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Theorem sconpht2 27139
Description: Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015.)
Hypotheses
Ref Expression
sconpht2.1  |-  ( ph  ->  J  e. SCon )
sconpht2.2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
sconpht2.3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
sconpht2.4  |-  ( ph  ->  ( F `  0
)  =  ( G `
 0 ) )
sconpht2.5  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
Assertion
Ref Expression
sconpht2  |-  ( ph  ->  F (  ~=ph  `  J
) G )

Proof of Theorem sconpht2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sconpht2.1 . . . 4  |-  ( ph  ->  J  e. SCon )
2 sconpht2.2 . . . . 5  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
3 sconpht2.3 . . . . . . 7  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
4 eqid 2443 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  |->  ( G `
 ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) )
54pcorevcl 20609 . . . . . . 7  |-  ( G  e.  ( II  Cn  J )  ->  (
( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) )  e.  ( II  Cn  J )  /\  ( ( x  e.  ( 0 [,] 1 )  |->  ( G `
 ( 1  -  x ) ) ) `
 0 )  =  ( G `  1
)  /\  ( (
x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) `  1 )  =  ( G ` 
0 ) ) )
63, 5syl 16 . . . . . 6  |-  ( ph  ->  ( ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) )  e.  ( II  Cn  J
)  /\  ( (
x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) `  0 )  =  ( G ` 
1 )  /\  (
( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) `  1
)  =  ( G `
 0 ) ) )
76simp1d 1000 . . . . 5  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) )  e.  ( II  Cn  J ) )
8 sconpht2.5 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
96simp2d 1001 . . . . . 6  |-  ( ph  ->  ( ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ` 
0 )  =  ( G `  1 ) )
108, 9eqtr4d 2478 . . . . 5  |-  ( ph  ->  ( F `  1
)  =  ( ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) `  0 ) )
112, 7, 10pcocn 20601 . . . 4  |-  ( ph  ->  ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) )  e.  ( II  Cn  J ) )
122, 7pco0 20598 . . . . 5  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
0 )  =  ( F `  0 ) )
132, 7pco1 20599 . . . . . 6  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
1 )  =  ( ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) `  1
) )
14 sconpht2.4 . . . . . . 7  |-  ( ph  ->  ( F `  0
)  =  ( G `
 0 ) )
156simp3d 1002 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ` 
1 )  =  ( G `  0 ) )
1614, 15eqtr4d 2478 . . . . . 6  |-  ( ph  ->  ( F `  0
)  =  ( ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) `  1 ) )
1713, 16eqtr4d 2478 . . . . 5  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
1 )  =  ( F `  0 ) )
1812, 17eqtr4d 2478 . . . 4  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
0 )  =  ( ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) ) `  1
) )
19 sconpht 27130 . . . 4  |-  ( ( J  e. SCon  /\  ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `
 ( 1  -  x ) ) ) )  e.  ( II 
Cn  J )  /\  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
0 )  =  ( ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) ) `  1
) )  ->  ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `
 ( 1  -  x ) ) ) ) (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( ( F ( *p `  J
) ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ) `
 0 ) } ) )
201, 11, 18, 19syl3anc 1218 . . 3  |-  ( ph  ->  ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `
 ( 1  -  x ) ) ) ) `  0 ) } ) )
2112sneqd 3901 . . . 4  |-  ( ph  ->  { ( ( F ( *p `  J
) ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ) `
 0 ) }  =  { ( F `
 0 ) } )
2221xpeq2d 4876 . . 3  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
0 ) } )  =  ( ( 0 [,] 1 )  X. 
{ ( F ` 
0 ) } ) )
2320, 22breqtrd 4328 . 2  |-  ( ph  ->  ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } ) )
24 eqid 2443 . . 3  |-  ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } )
254, 24, 2, 3, 14, 8pcophtb 20613 . 2  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ( 
~=ph  `  J ) ( ( 0 [,] 1
)  X.  { ( F `  0 ) } )  <->  F (  ~=ph  `  J ) G ) )
2623, 25mpbid 210 1  |-  ( ph  ->  F (  ~=ph  `  J
) G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   {csn 3889   class class class wbr 4304    e. cmpt 4362    X. cxp 4850   ` cfv 5430  (class class class)co 6103   0cc0 9294   1c1 9295    - cmin 9607   [,]cicc 11315    Cn ccn 18840   IIcii 20463    ~=ph cphtpc 20553   *pcpco 20584  SConcscon 27121
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-fi 7673  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-ioo 11316  df-icc 11319  df-fz 11450  df-fzo 11561  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-rest 14373  df-topn 14374  df-0g 14392  df-gsum 14393  df-topgen 14394  df-pt 14395  df-prds 14398  df-xrs 14452  df-qtop 14457  df-imas 14458  df-xps 14460  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-submnd 15477  df-mulg 15560  df-cntz 15847  df-cmn 16291  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-cnfld 17831  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-cld 18635  df-cn 18843  df-cnp 18844  df-tx 19147  df-hmeo 19340  df-xms 19907  df-ms 19908  df-tms 19909  df-ii 20465  df-htpy 20554  df-phtpy 20555  df-phtpc 20576  df-pco 20589  df-scon 27123
This theorem is referenced by:  cvmlift3lem1  27220
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