Theorem phtpcer 22794

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.) (Proof shortened by AV, 1-May-2021.)

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.

Step	Hyp	Ref	Expression
1		phtpcrel 22792	. 2 |- Rel ( ~=ph ` J )
2		isphtpc 22793	. . . 4 |- ( x ( ~=ph ` J ) y <-> ( x e. ( II Cn J ) /\ y e. ( II Cn J ) /\ ( x ( PHtpy ` J ) y ) =/= (/) ) )
3	2	simp2bi 1077	. . 3 |- ( x ( ~=ph ` J ) y -> y e. ( II Cn J ) )
4	2	simp1bi 1076	. . 3 |- ( x ( ~=ph ` J ) y -> x e. ( II Cn J ) )
5	2	simp3bi 1078	. . . . 5 |- ( x ( ~=ph ` J ) y -> ( x ( PHtpy ` J ) y ) =/= (/) )
6		n0 3931	. . . . 5 |- ( ( x ( PHtpy ` J ) y ) =/= (/) <-> E. f f e. ( x ( PHtpy ` J ) y ) )
7	5, 6	sylib 208	. . . 4 |- ( x ( ~=ph ` J ) y -> E. f f e. ( x ( PHtpy ` J ) y ) )
8	4	adantr 481	. . . . . 6 |- ( ( x ( ~=ph ` J ) y /\ f e. ( x ( PHtpy ` J ) y ) ) -> x e. ( II Cn J ) )
9	3	adantr 481	. . . . . 6 |- ( ( x ( ~=ph ` J ) y /\ f e. ( x ( PHtpy ` J ) y ) ) -> y e. ( II Cn J ) )
10		eqid 2622	. . . . . 6 |- ( u e. ( 0 [,] 1 ) , v e. ( 0 [,] 1 ) |-> ( u f ( 1 - v ) ) ) = ( u e. ( 0 [,] 1 ) , v e. ( 0 [,] 1 ) |-> ( u f ( 1 - v ) ) )
11		simpr 477	. . . . . 6 |- ( ( x ( ~=ph ` J ) y /\ f e. ( x ( PHtpy ` J ) y ) ) -> f e. ( x ( PHtpy ` J ) y ) )
12	8, 9, 10, 11	phtpycom 22787	. . . . 5 |- ( ( 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 ) )
13		ne0i 3921	. . . . 5 |- ( ( u e. ( 0 [,] 1 ) , v e. ( 0 [,] 1 ) |-> ( u f ( 1 - v ) ) ) e. ( y ( PHtpy ` J ) x ) -> ( y ( PHtpy ` J ) x ) =/= (/) )
14	12, 13	syl 17	. . . 4 |- ( ( x ( ~=ph ` J ) y /\ f e. ( x ( PHtpy ` J ) y ) ) -> ( y ( PHtpy ` J ) x ) =/= (/) )
15	7, 14	exlimddv 1863	. . 3 |- ( x ( ~=ph ` J ) y -> ( y ( PHtpy ` J ) x ) =/= (/) )
16		isphtpc 22793	. . 3 |- ( y ( ~=ph ` J ) x <-> ( y e. ( II Cn J ) /\ x e. ( II Cn J ) /\ ( y ( PHtpy ` J ) x ) =/= (/) ) )
17	3, 4, 15, 16	syl3anbrc 1246	. 2 |- ( x ( ~=ph ` J ) y -> y ( ~=ph ` J ) x )
18	4	adantr 481	. . 3 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> x e. ( II Cn J ) )
19		simpr 477	. . . . 5 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> y ( ~=ph ` J ) z )
20		isphtpc 22793	. . . . 5 |- ( y ( ~=ph ` J ) z <-> ( y e. ( II Cn J ) /\ z e. ( II Cn J ) /\ ( y ( PHtpy ` J ) z ) =/= (/) ) )
21	19, 20	sylib 208	. . . 4 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> ( y e. ( II Cn J ) /\ z e. ( II Cn J ) /\ ( y ( PHtpy ` J ) z ) =/= (/) ) )
22	21	simp2d 1074	. . 3 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> z e. ( II Cn J ) )
23	5	adantr 481	. . . . . 6 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> ( x ( PHtpy ` J ) y ) =/= (/) )
24	23, 6	sylib 208	. . . . 5 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> E. f f e. ( x ( PHtpy ` J ) y ) )
25	21	simp3d 1075	. . . . . 6 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> ( y ( PHtpy ` J ) z ) =/= (/) )
26		n0 3931	. . . . . 6 |- ( ( y ( PHtpy ` J ) z ) =/= (/) <-> E. g g e. ( y ( PHtpy ` J ) z ) )
27	25, 26	sylib 208	. . . . 5 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> E. g g e. ( y ( PHtpy ` J ) z ) )
28		eeanv 2182	. . . . 5 |- ( 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 ) ) )
29	24, 27, 28	sylanbrc 698	. . . 4 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> E. f E. g ( f e. ( x ( PHtpy ` J ) y ) /\ g e. ( y ( PHtpy ` J ) z ) ) )
30		eqid 2622	. . . . . . . 8 |- ( 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 ) ) ) )
31	18	adantr 481	. . . . . . . 8 |- ( ( ( 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 ) )
