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Theorem cvmliftpht 28595
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 21360 . . . . . 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 1008 . . . 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 28578 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
1110simp1d 1008 . 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 1009 . . . 4  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
14 phtpc01 21362 . . . . . . 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 2508 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
181, 12, 3, 13, 8, 17cvmliftiota 28578 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1918simp1d 1008 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
206simp3d 1010 . . . 4  |-  ( ph  ->  ( G ( PHtpy `  J ) H )  =/=  (/) )
21 n0 3799 . . . 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 21349 . . . . . . 7  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2524sselda 3509 . . . . . 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 11650 . . . . . . . . 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 21350 . . . . . . . . 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 2511 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( 0 g 0 ) )
361, 23, 25, 26, 35cvmlift2 28593 . . . . 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 3083 . . . . 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 28594 . . . . 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 3796 . . . . 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 2963 . . 3  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( M (
PHtpy `  C ) N )  =/=  (/) )
5222, 51exlimddv 1702 . 2  |-  ( ph  ->  ( M ( PHtpy `  C ) N )  =/=  (/) )
53 isphtpc 21360 . 2  |-  ( M (  ~=ph  `  C ) N  <->  ( M  e.  ( II  Cn  C
)  /\  N  e.  ( II  Cn  C
)  /\  ( M
( PHtpy `  C ) N )  =/=  (/) ) )
5411, 19, 52, 53syl3anbrc 1180 1  |-  ( ph  ->  M (  ~=ph  `  C
) N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2818   E!wreu 2819   (/)c0 3790   U.cuni 4251   class class class wbr 4453    o. ccom 5009   ` cfv 5594   iota_crio 6255  (class class class)co 6295   0cc0 9504   1c1 9505   [,]cicc 11544    Cn ccn 19591    tX ctx 19927   IIcii 21245   PHtpycphtpy 21334    ~=ph cphtpc 21335   CovMap ccvm 28532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-rest 14694  df-topn 14695  df-0g 14713  df-gsum 14714  df-topgen 14715  df-pt 14716  df-prds 14719  df-xrs 14773  df-qtop 14778  df-imas 14779  df-xps 14781  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-submnd 15839  df-mulg 15931  df-cntz 16226  df-cmn 16671  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-cn 19594  df-cnp 19595  df-cmp 19753  df-con 19779  df-lly 19833  df-nlly 19834  df-tx 19929  df-hmeo 20122  df-xms 20689  df-ms 20690  df-tms 20691  df-ii 21247  df-htpy 21336  df-phtpy 21337  df-phtpc 21358  df-pcon 28498  df-scon 28499  df-cvm 28533
This theorem is referenced by:  cvmlift3lem1  28596
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