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

Theorem cvmliftpht 24133
Description: If  G and  H are path-homotopic, then their lifts  M and  N are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
cvmliftpht.b  |-  B  = 
U. C
cvmliftpht.m  |-  M  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
cvmliftpht.n  |-  N  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )
cvmliftpht.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmliftpht.p  |-  ( ph  ->  P  e.  B )
cvmliftpht.e  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
cvmliftpht.g  |-  ( ph  ->  G (  ~=ph  `  J
) H )
Assertion
Ref Expression
cvmliftpht  |-  ( ph  ->  M (  ~=ph  `  C
) N )
Distinct variable groups:    B, f    f, F    f, J    C, f    f, G    f, H    P, f
Allowed substitution hints:    ph( f)    M( f)    N( f)

Proof of Theorem cvmliftpht
Dummy variables  h  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmliftpht.b . . . 4  |-  B  = 
U. C
2 cvmliftpht.m . . . 4  |-  M  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
3 cvmliftpht.f . . . 4  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
4 cvmliftpht.g . . . . . 6  |-  ( ph  ->  G (  ~=ph  `  J
) H )
5 isphtpc 18545 . . . . . 6  |-  ( G (  ~=ph  `  J ) H  <->  ( G  e.  ( II  Cn  J
)  /\  H  e.  ( II  Cn  J
)  /\  ( G
( PHtpy `  J ) H )  =/=  (/) ) )
64, 5sylib 188 . . . . 5  |-  ( ph  ->  ( G  e.  ( II  Cn  J )  /\  H  e.  ( II  Cn  J )  /\  ( G (
PHtpy `  J ) H )  =/=  (/) ) )
76simp1d 967 . . . 4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
8 cvmliftpht.p . . . 4  |-  ( ph  ->  P  e.  B )
9 cvmliftpht.e . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
101, 2, 3, 7, 8, 9cvmliftiota 24116 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
1110simp1d 967 . 2  |-  ( ph  ->  M  e.  ( II 
Cn  C ) )
12 cvmliftpht.n . . . 4  |-  N  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  H  /\  ( f ` 
0 )  =  P ) )
136simp2d 968 . . . 4  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
14 phtpc01 18547 . . . . . . 7  |-  ( G (  ~=ph  `  J ) H  ->  ( ( G `  0 )  =  ( H ` 
0 )  /\  ( G `  1 )  =  ( H ` 
1 ) ) )
154, 14syl 15 . . . . . 6  |-  ( ph  ->  ( ( G ` 
0 )  =  ( H `  0 )  /\  ( G ` 
1 )  =  ( H `  1 ) ) )
1615simpld 445 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  ( H `
 0 ) )
179, 16eqtrd 2348 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
181, 12, 3, 13, 8, 17cvmliftiota 24116 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1918simp1d 967 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
206simp3d 969 . . . 4  |-  ( ph  ->  ( G ( PHtpy `  J ) H )  =/=  (/) )
21 n0 3498 . . . 4  |-  ( ( G ( PHtpy `  J
) H )  =/=  (/) 
<->  E. g  g  e.  ( G ( PHtpy `  J ) H ) )
2220, 21sylib 188 . . 3  |-  ( ph  ->  E. g  g  e.  ( G ( PHtpy `  J ) H ) )
233adantr 451 . . . . . . . 8  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  F  e.  ( C CovMap  J ) )
247, 13phtpycn 18534 . . . . . . . . 9  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2524sselda 3214 . . . . . . . 8  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  g  e.  ( ( II  tX  II )  Cn  J ) )
268adantr 451 . . . . . . . 8  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  P  e.  B
)
279adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( G `  0 ) )
28 0elunit 10801 . . . . . . . . . . 11  |-  0  e.  ( 0 [,] 1
)
297adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  G  e.  ( II  Cn  J ) )
3013adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  H  e.  ( II  Cn  J ) )
31 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  g  e.  ( G ( PHtpy `  J
) H ) )
3229, 30, 31phtpyi 18535 . . . . . . . . . . 11  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  0  e.  ( 0 [,] 1
) )  ->  (
