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Theorem phtpcer 21227
Description: Path homotopy is an equivalence relation. Proposition 1.2 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 6-Jul-2015.)
Assertion
Ref Expression
phtpcer  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )

Proof of Theorem phtpcer
Dummy variables  f 
g  u  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phtpcrel 21225 . . . 4  |-  Rel  (  ~=ph  `  J )
21a1i 11 . . 3  |-  ( T. 
->  Rel  (  ~=ph  `  J
) )
3 isphtpc 21226 . . . . . 6  |-  ( x (  ~=ph  `  J ) y  <->  ( x  e.  ( II  Cn  J
)  /\  y  e.  ( II  Cn  J
)  /\  ( x
( PHtpy `  J )
y )  =/=  (/) ) )
43simp2bi 1012 . . . . 5  |-  ( x (  ~=ph  `  J ) y  ->  y  e.  ( II  Cn  J
) )
53simp1bi 1011 . . . . 5  |-  ( x (  ~=ph  `  J ) y  ->  x  e.  ( II  Cn  J
) )
63simp3bi 1013 . . . . . . 7  |-  ( x (  ~=ph  `  J ) y  ->  ( x
( PHtpy `  J )
y )  =/=  (/) )
7 n0 3794 . . . . . . 7  |-  ( ( x ( PHtpy `  J
) y )  =/=  (/) 
<->  E. f  f  e.  ( x ( PHtpy `  J ) y ) )
86, 7sylib 196 . . . . . 6  |-  ( x (  ~=ph  `  J ) y  ->  E. f 
f  e.  ( x ( PHtpy `  J )
y ) )
95adantr 465 . . . . . . . 8  |-  ( ( x (  ~=ph  `  J
) y  /\  f  e.  ( x ( PHtpy `  J ) y ) )  ->  x  e.  ( II  Cn  J
) )
104adantr 465 . . . . . . . 8  |-  ( ( x (  ~=ph  `  J
) y  /\  f  e.  ( x ( PHtpy `  J ) y ) )  ->  y  e.  ( II  Cn  J
) )
11 eqid 2467 . . . . . . . 8  |-  ( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  ( u f ( 1  -  v ) ) )  =  ( u  e.  ( 0 [,] 1
) ,  v  e.  ( 0 [,] 1
)  |->  ( u f ( 1  -  v
) ) )
12 simpr 461 . . . . . . . 8  |-  ( ( x (  ~=ph  `  J
) y  /\  f  e.  ( x ( PHtpy `  J ) y ) )  ->  f  e.  ( x ( PHtpy `  J ) y ) )
139, 10, 11, 12phtpycom 21220 . . . . . . 7  |-  ( ( x (  ~=ph  `  J
) y  /\  f  e.  ( x ( PHtpy `  J ) y ) )  ->  ( u  e.  ( 0 [,] 1
) ,  v  e.  ( 0 [,] 1
)  |->  ( u f ( 1  -  v
) ) )  e.  ( y ( PHtpy `  J ) x ) )
14 ne0i 3791 . . . . . . 7  |-  ( ( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  ( u f ( 1  -  v ) ) )  e.  ( y ( PHtpy `  J )
x )  ->  (
y ( PHtpy `  J
) x )  =/=  (/) )
1513, 14syl 16 . . . . . 6  |-  ( ( x (  ~=ph  `  J
) y  /\  f  e.  ( x ( PHtpy `  J ) y ) )  ->  ( y
( PHtpy `  J )
x )  =/=  (/) )
168, 15exlimddv 1702 . . . . 5  |-  ( x (  ~=ph  `  J ) y  ->  ( y
( PHtpy `  J )
x )  =/=  (/) )
17 isphtpc 21226 . . . . 5  |-  ( y (  ~=ph  `  J ) x  <->  ( y  e.  ( II  Cn  J
)  /\  x  e.  ( II  Cn  J
)  /\  ( y
( PHtpy `  J )
x )  =/=  (/) ) )
184, 5, 16, 17syl3anbrc 1180 . . . 4  |-  ( x (  ~=ph  `  J ) y  ->  y (  ~=ph  `  J ) x )
1918adantl 466 . . 3  |-  ( ( T.  /\  x ( 
