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Theorem htpyco1 18993
Description: Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypotheses
Ref Expression
htpyco1.n  |-  N  =  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( ( P `  x ) H y ) )
htpyco1.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
htpyco1.p  |-  ( ph  ->  P  e.  ( J  Cn  K ) )
htpyco1.f  |-  ( ph  ->  F  e.  ( K  Cn  L ) )
htpyco1.g  |-  ( ph  ->  G  e.  ( K  Cn  L ) )
htpyco1.h  |-  ( ph  ->  H  e.  ( F ( K Htpy  L ) G ) )
Assertion
Ref Expression
htpyco1  |-  ( ph  ->  N  e.  ( ( F  o.  P ) ( J Htpy  L ) ( G  o.  P
) ) )
Distinct variable groups:    x, y, H    x, K, y    x, L, y    ph, x, y   
x, J, y    x, P, y    x, X, y
Allowed substitution hints:    F( x, y)    G( x, y)    N( x, y)

Proof of Theorem htpyco1
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 htpyco1.j . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 htpyco1.p . . 3  |-  ( ph  ->  P  e.  ( J  Cn  K ) )
3 htpyco1.f . . 3  |-  ( ph  ->  F  e.  ( K  Cn  L ) )
4 cnco 17320 . . 3  |-  ( ( P  e.  ( J  Cn  K )  /\  F  e.  ( K  Cn  L ) )  -> 
( F  o.  P
)  e.  ( J  Cn  L ) )
52, 3, 4syl2anc 643 . 2  |-  ( ph  ->  ( F  o.  P
)  e.  ( J  Cn  L ) )
6 htpyco1.g . . 3  |-  ( ph  ->  G  e.  ( K  Cn  L ) )
7 cnco 17320 . . 3  |-  ( ( P  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  P
)  e.  ( J  Cn  L ) )
82, 6, 7syl2anc 643 . 2  |-  ( ph  ->  ( G  o.  P
)  e.  ( J  Cn  L ) )
9 htpyco1.n . . 3  |-  N  =  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( ( P `  x ) H y ) )
10 iitopon 18899 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1110a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
121, 11cnmpt1st 17690 . . . . 5  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  x )  e.  ( ( J  tX  II )  Cn  J ) )
131, 11, 12, 2cnmpt21f 17694 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( P `  x
) )  e.  ( ( J  tX  II )  Cn  K ) )
141, 11cnmpt2nd 17691 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  y )  e.  ( ( J  tX  II )  Cn  II ) )
15 cntop2 17295 . . . . . . . 8  |-  ( P  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
17 eqid 2435 . . . . . . . 8  |-  U. K  =  U. K
1817toptopon 16988 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
1916, 18sylib 189 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
2019, 3, 6htpycn 18988 . . . . 5  |-  ( ph  ->  ( F ( K Htpy 
L ) G ) 
C_  ( ( K 
tX  II )  Cn  L ) )
21 htpyco1.h . . . . 5  |-  ( ph  ->  H  e.  ( F ( K Htpy  L ) G ) )
2220, 21sseldd 3341 . . . 4  |-  ( ph  ->  H  e.  ( ( K  tX  II )  Cn  L ) )
231, 11, 13, 14, 22cnmpt22f 17697 . . 3  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( ( P `  x ) H y ) )  e.  ( ( J  tX  II )  Cn  L ) )
249, 23syl5eqel 2519 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  L ) )
25 cnf2 17303 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  P  e.  ( J  Cn  K ) )  ->  P : X
--> U. K )
261, 19, 2, 25syl3anc 1184 . . . . . 6  |-  ( ph  ->  P : X --> U. K
)
2726ffvelrnda 5862 . . . . 5  |-  ( (
ph  /\  s  e.  X )  ->  ( P `  s )  e.  U. K )
2819, 3, 6, 21htpyi 18989 . . . . 5  |-  ( (
ph  /\  ( P `  s )  e.  U. K )  ->  (
( ( P `  s ) H 0 )  =  ( F `
 ( P `  s ) )  /\  ( ( P `  s ) H 1 )  =  ( G `
 ( P `  s ) ) ) )
2927, 28syldan 457 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  (
( ( P `  s ) H 0 )  =  ( F `
 ( P `  s ) )  /\  ( ( P `  s ) H 1 )  =  ( G `
 ( P `  s ) ) ) )
3029simpld 446 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( P `  s
) H 0 )  =  ( F `  ( P `  s ) ) )
31 simpr 448 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
32 0elunit 11005 . . . 4  |-  0  e.  ( 0 [,] 1
)
33 fveq2 5720 . . . . . 6  |-  ( x  =  s  ->  ( P `  x )  =  ( P `  s ) )
34 id 20 . . . . . 6  |-  ( y  =  0  ->  y  =  0 )
3533, 34oveqan12d 6092 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 0 ) )
36 ovex 6098 . . . . 5  |-  ( ( P `  s ) H 0 )  e. 
_V
3735, 9, 36ovmpt2a 6196 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( ( P `  s
) H 0 ) )
3831, 32, 37sylancl 644 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( ( P `
 s ) H 0 ) )
39 fvco3 5792 . . . 4  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( ( F  o.  P ) `  s )  =  ( F `  ( P `
 s ) ) )
4026, 39sylan 458 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( F  o.  P
) `  s )  =  ( F `  ( P `  s ) ) )
4130, 38, 403eqtr4d 2477 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( ( F  o.  P ) `  s ) )
4229simprd 450 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( P `  s
) H 1 )  =  ( G `  ( P `  s ) ) )
43 1elunit 11006 . . . 4  |-  1  e.  ( 0 [,] 1
)
44 id 20 . . . . . 6  |-  ( y  =  1  ->  y  =  1 )
4533, 44oveqan12d 6092 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 1 ) )
46 ovex 6098 . . . . 5  |-  ( ( P `  s ) H 1 )  e. 
_V
4745, 9, 46ovmpt2a 6196 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( ( P `  s
) H 1 ) )
4831, 43, 47sylancl 644 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( P `
 s ) H 1 ) )
49 fvco3 5792 . . . 4  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( ( G  o.  P ) `  s )  =  ( G `  ( P `
 s ) ) )
5026, 49sylan 458 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( G  o.  P
) `  s )  =  ( G `  ( P `  s ) ) )
5142, 48, 503eqtr4d 2477 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( G  o.  P ) `  s ) )
521, 5, 8, 24, 41, 51ishtpyd 18990 1  |-  ( ph  ->  N  e.  ( ( F  o.  P ) ( J Htpy  L ) ( G  o.  P
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   U.cuni 4007    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   0cc0 8980   1c1 8981   [,]cicc 10909   Topctop 16948  TopOnctopon 16949    Cn ccn 17278    tX ctx 17582   IIcii 18895   Htpy chtpy 18982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-icc 10913  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-topgen 13657  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-top 16953  df-bases 16955  df-topon 16956  df-cn 17281  df-tx 17584  df-ii 18897  df-htpy 18985
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