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Theorem cvmliftpht 28952
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 21579 . . . . . 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 1006 . . . 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 28935 . . 3  |-  ( ph  ->  ( M  e.  ( II  Cn  C )  /\  ( F  o.  M )  =  G  /\  ( M ` 
0 )  =  P ) )
1110simp1d 1006 . 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 1007 . . . 4  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
14 phtpc01 21581 . . . . . . 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 457 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  ( H `
 0 ) )
179, 16eqtrd 2423 . . . 4  |-  ( ph  ->  ( F `  P
)  =  ( H `
 0 ) )
181, 12, 3, 13, 8, 17cvmliftiota 28935 . . 3  |-  ( ph  ->  ( N  e.  ( II  Cn  C )  /\  ( F  o.  N )  =  H  /\  ( N ` 
0 )  =  P ) )
1918simp1d 1006 . 2  |-  ( ph  ->  N  e.  ( II 
Cn  C ) )
206simp3d 1008 . . . 4  |-  ( ph  ->  ( G ( PHtpy `  J ) H )  =/=  (/) )
21 n0 3721 . . . 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 463 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  F  e.  ( C CovMap  J ) )
247, 13phtpycn 21568 . . . . . . 7  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2524sselda 3417 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  g  e.  ( ( II  tX  II )  Cn  J ) )
268adantr 463 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  P  e.  B
)
279adantr 463 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( G `  0 ) )
28 0elunit 11559 . . . . . . . . 9  |-  0  e.  ( 0 [,] 1
)
297adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  G  e.  ( II  Cn  J ) )
3013adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  H  e.  ( II  Cn  J ) )
31 simpr 459 . . . . . . . . . 10  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  g  e.  ( G ( PHtpy `  J
) H ) )
3229, 30, 31phtpyi 21569 . . . . . . . . 9  |-  ( ( ( ph  /\  g  e.  ( G ( PHtpy `  J ) H ) )  /\  0  e.  ( 0 [,] 1
) )  ->  (
( 0 g 0 )  =  ( G `
 0 )  /\  ( 1 g 0 )  =  ( G `
 1 ) ) )
3328, 32mpan2 669 . . . . . . . 8  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( ( 0 g 0 )  =  ( G `  0
)  /\  ( 1 g 0 )  =  ( G `  1
) ) )
3433simpld 457 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( 0 g 0 )  =  ( G `  0 ) )
3527, 34eqtr4d 2426 . . . . . 6  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( F `  P )  =  ( 0 g 0 ) )
361, 23, 25, 26, 35cvmlift2 28950 . . . . 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 2999 . . . . 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 723 . . . . . 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 723 . . . . . 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 723 . . . . . 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 723 . . . . . 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 723 . . . . . 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 753 . . . . . 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 754 . . . . . 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 28951 . . . . 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 3717 . . . . 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 2878 . . 3  |-  ( (
ph  /\  g  e.  ( G ( PHtpy `  J
) H ) )  ->  ( M (
PHtpy `  C ) N )  =/=  (/) )
5222, 51exlimddv 1734 . 2  |-  ( ph  ->  ( M ( PHtpy `  C ) N )  =/=  (/) )
53 isphtpc 21579 . 2  |-  ( M (  ~=ph  `  C ) N  <->  ( M  e.  ( II  Cn  C
)  /\  N  e.  ( II  Cn  C
)  /\  ( M
( PHtpy `  C ) N )  =/=  (/) ) )
5411, 19, 52, 53syl3anbrc 1178 1  |-  ( ph  ->  M (  ~=ph  `  C
) N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826    =/= wne 2577   E.wrex 2733   E!wreu 2734   (/)c0 3711   U.cuni 4163   class class class wbr 4367    o. ccom 4917   ` cfv 5496   iota_crio 6157  (class class class)co 6196   0cc0 9403   1c1 9404   [,]cicc 11453    Cn ccn 19811    tX ctx 20146   IIcii 21464   PHtpycphtpy 21553    ~=ph cphtpc 21554   CovMap ccvm 28889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-ec 7231  df-map 7340  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-sum 13511  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-cn 19814  df-cnp 19815  df-cmp 19973  df-con 19998  df-lly 20052  df-nlly 20053  df-tx 20148  df-hmeo 20341  df-xms 20908  df-ms 20909  df-tms 20910  df-ii 21466  df-htpy 21555  df-phtpy 21556  df-phtpc 21577  df-pcon 28855  df-scon 28856  df-cvm 28890
This theorem is referenced by:  cvmlift3lem1  28953
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