~=ph  `  J ) y )  ->  y (  ~=ph  `  J ) x )
205adantr 465 . . . . 5  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  x  e.  ( II  Cn  J
) )
21 simpr 461 . . . . . . 7  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  y
(  ~=ph  `  J )
z )
22 isphtpc 21226 . . . . . . 7  |-  ( y (  ~=ph  `  J ) z  <->  ( y  e.  ( II  Cn  J
)  /\  z  e.  ( II  Cn  J
)  /\  ( y
( PHtpy `  J )
z )  =/=  (/) ) )
2321, 22sylib 196 . . . . . 6  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  (
y  e.  ( II 
Cn  J )  /\  z  e.  ( II  Cn  J )  /\  (
y ( PHtpy `  J
) z )  =/=  (/) ) )
2423simp2d 1009 . . . . 5  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  z  e.  ( II  Cn  J
) )
256adantr 465 . . . . . . . 8  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  (
x ( PHtpy `  J
) y )  =/=  (/) )
2625, 7sylib 196 . . . . . . 7  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  E. f 
f  e.  ( x ( PHtpy `  J )
y ) )
2723simp3d 1010 . . . . . . . 8  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  (
y ( PHtpy `  J
) z )  =/=  (/) )
28 n0 3794 . . . . . . . 8  |-  ( ( y ( PHtpy `  J
) z )  =/=  (/) 
<->  E. g  g  e.  ( y ( PHtpy `  J ) z ) )
2927, 28sylib 196 . . . . . . 7  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  E. g 
g  e.  ( y ( PHtpy `  J )
z ) )
30 eeanv 1957 . . . . . . 7  |-  ( E. f E. g ( f  e.  ( x ( PHtpy `  J )
y )  /\  g  e.  ( y ( PHtpy `  J ) z ) )  <->  ( E. f 
f  e.  ( x ( PHtpy `  J )
y )  /\  E. g  g  e.  (
y ( PHtpy `  J
) z ) ) )
3126, 29, 30sylanbrc 664 . . . . . 6  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  E. f E. g ( f  e.  ( x ( PHtpy `  J ) y )  /\  g  e.  ( y ( PHtpy `  J
) z ) ) )
32 eqid 2467 . . . . . . . . . 10  |-  ( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  if ( v  <_  ( 1  /  2 ) ,  ( u f ( 2  x.  v ) ) ,  ( u g ( ( 2  x.  v )  - 
1 ) ) ) )  =  ( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  if ( v  <_  ( 1  /  2 ) ,  ( u f ( 2  x.  v ) ) ,  ( u g ( ( 2  x.  v )  - 
1 ) ) ) )
3320adantr 465 . . . . . . . . . 10  |-  ( ( ( x (  ~=ph  `  J ) y  /\  y (  ~=ph  `  J
) z )  /\  ( f  e.  ( x ( PHtpy `  J
) y )  /\  g  e.  ( y
( PHtpy `  J )
z ) ) )  ->  x  e.  ( II  Cn  J ) )
3423simp1d 1008 . . . . . . . . . . 11  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  y  e.  ( II  Cn  J
) )
3534adantr 465 . . . . . . . . . 10  |-  ( ( ( x (  ~=ph  `  J ) y  /\  y (  ~=ph  `  J
) z )  /\  ( f  e.  ( x ( PHtpy `  J
) y )  /\  g  e.  ( y
( PHtpy `  J )
z ) ) )  ->  y  e.  ( II  Cn  J ) )
3624adantr 465 . . . . . . . . . 10  |-  ( ( ( x (  ~=ph  `  J ) y  /\  y (  ~=ph  `  J
