvriotav 6281 . 2  |-  ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1
)  |->  ( w G 0 ) )  /\  ( h `  0
)  =  P ) )  =  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1
)  |->  ( z G 0 ) )  /\  ( g `  0
)  =  P ) )
15 coeq2 4998 . . . . . . . 8  |-  ( k  =  g  ->  ( F  o.  k )  =  ( F  o.  g ) )
16 oveq2 6316 . . . . . . . . . 10  |-  ( w  =  z  ->  (
u G w )  =  ( u G z ) )
1716cbvmptv 4488 . . . . . . . . 9  |-  ( w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( u G z ) )
1817a1i 11 . . . . . . . 8  |-  ( k  =  g  ->  (
w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( u G z ) ) )
1915, 18eqeq12d 2486 . . . . . . 7  |-  ( k  =  g  ->  (
( F  o.  k
)  =  ( w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  <-> 
( F  o.  g
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( u G z ) ) ) )
20 fveq1 5878 . . . . . . . 8  |-  ( k  =  g  ->  (
k `  0 )  =  ( g ` 
0 ) )
2120eqeq1d 2473 . . . . . . 7  |-  ( k  =  g  ->  (
( k `  0
)  =  ( (
iota_ h  e.  (
II  Cn  C )
( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  u )  <->  ( g `  0 )  =  ( ( iota_ h  e.  ( II  Cn  C
) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 ) 
|->  ( w G 0 ) )  /\  (
h `  0 )  =  P ) ) `  u ) ) )
2219, 21anbi12d 725 . . . . . 6  |-  ( k  =  g  ->  (
( ( F  o.  k )  =  ( w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  /\  ( k `
 0 )  =  ( ( iota_ h  e.  ( II  Cn  C
) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 ) 
|->  ( w G 0 ) )  /\  (
h `  0 )  =  P ) ) `  u ) )  <->  ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1
)  |->  ( u G z ) )  /\  ( g `  0
)  =  ( (
iota_ h  e.  (
II  Cn  C )
( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  u ) ) ) )
2322cbvriotav 6281 . . . . 5  |-  ( iota_ k  e.  ( II  Cn  C ) ( ( F  o.  k )  =  ( w  e.  ( 0 [,] 1
)  |->  ( u G w ) )  /\  ( k `  0
)  =  ( (
iota_ h  e.  (
II  Cn  C )
( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  u ) ) )  =  ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( u G z ) )  /\  (
g `  0 )  =  ( ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1
)  |->  ( w G 0 ) )  /\  ( h `  0
)  =  P ) ) `  u ) ) )
24 oveq1 6315 . . . . . . . . 9  |-  ( u  =  x  ->  (
u G z )  =  ( x G z ) )
2524mpteq2dv 4483 . . . . . . . 8  |-  ( u  =  x  ->  (
z  e.  ( 0 [,] 1 )  |->  ( u G z ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) )
2625eqeq2d 2481 . . . . . . 7  |-  ( u  =  x  ->  (
( F  o.  g
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( u G z ) )  <-> 
( F  o.  g
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) ) )
27 fveq2 5879 . . . . . . . 8  |-  ( u  =  x  ->  (
( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  u )  =  ( ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  x ) )
2827eqeq2d 2481 . . . . . . 7  |-  ( u  =  x  ->  (
( g `  0
)  =  ( (
iota_ h  e.  (
II  Cn  C )
( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  u )  <->  ( g `  0 )  =  ( ( iota_ h  e.  ( II  Cn  C
) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 ) 
|->  ( w G 0 ) )  /\  (
h `  0 )  =  P ) ) `  x ) ) )
2926, 28anbi12d 725 . . . . . 6  |-  ( u  =  x  ->  (
( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1 )  |->  ( u G z ) )  /\  ( g `
 0 )  =  ( ( iota_ h  e.  ( II  Cn  C
) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 ) 
|->  ( w G 0 ) )  /\  (
h `  0 )  =  P ) ) `  u ) )  <->  ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )  /\  ( g `  0
)  =  ( (
iota_ h  e.  (
II  Cn  C )
( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  x ) ) ) )
3029riotabidv 6272 . . . . 5  |-  ( u  =  x  ->  ( iota_ g  e.  ( II 
Cn  C ) ( ( F  o.  g
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( u G z ) )  /\  ( g ` 
0 )  =  ( ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  u ) ) )  =  ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
g `  0 )  =  ( ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1
)  |->  ( w G 0 ) )  /\  ( h `  0
)  =  P ) ) `  x ) ) ) )
3123, 30syl5eq 2517 . . . 4  |-  ( u  =  x  ->  ( iota_ k  e.  ( II 
Cn  C ) ( ( F  o.  k
)  =  ( w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  /\  ( k ` 
0 )  =  ( ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  u ) ) )  =  ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
g `  0 )  =  ( ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1
)  |->  ( w G 0 ) )  /\  ( h `  0
)  =  P ) ) `  x ) ) ) )
3231fveq1d 5881 . . 3  |-  ( u  =  x  ->  (
( iota_ k  e.  ( II  Cn  C ) ( ( F  o.  k )  =  ( w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  /\  ( k `
 0 )  =  ( ( iota_ h  e.  ( II  Cn  C
) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 ) 
|->  ( w G 0 ) )  /\  (
h `  0 )  =  P ) ) `  u ) ) ) `
 v )  =  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
g `  0 )  =  ( ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1
)  |->  ( w G 0 ) )  /\  ( h `  0
)  =  P ) ) `  x ) ) ) `  v
) )
33 fveq2 5879 . . 3  |-  ( v  =  y  ->  (
( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( g `
 0 )  =  ( ( iota_ h  e.  ( II  Cn  C
) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 ) 
|->  ( w G 0 ) )  /\  (
h `  0 )  =  P ) ) `  x ) ) ) `
 v )  =  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
g `  0 )  =  ( ( iota_ h  e.  ( II  Cn  C ) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1
)  |->  ( w G 0 ) )  /\  ( h `  0
)  =  P ) ) `  x ) ) ) `  y
) )
3432, 33cbvmpt2v 6390 . 2  |-  ( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  ( (
iota_ k  e.  (
II  Cn  C )
( ( F  o.  k )  =  ( w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  /\  ( k `
 0 )  =  ( ( iota_ h  e.  ( II  Cn  C
) ( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 ) 
|->  ( w G 0 ) )  /\  (
h `  0 )  =  P ) ) `  u ) ) ) `
 v ) )  =  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )  /\  ( g `  0
)  =  ( (
iota_ h  e.  (
II  Cn  C )
( ( F  o.  h )  =  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  /\  ( h `
 0 )  =  P ) ) `  x ) ) ) `
 y ) )
351, 2, 3, 4, 5, 14, 34cvmlift2lem13 30110 1  |-  ( ph  ->  E! f  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  f
)  =  G  /\  ( 0 f 0 )  =  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   E!wreu 2758   U.cuni 4190    |-> cmpt 4454    o. ccom 4843   ` cfv 5589   iota_crio 6269  (class class class)co 6308    |-> cmpt2 6310   0cc0 9557   1c1 9558   [,]cicc 11663    Cn ccn 20317    tX ctx 20652   IIcii 21985   CovMap ccvm 30050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-ec 7383  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-cmp 20479  df-con 20504  df-lly 20558  df-nlly 20559  df-tx 20654  df-hmeo 20847  df-xms 21413  df-ms 21414  df-tms 21415  df-ii 21987  df-htpy 22079  df-phtpy 22080  df-phtpc 22101  df-pcon 30016  df-scon 30017  df-cvm 30051
This theorem is referenced by:  cvmliftpht  30113
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