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Theorem htpyco1 18875
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 17253 . . 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 17253 . . 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 18781 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1110a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
121, 11cnmpt1st 17622 . . . . 5  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  x )  e.  ( ( J  tX  II )  Cn  J ) )
131, 11, 12, 2cnmpt21f 17626 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( P `  x
) )  e.  ( ( J  tX  II )  Cn  K ) )
141, 11cnmpt2nd 17623 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  y )  e.  ( ( J  tX  II )  Cn  II ) )
15 cntop2 17228 . . . . . . . 8  |-  ( P  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
17 eqid 2388 . . . . . . . 8  |-  U. K  =  U. K
1817toptopon 16922 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
1916, 18sylib 189 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
2019, 3, 6htpycn 18870 . . . . 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 3293 . . . 4  |-  ( ph  ->  H  e.  ( ( K  tX  II )  Cn  L ) )
231, 11, 13, 14, 22cnmpt22f 17629 . . 3  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( ( P `  x ) H y ) )  e.  ( ( J  tX  II )  Cn  L ) )
249, 23syl5eqel 2472 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  L ) )
25 cnf2 17236 . . . . . . 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 5810 . . . . 5  |-  ( (
ph  /\  s  e.  X )  ->  ( P `  s )  e.  U. K )
2819, 3, 6, 21htpyi 18871 . . . . 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 10948 . . . 4  |-  0  e.  ( 0 [,] 1
)
33 fveq2 5669 . . . . . 6  |-  ( x  =  s  ->  ( P `  x )  =  ( P `  s ) )
34 id 20 . . . . . 6  |-  ( y  =  0  ->  y  =  0 )
3533, 34oveqan12d 6040 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 0 ) )
36 ovex 6046 . . . . 5  |-  ( ( P `  s ) H 0 )  e. 
_V
3735, 9, 36ovmpt2a 6144 . . . 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 5740 . . . 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 2430 . 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 10949 . . . 4  |-  1  e.  ( 0 [,] 1
)
44 id 20 . . . . . 6  |-  ( y  =  1  ->  y  =  1 )
4533, 44oveqan12d 6040 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 1 ) )
46 ovex 6046 . . . . 5  |-  ( ( P `  s ) H 1 )  e. 
_V
4745, 9, 46ovmpt2a 6144 . . . 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 5740 . . . 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 2430 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( G  o.  P ) `  s ) )
521, 5, 8, 24, 41, 51ishtpyd 18872 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 1649    e. wcel 1717   U.cuni 3958    o. ccom 4823   -->wf 5391   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   0cc0 8924   1c1 8925   [,]cicc 10852   Topctop 16882  TopOnctopon 16883    Cn ccn 17211    tX ctx 17514   IIcii 18777   Htpy chtpy 18864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-icc 10856  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-topgen 13595  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-top 16887  df-bases 16889  df-topon 16890  df-cn 17214  df-tx 17516  df-ii 18779  df-htpy 18867
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