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Theorem sconpht2 29746
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 2420 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  |->  ( G `
 ( 1  -  x ) ) )  =  ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) )
54pcorevcl 21942 . . . . . . 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 17 . . . . . 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 1017 . . . . 5  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) )  e.  ( II  Cn  J ) )
8 sconpht2.5 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
96simp2d 1018 . . . . . 6  |-  ( ph  ->  ( ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ` 
0 )  =  ( G `  1 ) )
108, 9eqtr4d 2464 . . . . 5  |-  ( ph  ->  ( F `  1
)  =  ( ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) `  0 ) )
112, 7, 10pcocn 21934 . . . 4  |-  ( ph  ->  ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) )  e.  ( II  Cn  J ) )
122, 7pco0 21931 . . . . 5  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
0 )  =  ( F `  0 ) )
132, 7pco1 21932 . . . . . 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 1019 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ` 
1 )  =  ( G `  0 ) )
1614, 15eqtr4d 2464 . . . . . 6  |-  ( ph  ->  ( F `  0
)  =  ( ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) `  1 ) )
1713, 16eqtr4d 2464 . . . . 5  |-  ( ph  ->  ( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
1 )  =  ( F `  0 ) )
1812, 17eqtr4d 2464 . . . 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 29737 . . . 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 1264 . . 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 4005 . . . 4  |-  ( ph  ->  { ( ( F ( *p `  J
) ( x  e.  ( 0 [,] 1
)  |->  ( G `  ( 1  -  x
) ) ) ) `
 0 ) }  =  { ( F `
 0 ) } )
2221xpeq2d 4869 . . 3  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( ( F ( *p `  J ) ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) ) ) ` 
0 ) } )  =  ( ( 0 [,] 1 )  X. 
{ ( F ` 
0 ) } ) )
2320, 22breqtrd 4441 . 2  |-  ( ph  ->  ( F ( *p
`  J ) ( x  e.  ( 0 [,] 1 )  |->  ( G `  ( 1  -  x ) ) ) ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } ) )
24 eqid 2420 . . 3  |-  ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( F `
 0 ) } )
254, 24, 2, 3, 14, 8pcophtb 21946 . 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 213 1  |-  ( ph  ->  F (  ~=ph  `  J
) G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1867   {csn 3993   class class class wbr 4417    |-> cmpt 4475    X. cxp 4843   ` cfv 5592  (class class class)co 6296   0cc0 9528   1c1 9529    - cmin 9849   [,]cicc 11627    Cn ccn 20164   IIcii 21796    ~=ph cphtpc 21886   *pcpco 21917  SConcscon 29728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6917  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fsupp 7881  df-fi 7922  df-sup 7953  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ioo 11628  df-icc 11631  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-sca 15158  df-vsca 15159  df-ip 15160  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-hom 15166  df-cco 15167  df-rest 15273  df-topn 15274  df-0g 15292  df-gsum 15293  df-topgen 15294  df-pt 15295  df-prds 15298  df-xrs 15352  df-qtop 15357  df-imas 15358  df-xps 15360  df-mre 15436  df-mrc 15437  df-acs 15439  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-submnd 16527  df-mulg 16620  df-cntz 16915  df-cmn 17360  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cld 19958  df-cn 20167  df-cnp 20168  df-tx 20501  df-hmeo 20694  df-xms 21259  df-ms 21260  df-tms 21261  df-ii 21798  df-htpy 21887  df-phtpy 21888  df-phtpc 21909  df-pco 21922  df-scon 29730
This theorem is referenced by:  cvmlift3lem1  29827
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