) z )  /\  ( f  e.  ( x ( PHtpy `  J
) y )  /\  g  e.  ( y
( PHtpy `  J )
z ) ) )  ->  z  e.  ( II  Cn  J ) )
37 simprl 755 . . . . . . . . . 10  |-  ( ( ( x (  ~=ph  `  J ) y  /\  y (  ~=ph  `  J
) z )  /\  ( f  e.  ( x ( PHtpy `  J
) y )  /\  g  e.  ( y
( PHtpy `  J )
z ) ) )  ->  f  e.  ( x ( PHtpy `  J
) y ) )
38 simprr 756 . . . . . . . . . 10  |-  ( ( ( x (  ~=ph  `  J ) y  /\  y (  ~=ph  `  J
) z )  /\  ( f  e.  ( x ( PHtpy `  J
) y )  /\  g  e.  ( y
( PHtpy `  J )
z ) ) )  ->  g  e.  ( y ( PHtpy `  J
) z ) )
3932, 33, 35, 36, 37, 38phtpycc 21223 . . . . . . . . 9  |-  ( ( ( x (  ~=ph  `  J ) y  /\  y (  ~=ph  `  J
) z )  /\  ( f  e.  ( x ( PHtpy `  J
) y )  /\  g  e.  ( y
( PHtpy `  J )
z ) ) )  ->  ( u  e.  ( 0 [,] 1
) ,  v  e.  ( 0 [,] 1
)  |->  if ( v  <_  ( 1  / 
2 ) ,  ( u f ( 2  x.  v ) ) ,  ( u g ( ( 2  x.  v )  -  1 ) ) ) )  e.  ( x (
PHtpy `  J ) z ) )
40 ne0i 3791 . . . . . . . . 9  |-  ( ( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  if ( v  <_  (
1  /  2 ) ,  ( u f ( 2  x.  v
) ) ,  ( u g ( ( 2  x.  v )  -  1 ) ) ) )  e.  ( x ( PHtpy `  J
) z )  -> 
( x ( PHtpy `  J ) z )  =/=  (/) )
4139, 40syl 16 . . . . . . . 8  |-  ( ( ( x (  ~=ph  `  J ) y  /\  y (  ~=ph  `  J
) z )  /\  ( f  e.  ( x ( PHtpy `  J
) y )  /\  g  e.  ( y
( PHtpy `  J )
z ) ) )  ->  ( x (
PHtpy `  J ) z )  =/=  (/) )
4241ex 434 . . . . . . 7  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  (
( f  e.  ( x ( PHtpy `  J
) y )  /\  g  e.  ( y
( PHtpy `  J )
z ) )  -> 
( x ( PHtpy `  J ) z )  =/=  (/) ) )
4342exlimdvv 1701 . . . . . 6  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  ( E. f E. g ( f  e.  ( x ( PHtpy `  J )
y )  /\  g  e.  ( y ( PHtpy `  J ) z ) )  ->  ( x
( PHtpy `  J )
z )  =/=  (/) ) )
4431, 43mpd 15 . . . . 5  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  (
x ( PHtpy `  J
) z )  =/=  (/) )
45 isphtpc 21226 . . . . 5  |-  ( x (  ~=ph  `  J ) z  <->  ( x  e.  ( II  Cn  J
)  /\  z  e.  ( II  Cn  J
)  /\  ( x
( PHtpy `  J )
z )  =/=  (/) ) )
4620, 24, 44, 45syl3anbrc 1180 . . . 4  |-  ( ( x (  ~=ph  `  J
) y  /\  y
(  ~=ph  `  J )
z )  ->  x
(  ~=ph  `  J )
z )
4746adantl 466 . . 3  |-  ( ( T.  /\  ( x (  ~=ph  `  J ) y  /\  y ( 
~=ph  `  J ) z ) )  ->  x
(  ~=ph  `  J )
z )
48 eqid 2467 . . . . . . . . . 10  |-  ( y  e.  ( 0 [,] 1 ) ,  z  e.  ( 0 [,] 1 )  |->  ( x `
 y ) )  =  ( y  e.  ( 0 [,] 1
) ,  z  e.  ( 0 [,] 1
)  |->  ( x `  y ) )
49 id 22 . . . . . . . . . 10  |-  ( x  e.  ( II  Cn  J )  ->  x  e.  ( II  Cn  J
) )
5048, 49phtpyid 21221 . . . . . . . . 9  |-  ( x  e.  ( II  Cn  J )  ->  (
