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Theorem cvmlift2 25004
Description: A two-dimensional version of cvmlift 24987. 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 5032 . . . . 5  |-  ( h  =  g  ->  ( F  o.  h )  =  ( F  o.  g ) )
7 oveq1 6089 . . . . . . 7  |-  ( w  =  z  ->  (
w G 0 )  =  ( z G 0 ) )
87cbvmptv 4301 . . . . . 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 2451 . . . 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 5728 . . . . 5  |-  ( h  =  g  ->  (
h `  0 )  =  ( g ` 
0 ) )
1211eqeq1d 2445 . . . 4  |-  ( h  =  g  ->  (
( h `  0
)  =  P  <->  ( g `  0 )  =  P ) )
1310, 12anbi12d 693 . . 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 6562 . 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 5032 . . . . . . . 8  |-  ( k  =  g  ->  ( F  o.  k )  =  ( F  o.  g ) )
16 oveq2 6090 . . . . . . . . . 10  |-  ( w  =  z  ->  (
u G w )  =  ( u G z ) )
1716cbvmptv 4301 . . . . . . . . 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 2451 . . . . . . 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 5728 . . . . . . . 8  |-  ( k  =  g  ->  (
k `  0 )  =  ( g ` 
0 ) )
2120eqeq1d 2445 . . . . . . 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 693 . . . . . 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 6562 . . . . 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 6089 . . . . . . . . 9  |-  ( u  =  x  ->  (
u G z )  =  ( x G z ) )
2524mpteq2dv 4297 . . . . . . . 8  |-  ( u  =  x  ->  (
z  e.  ( 0 [,] 1 )  |->  ( u G z ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) )
2625eqeq2d 2448 . . . . . . 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 5729 . . . . . . . 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 2448 . . . . . . 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 693 . . . . . 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 6552 . . . . 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 2481 . . . 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 5731 . . 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 5729 . . 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 6153 . 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 25003 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 360    = wceq 1653    e. wcel 1726   E!wreu 2708   U.cuni 4016    e. cmpt 4267    o. ccom 4883   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   iota_crio 6543   0cc0 8991   1c1 8992   [,]cicc 10920    Cn ccn 17289    tX ctx 17593   IIcii 18906   CovMap ccvm 24943
This theorem is referenced by:  cvmliftpht  25006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-ec 6908  df-map 7021  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ico 10923  df-icc 10924  df-fz 11045  df-fzo 11137  df-fl 11203  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-sum 12481  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-hom 13554  df-cco 13555  df-rest 13651  df-topn 13652  df-topgen 13668  df-pt 13669  df-prds 13672  df-xrs 13727  df-0g 13728  df-gsum 13729  df-qtop 13734  df-imas 13735  df-xps 13737  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-submnd 14740  df-mulg 14816  df-cntz 15117  df-cmn 15415  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-cnfld 16705  df-top 16964  df-bases 16966  df-topon 16967  df-topsp 16968  df-cld 17084  df-ntr 17085  df-cls 17086  df-nei 17163  df-cn 17292  df-cnp 17293  df-cmp 17451  df-con 17476  df-lly 17530  df-nlly 17531  df-tx 17595  df-hmeo 17788  df-xms 18351  df-ms 18352  df-tms 18353  df-ii 18908  df-htpy 18996  df-phtpy 18997  df-phtpc 19018  df-pcon 24909  df-scon 24910  df-cvm 24944
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