Theorem phtpycc 22790

Description: Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)

Hypotheses

Ref	Expression
phtpycc.1	|- M = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> if ( y <_ ( 1 / 2 ) , ( x K ( 2 x. y ) ) , ( x L ( ( 2 x. y ) - 1 ) ) ) )
phtpycc.3	|- ( ph -> F e. ( II Cn J ) )
phtpycc.4	|- ( ph -> G e. ( II Cn J ) )
phtpycc.5	|- ( ph -> H e. ( II Cn J ) )
phtpycc.6	|- ( ph -> K e. ( F ( PHtpy ` J ) G ) )
phtpycc.7	|- ( ph -> L e. ( G ( PHtpy ` J ) H ) )

Assertion

Ref	Expression
phtpycc	|- ( ph -> M e. ( F ( PHtpy ` J ) H ) )

Distinct variable groups:   x, y, J   x, K, y   ph, x, y   x, L, y

Allowed substitution hints:   F( x, y)   G( x, y)   H( x, y)   M( x, y)

Proof of Theorem phtpycc

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		phtpycc.3	. 2 |- ( ph -> F e. ( II Cn J ) )
2		phtpycc.5	. 2 |- ( ph -> H e. ( II Cn J ) )
3		phtpycc.1	. . 3 |- M = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> if ( y <_ ( 1 / 2 ) , ( x K ( 2 x. y ) ) , ( x L ( ( 2 x. y ) - 1 ) ) ) )
4		iitopon 22682	. . . 4 |- II e. (TopOn ` ( 0 [,] 1 ) )
5	4	a1i 11	. . 3 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
6		phtpycc.4	. . 3 |- ( ph -> G e. ( II Cn J ) )
7	1, 6	phtpyhtpy 22781	. . . 4 |- ( ph -> ( F ( PHtpy ` J ) G ) C_ ( F ( II Htpy J ) G ) )
8		phtpycc.6	. . . 4 |- ( ph -> K e. ( F ( PHtpy ` J ) G ) )
9	7, 8	sseldd 3604	. . 3 |- ( ph -> K e. ( F ( II Htpy J ) G ) )
10	6, 2	phtpyhtpy 22781	. . . 4 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( G ( II Htpy J ) H ) )
11		phtpycc.7	. . . 4 |- ( ph -> L e. ( G ( PHtpy ` J ) H ) )
12	10, 11	sseldd 3604	. . 3 |- ( ph -> L e. ( G ( II Htpy J ) H ) )
13	3, 5, 1, 6, 2, 9, 12	htpycc 22779	. 2 |- ( ph -> M e. ( F ( II Htpy J ) H ) )
14		0elunit 12290	. . . 4 |- 0 e. ( 0 [,] 1 )
15		simpr 477	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> s e. ( 0 [,] 1 ) )
16		simpr 477	. . . . . . 7 |- ( ( x = 0 /\ y = s ) -> y = s )
17	16	breq1d 4663	. . . . . 6 |- ( ( x = 0 /\ y = s ) -> ( y <_ ( 1 / 2 ) <-> s <_ ( 1 / 2 ) ) )
18		simpl 473	. . . . . . 7 |- ( ( x = 0 /\ y = s ) -> x = 0 )
19	16	oveq2d 6666	. . . . . . 7 |- ( ( x = 0 /\ y = s ) -> ( 2 x. y ) = ( 2 x. s ) )
20	18, 19	oveq12d 6668	. . . . . 6 |- ( ( x = 0 /\ y = s ) -> ( x K ( 2 x. y ) ) = ( 0 K ( 2 x. s ) ) )
21	19	oveq1d 6665	. . . . . . 7 |- ( ( x = 0 /\ y = s ) -> ( ( 2 x. y ) - 1 ) = ( ( 2 x. s ) - 1 ) )
22	18, 21	oveq12d 6668	. . . . . 6 |- ( ( x = 0 /\ y = s ) -> ( x L ( ( 2 x. y ) - 1 ) ) = ( 0 L ( ( 2 x. s ) - 1 ) ) )
23	17, 20, 22	ifbieq12d 4113	. . . . 5 |- ( ( x = 0 /\ y = s ) -> if ( y <_ ( 1 / 2 ) , ( x K ( 2 x. y ) ) , ( x L ( ( 2 x. y ) - 1 ) ) ) = if ( s <_ ( 1 / 2 ) , ( 0 K ( 2 x. s ) ) , ( 0 L ( ( 2 x. s ) - 1 ) ) ) )
24		ovex 6678	. . . . . 6 |- ( 0 K ( 2 x. s ) ) e. _V
25		ovex 6678	. . . . . 6 |- ( 0 L ( ( 2 x. s ) - 1 ) ) e. _V