y  e.  ( 0 [,] 1 ) ,  z  e.  ( 0 [,] 1 )  |->  ( x `  y ) )  e.  ( x ( PHtpy `  J )
x ) )
51 ne0i 3791 . . . . . . . . 9  |-  ( ( y  e.  ( 0 [,] 1 ) ,  z  e.  ( 0 [,] 1 )  |->  ( x `  y ) )  e.  ( x ( PHtpy `  J )
x )  ->  (
x ( PHtpy `  J
) x )  =/=  (/) )
5250, 51syl 16 . . . . . . . 8  |-  ( x  e.  ( II  Cn  J )  ->  (
x ( PHtpy `  J
) x )  =/=  (/) )
5352ancli 551 . . . . . . 7  |-  ( x  e.  ( II  Cn  J )  ->  (
x  e.  ( II 
Cn  J )  /\  ( x ( PHtpy `  J ) x )  =/=  (/) ) )
5453pm4.71ri 633 . . . . . 6  |-  ( x  e.  ( II  Cn  J )  <->  ( (
x  e.  ( II 
Cn  J )  /\  ( x ( PHtpy `  J ) x )  =/=  (/) )  /\  x  e.  ( II  Cn  J
) ) )
55 df-3an 975 . . . . . 6  |-  ( ( x  e.  ( II 
Cn  J )  /\  ( x ( PHtpy `  J ) x )  =/=  (/)  /\  x  e.  ( II  Cn  J
) )  <->  ( (
x  e.  ( II 
Cn  J )  /\  ( x ( PHtpy `  J ) x )  =/=  (/) )  /\  x  e.  ( II  Cn  J
) ) )
56 3ancomb 982 . . . . . 6  |-  ( ( x  e.  ( II 
Cn  J )  /\  ( x ( PHtpy `  J ) x )  =/=  (/)  /\  x  e.  ( II  Cn  J
) )  <->  ( x  e.  ( II  Cn  J
)  /\  x  e.  ( II  Cn  J
)  /\  ( x
( PHtpy `  J )
x )  =/=  (/) ) )
5754, 55, 563bitr2i 273 . . . . 5  |-  ( x  e.  ( II  Cn  J )  <->  ( x  e.  ( II  Cn  J
)  /\  x  e.  ( II  Cn  J
)  /\  ( x
( PHtpy `  J )
x )  =/=  (/) ) )
58 isphtpc 21226 . . . . 5  |-  ( x (  ~=ph  `  J ) x  <->  ( x  e.  ( II  Cn  J
)  /\  x  e.  ( II  Cn  J
)  /\  ( x
( PHtpy `  J )
x )  =/=  (/) ) )
5957, 58bitr4i 252 . . . 4  |-  ( x  e.  ( II  Cn  J )  <->  x (  ~=ph  `  J ) x )
6059a1i 11 . . 3  |-  ( T. 
->  ( x  e.  ( II  Cn  J )  <-> 
x (  ~=ph  `  J
) x ) )
612, 19, 47, 60iserd 7334 . 2  |-  ( T. 
->  (  ~=ph  `  J
)  Er  ( II 
Cn  J ) )
6261trud 1388 1  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973   T. wtru 1380   E.wex 1596    e. wcel 1767    =/= wne 2662   (/)c0 3785   ifcif 3939   class class class wbr 4447   Rel wrel 5004   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    Er wer 7305   0cc0 9488   1c1 9489    x. cmul 9493    <_ cle 9625    - cmin 9801    / cdiv 10202   2c2 10581   [,]cicc 11528    Cn ccn 19488   IIcii 21111   PHtpycphtpy 21200    ~=ph cphtpc 21201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-cn 19491  df-cnp 19492  df-tx 19795  df-hmeo 19988  df-xms 20555  df-ms 20556  df-tms 20557  df-ii 21113  df-htpy 21202  df-phtpy 21203  df-phtpc 21224
This theorem is referenced by:  pcophtb  21261  pi1buni  21272  pi1addf  21279  pi1addval  21280  pi1grplem  21281  pi1inv  21284  pi1xfrf  21285  pi1xfr  21287  pi1xfrcnvlem  21288  pi1cof  21291  sconpi1  28321
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