Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift2 Structured version   Unicode version

Theorem cvmlift2 29827
Description: A two-dimensional version of cvmlift 29810. 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 5013 . . . . 5  |-  ( h  =  g  ->  ( F  o.  h )  =  ( F  o.  g ) )
7 oveq1 6312 . . . . . . 7  |-  ( w  =  z  ->  (
w G 0 )  =  ( z G 0 ) )
87cbvmptv 4518 . . . . . 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 5880 . . . . 5  |-  ( h  =  g  ->  (
h `  0 )  =  ( g ` 
0 ) )
1211eqeq1d 2431 . . . 4  |-  ( h  =  g  ->  (
( h `  0
)  =  P  <->  ( g `  0 )  =  P ) )
1310, 12anbi12d 715 . . 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 6278 . 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 5013 . . . . . . . 8  |-  ( k  =  g  ->  ( F  o.  k )  =  ( F  o.  g ) )
16 oveq2 6313 . . . . . . . . . 10  |-  ( w  =  z  ->  (
u G w )  =  ( u G z ) )
1716cbvmptv 4518 . . . . . . . . 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 5880 . . . . . . . 8  |-  ( k  =  g  ->  (
k `  0 )  =  ( g ` 
0 ) )
2120eqeq1d 2431 . . . . . . 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 715 . . . . . 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 6278 . . . . 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 6312 . . . . . . . . 9  |-  ( u  =  x  ->  (
u G z )  =  ( x G z ) )
2524mpteq2dv 4513 . . . . . . . 8  |-  ( u  =  x  ->  (
z  e.  ( 0 [,] 1 )  |->  ( u G z ) )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) )
2625eqeq2d 2443 . . . . . . 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 5881 . . . . . . . 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 2443 . . . . . . 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 715 . . . . . 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 6269 . . . . 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 2482 . . . 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 5883 . . 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 5881 . . 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 6385 . 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 29826 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 370    = wceq 1437    e. wcel 1870   E!wreu 2784   U.cuni 4222    |-> cmpt 4484    o. ccom 4858   ` cfv 5601   iota_crio 6266  (class class class)co 6305    |-> cmpt2 6307   0cc0 9538   1c1 9539   [,]cicc 11638    Cn ccn 20171    tX ctx 20506   IIcii 21803   CovMap ccvm 29766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ec 7373  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-cn 20174  df-cnp 20175  df-cmp 20333  df-con 20358  df-lly 20412  df-nlly 20413  df-tx 20508  df-hmeo 20701  df-xms 21266  df-ms 21267  df-tms 21268  df-ii 21805  df-htpy 21894  df-phtpy 21895  df-phtpc 21916  df-pcon 29732  df-scon 29733  df-cvm 29767
This theorem is referenced by:  cvmliftpht  29829
  Copyright terms: Public domain W3C validator