( 0 g 0 )  =  ( G `
 0 )  /\  ( 1 g 0 )  =  ( G `
 1 ) ) )
3328, 32mpan2 652 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( ( 0 g 0 )  =  ( G `  0
)  /\  ( 1 g 0 )  =  ( G `  1
) ) )
3433simpld 445 . . . . . . . . 9  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( 0 g 0 )  =  ( G `  0 ) )
3527, 34eqtr4d 2351 . . . . . . . 8  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( 0 g 0 ) )
361, 23, 25, 26, 35cvmlift2 24131 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  E! h  e.  ( ( II  tX  II )  Cn  C
) ( ( F  o.  h )  =  g  /\  ( 0 h 0 )  =  P ) )
37 reurex 2788 . . . . . . 7  |-  ( E! h  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  h )  =  g  /\  (
0 h 0 )  =  P )  ->  E. h  e.  (
( II  tX  II )  Cn  C ) ( ( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) )
3836, 37syl 15 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  E. h  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) )
393ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  F  e.  ( C CovMap  J ) )
408ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  P  e.  B )
419ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  ( F `  P )  =  ( G ` 
0 ) )
427ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  G  e.  ( II  Cn  J
) )
4313ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  H  e.  ( II  Cn  J
) )
44 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  g  e.  ( G ( PHtpy `  J ) H ) )
45 simprl 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  h  e.  ( ( II  tX  II )  Cn  C
) )
46 simprrl 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  ( F  o.  h )  =  g )
47 simprrr 741 . . . . . . . . . 10  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  (
0 h 0 )  =  P )
481, 2, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47cvmliftphtlem 24132 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  h  e.  ( M ( PHtpy `  C ) N ) )
49 ne0i 3495 . . . . . . . . 9  |-  ( h  e.  ( M (
PHtpy `  C ) N )  ->  ( M
( PHtpy `  C ) N )  =/=  (/) )
5048, 49syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  ( h  e.  ( ( II 
tX  II )  Cn  C )  /\  (
( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) ) )  ->  ( M ( PHtpy `  C
) N )  =/=  (/) )
5150expr 598 . . . . . . 7  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  h  e.  ( ( II  tX  II )  Cn  C
) )  ->  (
( ( F  o.  h )  =  g  /\  ( 0 h 0 )  =  P )  ->  ( M
( PHtpy `  C ) N )  =/=  (/) ) )
5251rexlimdva 2701 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( E. h  e.  ( ( II  tX  II )  Cn  C
) ( ( F  o.  h )  =  g  /\  ( 0 h 0 )  =  P )  ->  ( M ( PHtpy `  C
) N )  =/=  (/) ) )
5338, 52mpd 14 . . . . 5  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( M (
PHtpy `  C ) N )  =/=  (/) )
5453ex 423 . . . 4  |-  ( ph  ->  ( g  e.  ( G ( PHtpy `  J
) H )  -> 
( M ( PHtpy `  C ) N )  =/=  (/) ) )
5554exlimdv 1627 . . 3  |-  ( ph  ->  ( E. g  g  e.  ( G (
PHtpy `  J ) H )  ->  ( M
( PHtpy `  C ) N )  =/=  (/) ) )
5622, 55mpd 14 . 2  |-  ( ph  ->  ( M ( PHtpy `  C ) N )  =/=  (/) )
57 isphtpc 18545 . 2  |-  ( M (  ~=ph  `  C ) N  <->  ( M  e.  ( II  Cn  C
)  /\  N  e.  ( II  Cn  C
)  /\  ( M
( PHtpy `  C ) N )  =/=  (/) ) )
5811, 19, 56, 57syl3anbrc 1136 1  |-  ( ph  ->  M (  ~=ph  `  C
) N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578   E!wreu 2579   (/)c0 3489   U.cuni 3864   class class class wbr 4060    o. ccom 4730   ` cfv 5292  (class class class)co 5900   iota_crio 6339   0cc0 8782   1c1 8783   [,]cicc 10706    Cn ccn 17010    tX ctx 17311   IIcii 18431   PHtpycphtpy 18519    ~=ph cphtpc 18520   CovMap ccvm 24070
This theorem is referenced by:  cvmlift3lem1  24134
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
  Copyright terms: Public domain W3C validator