26	24, 25	ifex 4156	. . . . 5 |- if ( s <_ ( 1 / 2 ) , ( 0 K ( 2 x. s ) ) , ( 0 L ( ( 2 x. s ) - 1 ) ) ) e. _V
27	23, 3, 26	ovmpt2a 6791	. . . 4 |- ( ( 0 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( 0 M s ) = if ( s <_ ( 1 / 2 ) , ( 0 K ( 2 x. s ) ) , ( 0 L ( ( 2 x. s ) - 1 ) ) ) )
28	14, 15, 27	sylancr 695	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 M s ) = if ( s <_ ( 1 / 2 ) , ( 0 K ( 2 x. s ) ) , ( 0 L ( ( 2 x. s ) - 1 ) ) ) )
29		simpll 790	. . . . . 6 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ph )
30		elii1 22734	. . . . . . . 8 |- ( s e. ( 0 [,] ( 1 / 2 ) ) <-> ( s e. ( 0 [,] 1 ) /\ s <_ ( 1 / 2 ) ) )
31		iihalf1 22730	. . . . . . . 8 |- ( s e. ( 0 [,] ( 1 / 2 ) ) -> ( 2 x. s ) e. ( 0 [,] 1 ) )
32	30, 31	sylbir 225	. . . . . . 7 |- ( ( s e. ( 0 [,] 1 ) /\ s <_ ( 1 / 2 ) ) -> ( 2 x. s ) e. ( 0 [,] 1 ) )
33	32	adantll 750	. . . . . 6 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ( 2 x. s ) e. ( 0 [,] 1 ) )
34	1, 6, 8	phtpyi 22783	. . . . . 6 |- ( ( ph /\ ( 2 x. s ) e. ( 0 [,] 1 ) ) -> ( ( 0 K ( 2 x. s ) ) = ( F ` 0 ) /\ ( 1 K ( 2 x. s ) ) = ( F ` 1 ) ) )
35	29, 33, 34	syl2anc 693	. . . . 5 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ( ( 0 K ( 2 x. s ) ) = ( F ` 0 ) /\ ( 1 K ( 2 x. s ) ) = ( F ` 1 ) ) )
36	35	simpld 475	. . . 4 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ( 0 K ( 2 x. s ) ) = ( F ` 0 ) )
37		simpll 790	. . . . . . 7 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ph )
38		elii2 22735	. . . . . . . . 9 |- ( ( s e. ( 0 [,] 1 ) /\ -. s <_ ( 1 / 2 ) ) -> s e. ( ( 1 / 2 ) [,] 1 ) )
39		iihalf2 22732	. . . . . . . . 9 |- ( s e. ( ( 1 / 2 ) [,] 1 ) -> ( ( 2 x. s ) - 1 ) e. ( 0 [,] 1 ) )
40	38, 39	syl 17	. . . . . . . 8 |- ( ( s e. ( 0 [,] 1 ) /\ -. s <_ ( 1 / 2 ) ) -> ( ( 2 x. s ) - 1 ) e. ( 0 [,] 1 ) )
41	40	adantll 750	. . . . . . 7 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( ( 2 x. s ) - 1 ) e. ( 0 [,] 1 ) )
42	6, 2, 11	phtpyi 22783	. . . . . . 7 |- ( ( ph /\ ( ( 2 x. s ) - 1 ) e. ( 0 [,] 1 ) ) -> ( ( 0 L ( ( 2 x. s ) - 1 ) ) = ( G ` 0 ) /\ ( 1 L ( ( 2 x. s ) - 1 ) ) = ( G ` 1 ) ) )
43	37, 41, 42	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( ( 0 L ( ( 2 x. s ) - 1 ) ) = ( G ` 0 ) /\ ( 1 L ( ( 2 x. s ) - 1 ) ) = ( G ` 1 ) ) )
44	43	simpld 475	. . . . 5 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( 0 L ( ( 2 x. s ) - 1 ) ) = ( G ` 0 ) )
45	1, 6, 8	phtpy01 22784	. . . . . . 7 |- ( ph -> ( ( F ` 0 ) = ( G ` 0 ) /\ ( F ` 1 ) = ( G ` 1 ) ) )
46	45	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( ( F ` 0 ) = ( G ` 0 ) /\ ( F ` 1 ) = ( G ` 1 ) ) )
47	46	simpld 475	. . . . 5 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( F ` 0 ) = ( G ` 0 ) )
48	44, 47	eqtr4d 2659	. . . 4 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( 0 L ( ( 2 x. s ) - 1 ) ) = ( F ` 0 ) )
