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Theorem cvmlift2 24131
Description: A two-dimensional version of cvmlift 24114. 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 4879 . . . . 5  |-  ( h  =  g  ->  ( F  o.  h )  =  ( F  o.  g ) )
7 oveq1 5907 . . . . . . 7  |-  ( w  =  z  ->  (
w G 0 )  =  ( z G 0 ) )
87cbvmptv 4148 . . . . . 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 2330 . . . 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 5562 . . . . 5  |-  ( h  =  g  ->  (
h `  0 )  =  ( g ` 
0 ) )
1211eqeq1d 2324 . . . 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 6358 . 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 4879 . . . . . . . 8  |-  ( k  =  g  ->  ( F  o.  k )  =  ( F  o.  g ) )
16 oveq2 5908 . . . . . . . . . 10  |-  ( w  =  z  ->  (
u G w )  =  ( u G z ) )
1716cbvmptv 4148 . . . . . . . . 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 2330 . . . . . . 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 5562 . . . . . . . 8  |-  ( k  =  g  ->  (
k `  0 )  =  ( g ` 
0 ) )
2120eqeq1d 2324 . . . . . . 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 6358 . . . . 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 5907 . . . . . . . . 9  |-  ( u  =  x  ->  (
u G z )  =  ( x G z ) )
2524mpteq2dv 4144 . . . . . . . 8  |-  ( u  =  x  ->  (
z  e.  ( 0 [,] 1 )  |->  ( u G z ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) )
2625eqeq2d 2327 . . . . . . 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 5563 . . . . . . . 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 2327 . . . . . . 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 6348 . . . . 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 2360 . . . 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 5565 . . 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 5563 . . 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 5968 . 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 24130 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 1633    e. wcel 1701   E!wreu 2579   U.cuni 3864    e. cmpt 4114    o. ccom 4730   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   iota_crio 6339   0cc0 8782   1c1 8783   [,]cicc 10706    Cn ccn 17010    tX ctx 17311   IIcii 18431   CovMap ccvm 24070
This theorem is referenced by:  cvmliftpht  24133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-ec 6704  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009  df-sum 12206  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-mulg 14541  df-cntz 14842  df-cmn 15140  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-cn 17013  df-cnp 17014  df-cmp 17170  df-con 17194  df-lly 17248  df-nlly 17249  df-tx 17313  df-hmeo 17502  df-xms 17937  df-ms 17938  df-tms 17939  df-ii 18433  df-htpy 18521  df-phtpy 18522  df-phtpc 18543  df-pcon 24036  df-scon 24037  df-cvm 24071
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