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Theorem cvmliftpht 25005
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 19019 . . . . . 6  |-  ( G (  ~=ph  `  J ) H  <->  ( G  e.  ( II  Cn  J
)  /\  H  e.  ( II  Cn  J
)  /\  ( G
( PHtpy `  J ) H )  =/=  (/) ) )
64, 5sylib 189 . . . . 5  |-  ( ph  ->  ( G  e.  ( II  Cn  J )  /\  H  e.  ( II  Cn  J )  /\  ( G (
PHtpy `  J ) H )  =/=  (/) ) )
76simp1d 969 . . . 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 24988 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
1110simp1d 969 . 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 970 . . . 4  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
14 phtpc01 19021 . . . . . . 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 446 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  ( H `
 0 ) )
179, 16eqtrd 2468 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
181, 12, 3, 13, 8, 17cvmliftiota 24988 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1918simp1d 969 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
206simp3d 971 . . . 4  |-  ( ph  ->  ( G ( PHtpy `  J ) H )  =/=  (/) )
21 n0 3637 . . . 4  |-  ( ( G ( PHtpy `  J
) H )  =/=  (/) 
<->  E. g  g  e.  ( G ( PHtpy `  J ) H ) )
2220, 21sylib 189 . . 3  |-  ( ph  ->  E. g  g  e.  ( G ( PHtpy `  J ) H ) )
233adantr 452 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  F  e.  ( C CovMap  J ) )
247, 13phtpycn 19008 . . . . . . 7  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2524sselda 3348 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  g  e.  ( ( II  tX  II )  Cn  J ) )
268adantr 452 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  P  e.  B
)
279adantr 452 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( G `  0 ) )
28 0elunit 11015 . . . . . . . . 9  |-  0  e.  ( 0 [,] 1
)
297adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  G  e.  ( II  Cn  J ) )
3013adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  H  e.  ( II  Cn  J ) )
31 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  g  e.  ( G ( PHtpy `  J
) H ) )
3229, 30, 31phtpyi 19009 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  0  e.  ( 0 [,] 1
) )  ->  (
( 0 g 0 )  =  ( G `
 0 )  /\  ( 1 g 0 )  =  ( G `
 1 ) ) )
3328, 32mpan2 653 . . . . . . . 8  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( ( 0 g 0 )  =  ( G `  0
)  /\  ( 1 g 0 )  =  ( G `  1
) ) )
3433simpld 446 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( 0 g 0 )  =  ( G `  0 ) )
3527, 34eqtr4d 2471 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( 0 g 0 ) )
361, 23, 25, 26, 35cvmlift2 25003 . . . . 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 2922 . . . . 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 707 . . . . . 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 707 . . . . . 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 707 . . . . . 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 707 . . . . . 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 707 . . . . . 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 732 . . . . . 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 733 . . . . . 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 741 . . . . . 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 742 . . . . . 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 25004 . . . . 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 3634 . . . . 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 2834 . . 3  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( M (
PHtpy `  C ) N )  =/=  (/) )
5222, 51exlimddv 1648 . 2  |-  ( ph  ->  ( M ( PHtpy `  C ) N )  =/=  (/) )
53 isphtpc 19019 . 2  |-  ( M (  ~=ph  `  C ) N  <->  ( M  e.  ( II  Cn  C
)  /\  N  e.  ( II  Cn  C
)  /\  ( M
( PHtpy `  C ) N )  =/=  (/) ) )
5411, 19, 52, 53syl3anbrc 1138 1  |-  ( ph  ->  M (  ~=ph  `  C
) N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   E!wreu 2707   (/)c0 3628   U.cuni 4015   class class class wbr 4212    o. ccom 4882   ` cfv 5454  (class class class)co 6081   iota_crio 6542   0cc0 8990   1c1 8991   [,]cicc 10919    Cn ccn 17288    tX ctx 17592   IIcii 18905   PHtpycphtpy 18993    ~=ph cphtpc 18994   CovMap ccvm 24942
This theorem is referenced by:  cvmlift3lem1  25006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-ec 6907  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-cn 17291  df-cnp 17292  df-cmp 17450  df-con 17475  df-lly 17529  df-nlly 17530  df-tx 17594  df-hmeo 17787  df-xms 18350  df-ms 18351  df-tms 18352  df-ii 18907  df-htpy 18995  df-phtpy 18996  df-phtpc 19017  df-pcon 24908  df-scon 24909  df-cvm 24943
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