49	36, 48	ifeqda 4121	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> if ( s <_ ( 1 / 2 ) , ( 0 K ( 2 x. s ) ) , ( 0 L ( ( 2 x. s ) - 1 ) ) ) = ( F ` 0 ) )
50	28, 49	eqtrd 2656	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 M s ) = ( F ` 0 ) )
51		1elunit 12291	. . . 4 |- 1 e. ( 0 [,] 1 )
52		simpr 477	. . . . . . 7 |- ( ( x = 1 /\ y = s ) -> y = s )
53	52	breq1d 4663	. . . . . 6 |- ( ( x = 1 /\ y = s ) -> ( y <_ ( 1 / 2 ) <-> s <_ ( 1 / 2 ) ) )
54		simpl 473	. . . . . . 7 |- ( ( x = 1 /\ y = s ) -> x = 1 )
55	52	oveq2d 6666	. . . . . . 7 |- ( ( x = 1 /\ y = s ) -> ( 2 x. y ) = ( 2 x. s ) )
56	54, 55	oveq12d 6668	. . . . . 6 |- ( ( x = 1 /\ y = s ) -> ( x K ( 2 x. y ) ) = ( 1 K ( 2 x. s ) ) )
57	55	oveq1d 6665	. . . . . . 7 |- ( ( x = 1 /\ y = s ) -> ( ( 2 x. y ) - 1 ) = ( ( 2 x. s ) - 1 ) )
58	54, 57	oveq12d 6668	. . . . . 6 |- ( ( x = 1 /\ y = s ) -> ( x L ( ( 2 x. y ) - 1 ) ) = ( 1 L ( ( 2 x. s ) - 1 ) ) )
59	53, 56, 58	ifbieq12d 4113	. . . . 5 |- ( ( x = 1 /\ y = s ) -> if ( y <_ ( 1 / 2 ) , ( x K ( 2 x. y ) ) , ( x L ( ( 2 x. y ) - 1 ) ) ) = if ( s <_ ( 1 / 2 ) , ( 1 K ( 2 x. s ) ) , ( 1 L ( ( 2 x. s ) - 1 ) ) ) )
60		ovex 6678	. . . . . 6 |- ( 1 K ( 2 x. s ) ) e. _V
61		ovex 6678	. . . . . 6 |- ( 1 L ( ( 2 x. s ) - 1 ) ) e. _V
62	60, 61	ifex 4156	. . . . 5 |- if ( s <_ ( 1 / 2 ) , ( 1 K ( 2 x. s ) ) , ( 1 L ( ( 2 x. s ) - 1 ) ) ) e. _V
63	59, 3, 62	ovmpt2a 6791	. . . 4 |- ( ( 1 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( 1 M s ) = if ( s <_ ( 1 / 2 ) , ( 1 K ( 2 x. s ) ) , ( 1 L ( ( 2 x. s ) - 1 ) ) ) )
64	51, 15, 63	sylancr 695	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 M s ) = if ( s <_ ( 1 / 2 ) , ( 1 K ( 2 x. s ) ) , ( 1 L ( ( 2 x. s ) - 1 ) ) ) )
65	35	simprd 479	. . . 4 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ( 1 K ( 2 x. s ) ) = ( F ` 1 ) )
66	43	simprd 479	. . . . 5 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( 1 L ( ( 2 x. s ) - 1 ) ) = ( G ` 1 ) )
67	46	simprd 479	. . . . 5 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( F ` 1 ) = ( G ` 1 ) )
68	66, 67	eqtr4d 2659	. . . 4 |- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( 1 L ( ( 2 x. s ) - 1 ) ) = ( F ` 1 ) )
69	65, 68	ifeqda 4121	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> if ( s <_ ( 1 / 2 ) , ( 1 K ( 2 x. s ) ) , ( 1 L ( ( 2 x. s ) - 1 ) ) ) = ( F ` 1 ) )
70	64, 69	eqtrd 2656	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 M s ) = ( F ` 1 ) )
71	1, 2, 13, 50, 70	isphtpyd 22785	1 |- ( ph -> M e. ( F ( PHtpy ` J ) H ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   [,]cicc 12178  TopOnctopon 20715   Cn ccn 21028   IIcii 22678   Htpy chtpy 22766   PHtpycphtpy 22767

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

This theorem is referenced by:  phtpcer  22794  phtpcerOLD  22795