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Theorem phtpycc 21657
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.
StepHypRef 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 21549 . . . 4  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
54a1i 11 . . 3  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
6 phtpycc.4 . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
71, 6phtpyhtpy 21648 . . . 4  |-  ( ph  ->  ( F ( PHtpy `  J ) G ) 
C_  ( F ( II Htpy  J ) G ) )
8 phtpycc.6 . . . 4  |-  ( ph  ->  K  e.  ( F ( PHtpy `  J ) G ) )
97, 8sseldd 3490 . . 3  |-  ( ph  ->  K  e.  ( F ( II Htpy  J ) G ) )
106, 2phtpyhtpy 21648 . . . 4  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( G ( II Htpy  J ) H ) )
11 phtpycc.7 . . . 4  |-  ( ph  ->  L  e.  ( G ( PHtpy `  J ) H ) )
1210, 11sseldd 3490 . . 3  |-  ( ph  ->  L  e.  ( G ( II Htpy  J ) H ) )
133, 5, 1, 6, 2, 9, 12htpycc 21646 . 2  |-  ( ph  ->  M  e.  ( F ( II Htpy  J ) H ) )
14 0elunit 11641 . . . 4  |-  0  e.  ( 0 [,] 1
)
15 simpr 459 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
16 simpr 459 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
1716breq1d 4449 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
18 simpl 455 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
1916oveq2d 6286 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
2018, 19oveq12d 6288 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 0 K ( 2  x.  s ) ) )
2119oveq1d 6285 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
2218, 21oveq12d 6288 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 0 L ( ( 2  x.  s )  -  1 ) ) )
2317, 20, 22ifbieq12d 3956 . . . . 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 6298 . . . . . 6  |-  ( 0 K ( 2  x.  s ) )  e. 
_V
25 ovex 6298 . . . . . 6  |-  ( 0 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
2624, 25ifex 3997 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
2723, 3, 26ovmpt2a 6406 . . . 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 ) ) ) )
2814, 15, 27sylancr 661 . . 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 751 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  ph )
30 elii1 21601 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  <->  ( s  e.  ( 0 [,] 1
)  /\  s  <_  ( 1  /  2 ) ) )
31 iihalf1 21597 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
3230, 31sylbir 213 . . . . . . 7  |-  ( ( s  e.  ( 0 [,] 1 )  /\  s  <_  ( 1  / 
2 ) )  -> 
( 2  x.  s
)  e.  ( 0 [,] 1 ) )
3332adantll 711 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
341, 6, 8phtpyi 21650 . . . . . 6  |-  ( (
ph  /\  ( 2  x.  s )  e.  ( 0 [,] 1
) )  ->  (
( 0 K ( 2  x.  s ) )  =  ( F `
 0 )  /\  ( 1 K ( 2  x.  s ) )  =  ( F `
 1 ) ) )
3529, 33, 34syl2anc 659 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
( 0 K ( 2  x.  s ) )  =  ( F `
 0 )  /\  ( 1 K ( 2  x.  s ) )  =  ( F `
 1 ) ) )
3635simpld 457 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
0 K ( 2  x.  s ) )  =  ( F ` 
0 ) )
37 simpll 751 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  ph )
38 elii2 21602 . . . . . . . . 9  |-  ( ( s  e.  ( 0 [,] 1 )  /\  -.  s  <_  ( 1  /  2 ) )  ->  s  e.  ( ( 1  /  2
) [,] 1 ) )
39 iihalf2 21599 . . . . . . . . 9  |-  ( s  e.  ( ( 1  /  2 ) [,] 1 )  ->  (
( 2  x.  s
)  -  1 )  e.  ( 0 [,] 1 ) )
4038, 39syl 16 . . . . . . . 8  |-  ( ( s  e.  ( 0 [,] 1 )  /\  -.  s  <_  ( 1  /  2 ) )  ->  ( ( 2  x.  s )  - 
1 )  e.  ( 0 [,] 1 ) )
4140adantll 711 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( 2  x.  s )  -  1 )  e.  ( 0 [,] 1 ) )
426, 2, 11phtpyi 21650 . . . . . . 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 ) ) )
4337, 41, 42syl2anc 659 . . . . . 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 ) ) )
4443simpld 457 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 0 ) )
451, 6, 8phtpy01 21651 . . . . . . 7  |-  ( ph  ->  ( ( F ` 
0 )  =  ( G `  0 )  /\  ( F ` 
1 )  =  ( G `  1 ) ) )
4645ad2antrr 723 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( F ` 
0 )  =  ( G `  0 )  /\  ( F ` 
1 )  =  ( G `  1 ) ) )
4746simpld 457 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( F `  0
)  =  ( G `
 0 ) )
4844, 47eqtr4d 2498 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 0 ) )
4936, 48ifeqda 3962 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( 0 K ( 2  x.  s
) ) ,  ( 0 L ( ( 2  x.  s )  -  1 ) ) )  =  ( F `
 0 ) )
5028, 49eqtrd 2495 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 M s )  =  ( F ` 
0 ) )
51 1elunit 11642 . . . 4  |-  1  e.  ( 0 [,] 1
)
52 simpr 459 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
5352breq1d 4449 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
54 simpl 455 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
5552oveq2d 6286 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
5654, 55oveq12d 6288 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 1 K ( 2  x.  s ) ) )
5755oveq1d 6285 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
5854, 57oveq12d 6288 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 1 L ( ( 2  x.  s )  -  1 ) ) )
5953, 56, 58ifbieq12d 3956 . . . . 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 6298 . . . . . 6  |-  ( 1 K ( 2  x.  s ) )  e. 
_V
61 ovex 6298 . . . . . 6  |-  ( 1 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
6260, 61ifex 3997 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
6359, 3, 62ovmpt2a 6406 . . . 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 ) ) ) )
6451, 15, 63sylancr 661 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 M s )  =  if ( s  <_  ( 1  / 
2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  -  1 ) ) ) )
6535simprd 461 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
1 K ( 2  x.  s ) )  =  ( F ` 
1 ) )
6643simprd 461 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 1 ) )
6746simprd 461 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( F `  1
)  =  ( G `
 1 ) )
6866, 67eqtr4d 2498 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 1 ) )
6965, 68ifeqda 3962 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( 1 K ( 2  x.  s
) ) ,  ( 1 L ( ( 2  x.  s )  -  1 ) ) )  =  ( F `
 1 ) )
7064, 69eqtrd 2495 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 M s )  =  ( F ` 
1 ) )
711, 2, 13, 50, 70isphtpyd 21652 1  |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ifcif 3929   class class class wbr 4439   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   0cc0 9481   1c1 9482    x. cmul 9486    <_ cle 9618    - cmin 9796    / cdiv 10202   2c2 10581   [,]cicc 11535  TopOnctopon 19562    Cn ccn 19892   IIcii 21545   Htpy chtpy 21633   PHtpycphtpy 21634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  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 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-cn 19895  df-cnp 19896  df-tx 20229  df-hmeo 20422  df-xms 20989  df-ms 20990  df-tms 20991  df-ii 21547  df-htpy 21636  df-phtpy 21637
This theorem is referenced by:  phtpcer  21661
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