32	21	simp1d 1073	. . . . . . . . 9 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> y e. ( II Cn J ) )
33	32	adantr 481	. . . . . . . 8 |- ( ( ( 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 ) )
34	22	adantr 481	. . . . . . . 8 |- ( ( ( 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 ) )
35		simprl 794	. . . . . . . 8 |- ( ( ( 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 ) )
36		simprr 796	. . . . . . . 8 |- ( ( ( 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 ) )
37	30, 31, 33, 34, 35, 36	phtpycc 22790	. . . . . . 7 |- ( ( ( 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 ) )
38		ne0i 3921	. . . . . . 7 |- ( ( 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 ) =/= (/) )
39	37, 38	syl 17	. . . . . 6 |- ( ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) /\ ( f e. ( x ( PHtpy ` J ) y ) /\ g e. ( y ( PHtpy ` J ) z ) ) ) -> ( x ( PHtpy ` J ) z ) =/= (/) )
40	39	ex 450	. . . . 5 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> ( ( f e. ( x ( PHtpy ` J ) y ) /\ g e. ( y ( PHtpy ` J ) z ) ) -> ( x ( PHtpy ` J ) z ) =/= (/) ) )
41	40	exlimdvv 1862	. . . 4 |- ( ( 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 ) =/= (/) ) )
42	29, 41	mpd 15	. . 3 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> ( x ( PHtpy ` J ) z ) =/= (/) )
43		isphtpc 22793	. . 3 |- ( x ( ~=ph ` J ) z <-> ( x e. ( II Cn J ) /\ z e. ( II Cn J ) /\ ( x ( PHtpy ` J ) z ) =/= (/) ) )
44	18, 22, 42, 43	syl3anbrc 1246	. 2 |- ( ( x ( ~=ph ` J ) y /\ y ( ~=ph ` J ) z ) -> x ( ~=ph ` J ) z )
45		eqid 2622	. . . . . . . 8 |- ( y e. ( 0 [,] 1 ) , z e. ( 0 [,] 1 ) |-> ( x ` y ) ) = ( y e. ( 0 [,] 1 ) , z e. ( 0 [,] 1 ) |-> ( x ` y ) )
46		id 22	. . . . . . . 8 |- ( x e. ( II Cn J ) -> x e. ( II Cn J ) )
47	45, 46	phtpyid 22788	. . . . . . 7 |- ( x e. ( II Cn J ) -> ( y e. ( 0 [,] 1 ) , z e. ( 0 [,] 1 ) |-> ( x ` y ) ) e. ( x ( PHtpy ` J ) x ) )
48		ne0i 3921	. . . . . . 7 |- ( ( y e. ( 0 [,] 1 ) , z e. ( 0 [,] 1 ) |-> ( x ` y ) ) e. ( x ( PHtpy ` J ) x ) -> ( x ( PHtpy ` J ) x ) =/= (/) )
49	47, 48	syl 17	. . . . . 6 |- ( x e. ( II Cn J ) -> ( x ( PHtpy ` J ) x ) =/= (/) )
50	49	ancli 574	. . . . 5 |- ( x e. ( II Cn J ) -> ( x e. ( II Cn J ) /\ ( x ( PHtpy ` J ) x ) =/= (/) ) )
51	50	pm4.71ri 665	. . . 4 |- ( x e. ( II Cn J ) <-> ( ( x e. ( II Cn J ) /\ ( x ( PHtpy ` J ) x ) =/= (/) ) /\ x e. ( II Cn J ) ) )
52		df-3an 1039	. . . 4 |- ( ( 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 ) ) )
53		3ancomb 1047	. . . 4 |- ( ( 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 ) =/= (/) ) )
54	51, 52, 53	3bitr2i 288	. . 3 |- ( x e. ( II Cn J ) <-> ( x e. ( II Cn J ) /\ x e. ( II Cn J ) /\ ( x ( PHtpy ` J ) x ) =/= (/) ) )
55		isphtpc 22793	. . 3 |- ( x ( ~=ph ` J ) x <-> ( x e. ( II Cn J ) /\ x e. ( II Cn J ) /\ ( x ( PHtpy ` J ) x ) =/= (/) ) )
56	54, 55	bitr4i 267	. 2 |- ( x e. ( II Cn J ) <-> x ( ~=ph ` J ) x )
57	1, 17, 44, 56	iseri 7769	1 |- ( ~=ph ` J ) Er ( II Cn J )

Syntax hints:   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)c0 3915   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Er wer 7739   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   [,]cicc 12178   Cn ccn 21028   IIcii 22678   PHtpycphtpy 22767   ~=ph cphtpc 22768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791

This theorem is referenced by:  pcophtb  22829  pi1buni  22840  pi1addf  22847  pi1addval  22848  pi1grplem  22849  pi1inv  22852  pi1xfrf  22853  pi1xfr  22855  pi1xfrcnvlem  22856  pi1cof  22859  sconnpi1  31221