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Theorem cvmlift2 23847
Description: A two-dimensional version of cvmlift 23830. 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 4842 . . . . 5  |-  ( h  =  g  ->  ( F  o.  h )  =  ( F  o.  g ) )
7 oveq1 5865 . . . . . . 7  |-  ( w  =  z  ->  (
w G 0 )  =  ( z G 0 ) )
87cbvmptv 4111 . . . . . 6  |-  ( w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( z G 0 ) )
98a1i 10 . . . . 5  |-  ( h  =  g  ->  (
w  e.  ( 0 [,] 1 )  |->  ( w G 0 ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) )
106, 9eqeq12d 2297 . . . 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 5524 . . . . 5  |-  ( h  =  g  ->  (
h `  0 )  =  ( g ` 
0 ) )
1211eqeq1d 2291 . . . 4  |-  ( h  =  g  ->  (
( h `  0
)  =  P  <->  ( g `  0 )  =  P ) )
1310, 12anbi12d 691 . . 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 6316 . 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 4842 . . . . . . . 8  |-  ( k  =  g  ->  ( F  o.  k )  =  ( F  o.  g ) )
16 oveq2 5866 . . . . . . . . . 10  |-  ( w  =  z  ->  (
u G w )  =  ( u G z ) )
1716cbvmptv 4111 . . . . . . . . 9  |-  ( w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( u G z ) )
1817a1i 10 . . . . . . . 8  |-  ( k  =  g  ->  (
w  e.  ( 0 [,] 1 )  |->  ( u G w ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( u G z ) ) )
1915, 18eqeq12d 2297 . . . . . . 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 5524 . . . . . . . 8  |-  ( k  =  g  ->  (
k `  0 )  =  ( g ` 
0 ) )
2120eqeq1d 2291 . . . . . . 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 691 . . . . . 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 6316 . . . . 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 5865 . . . . . . . . 9  |-  ( u  =  x  ->  (
u G z )  =  ( x G z ) )
2524mpteq2dv 4107 . . . . . . . 8  |-  ( u  =  x  ->  (
z  e.  ( 0 [,] 1 )  |->  ( u G z ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) )
2625eqeq2d 2294 . . . . . . 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 5525 . . . . . . . 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 2294 . . . . . . 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 691 . . . . . 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 6306 . . . . 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 2327 . . . 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 5527 . . 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 5525 . . 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 5926 . 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 23846 1  |-  ( ph  ->  E! f  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  f
)  =  G  /\  ( 0 f 0 )  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E!wreu 2545   U.cuni 3827    e. cmpt 4077    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297   0cc0 8737   1c1 8738   [,]cicc 10659    Cn ccn 16954    tX ctx 17255   IIcii 18379   CovMap ccvm 23786
This theorem is referenced by:  cvmliftpht  23849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-cmp 17114  df-con 17138  df-lly 17192  df-nlly 17193  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-ii 18381  df-htpy 18468  df-phtpy 18469  df-phtpc 18490  df-pcon 23752  df-scon 23753  df-cvm 23787
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