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Theorem sconpht2 23784
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 2296 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  |->  ( G `
 ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) )
54pcorevcl 18539 . . . . . . 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 15 . . . . . 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 967 . . . . 5  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) )  e.  ( II  Cn  J ) )
8 sconpht2.5 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
96simp2d 968 . . . . . 6  |-  ( ph  ->  ( ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ` 
0 )  =  ( G `  1 ) )
108, 9eqtr4d 2331 . . . . 5  |-  ( ph  ->  ( F `  1
)  =  ( ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) `  0 ) )
112, 7, 10pcocn 18531 . . . 4  |-  ( ph  ->  ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) )  e.  ( II  Cn  J ) )
122, 7pco0 18528 . . . . 5  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
0 )  =  ( F `  0 ) )
132, 7pco1 18529 . . . . . 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 969 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ` 
1 )  =  ( G `  0 ) )
1614, 15eqtr4d 2331 . . . . . 6  |-  ( ph  ->  ( F `  0
)  =  ( ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) `  1 ) )
1713, 16eqtr4d 2331 . . . . 5  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
1 )  =  ( F `  0 ) )
1812, 17eqtr4d 2331 . . . 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 23775 . . . 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 1182 . . 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 3666 . . . 4  |-  ( ph  ->  { ( ( F ( *p `  J
) ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ) `
 0 ) }  =  { ( F `
 0 ) } )
2221xpeq2d 4729 . . 3  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
0 ) } )  =  ( ( 0 [,] 1 )  X. 
{ ( F ` 
0 ) } ) )
2320, 22breqtrd 4063 . 2  |-  ( ph  ->  ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } ) )
24 eqid 2296 . . 3  |-  ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } )
254, 24, 2, 3, 14, 8pcophtb 18543 . 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 201 1  |-  ( ph  ->  F (  ~=ph  `  J
) G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    - cmin 9053   [,]cicc 10675    Cn ccn 16970   IIcii 18395    ~=ph cphtpc 18483   *pcpco 18514  SConcscon 23766
This theorem is referenced by:  cvmlift3lem1  23865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-ii 18397  df-htpy 18484  df-phtpy 18485  df-phtpc 18506  df-pco 18519  df-scon 23768
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