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Theorem cvmliftpht 27222
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 20581 . . . . . 6  |-  ( G (  ~=ph  `  J ) H  <->  ( G  e.  ( II  Cn  J
)  /\  H  e.  ( II  Cn  J
)  /\  ( G
( PHtpy `  J ) H )  =/=  (/) ) )
64, 5sylib 196 . . . . 5  |-  ( ph  ->  ( G  e.  ( II  Cn  J )  /\  H  e.  ( II  Cn  J )  /\  ( G (
PHtpy `  J ) H )  =/=  (/) ) )
76simp1d 1000 . . . 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 27205 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
1110simp1d 1000 . 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 1001 . . . 4  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
14 phtpc01 20583 . . . . . . 7  |-  ( G (  ~=ph  `  J ) H  ->  ( ( G `  0 )  =  ( H ` 
0 )  /\  ( G `  1 )  =  ( H ` 
1 ) ) )
154, 14syl 16 . . . . . 6  |-  ( ph  ->  ( ( G ` 
0 )  =  ( H `  0 )  /\  ( G ` 
1 )  =  ( H `  1 ) ) )
1615simpld 459 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  ( H `
 0 ) )
179, 16eqtrd 2475 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
181, 12, 3, 13, 8, 17cvmliftiota 27205 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1918simp1d 1000 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
206simp3d 1002 . . . 4  |-  ( ph  ->  ( G ( PHtpy `  J ) H )  =/=  (/) )
21 n0 3661 . . . 4  |-  ( ( G ( PHtpy `  J
) H )  =/=  (/) 
<->  E. g  g  e.  ( G ( PHtpy `  J ) H ) )
2220, 21sylib 196 . . 3  |-  ( ph  ->  E. g  g  e.  ( G ( PHtpy `  J ) H ) )
233adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  F  e.  ( C CovMap  J ) )
247, 13phtpycn 20570 . . . . . . 7  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2524sselda 3371 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  g  e.  ( ( II  tX  II )  Cn  J ) )
268adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  P  e.  B
)
279adantr 465 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( G `  0 ) )
28 0elunit 11418 . . . . . . . . 9  |-  0  e.  ( 0 [,] 1
)
297adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  G  e.  ( II  Cn  J ) )
3013adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  H  e.  ( II  Cn  J ) )
31 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  g  e.  ( G ( PHtpy `  J
) H ) )
3229, 30, 31phtpyi 20571 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  0  e.  ( 0 [,] 1
) )  ->  (
( 0 g 0 )  =  ( G `
 0 )  /\  ( 1 g 0 )  =  ( G `
 1 ) ) )
3328, 32mpan2 671 . . . . . . . 8  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( ( 0 g 0 )  =  ( G `  0
)  /\  ( 1 g 0 )  =  ( G `  1
) ) )
3433simpld 459 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( 0 g 0 )  =  ( G `  0 ) )
3527, 34eqtr4d 2478 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( 0 g 0 ) )
361, 23, 25, 26, 35cvmlift2 27220 . . . . 5  |-  ( (
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 2952 . . . . 5  |-  ( 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 16 . . . 4  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  E. h  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  h
)  =  g  /\  ( 0 h 0 )  =  P ) )
393ad2antrr 725 . . . . . 6  |-  ( ( ( 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 725 . . . . . 6  |-  ( ( ( 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 725 . . . . . 6  |-  ( ( ( 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 725 . . . . . 6  |-  ( ( ( 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 725 . . . . . 6  |-  ( ( ( 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 754 . . . . . 6  |-  ( ( ( 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 755 . . . . . 6  |-  ( ( ( 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 763 . . . . . 6  |-  ( ( ( 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 764 . . . . . 6  |-  ( ( ( 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 27221 . . . . 5  |-  ( ( ( 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 3658 . . . . 5  |-  ( h  e.  ( M (
PHtpy `  C ) N )  ->  ( M
( PHtpy `  C ) N )  =/=  (/) )
5048, 49syl 16 . . . 4  |-  ( ( ( 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 )  =/=  (/) )
5138, 50rexlimddv 2860 . . 3  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( M (
PHtpy `  C ) N )  =/=  (/) )
5222, 51exlimddv 1692 . 2  |-  ( ph  ->  ( M ( PHtpy `  C ) N )  =/=  (/) )
53 isphtpc 20581 . 2  |-  ( M (  ~=ph  `  C ) N  <->  ( M  e.  ( II  Cn  C
)  /\  N  e.  ( II  Cn  C
)  /\  ( M
( PHtpy `  C ) N )  =/=  (/) ) )
5411, 19, 52, 53syl3anbrc 1172 1  |-  ( ph  ->  M (  ~=ph  `  C
) N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2620   E.wrex 2731   E!wreu 2732   (/)c0 3652   U.cuni 4106   class class class wbr 4307    o. ccom 4859   ` cfv 5433   iota_crio 6066  (class class class)co 6106   0cc0 9297   1c1 9298   [,]cicc 11318    Cn ccn 18843    tX ctx 19148   IIcii 20466   PHtpycphtpy 20555    ~=ph cphtpc 20556   CovMap ccvm 27159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376  ax-mulf 9377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-supp 6706  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-ec 7118  df-map 7231  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fsupp 7636  df-fi 7676  df-sup 7706  df-oi 7739  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-q 10969  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-ioo 11319  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-fl 11657  df-seq 11822  df-exp 11881  df-hash 12119  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-clim 12981  df-sum 13179  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-sca 14269  df-vsca 14270  df-ip 14271  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-hom 14277  df-cco 14278  df-rest 14376  df-topn 14377  df-0g 14395  df-gsum 14396  df-topgen 14397  df-pt 14398  df-prds 14401  df-xrs 14455  df-qtop 14460  df-imas 14461  df-xps 14463  df-mre 14539  df-mrc 14540  df-acs 14542  df-mnd 15430  df-submnd 15480  df-mulg 15563  df-cntz 15850  df-cmn 16294  df-psmet 17824  df-xmet 17825  df-met 17826  df-bl 17827  df-mopn 17828  df-cnfld 17834  df-top 18518  df-bases 18520  df-topon 18521  df-topsp 18522  df-cld 18638  df-ntr 18639  df-cls 18640  df-nei 18717  df-cn 18846  df-cnp 18847  df-cmp 19005  df-con 19031  df-lly 19085  df-nlly 19086  df-tx 19150  df-hmeo 19343  df-xms 19910  df-ms 19911  df-tms 19912  df-ii 20468  df-htpy 20557  df-phtpy 20558  df-phtpc 20579  df-pcon 27125  df-scon 27126  df-cvm 27160
This theorem is referenced by:  cvmlift3lem1  27223
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