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Theorem htpyco1 20555
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 18875 . . 3  |-  ( ( P  e.  ( J  Cn  K )  /\  F  e.  ( K  Cn  L ) )  -> 
( F  o.  P
)  e.  ( J  Cn  L ) )
52, 3, 4syl2anc 661 . 2  |-  ( ph  ->  ( F  o.  P
)  e.  ( J  Cn  L ) )
6 htpyco1.g . . 3  |-  ( ph  ->  G  e.  ( K  Cn  L ) )
7 cnco 18875 . . 3  |-  ( ( P  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  P
)  e.  ( J  Cn  L ) )
82, 6, 7syl2anc 661 . 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 20460 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1110a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
121, 11cnmpt1st 19246 . . . . 5  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  x )  e.  ( ( J  tX  II )  Cn  J ) )
131, 11, 12, 2cnmpt21f 19250 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( P `  x
) )  e.  ( ( J  tX  II )  Cn  K ) )
141, 11cnmpt2nd 19247 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  y )  e.  ( ( J  tX  II )  Cn  II ) )
15 cntop2 18850 . . . . . . . 8  |-  ( P  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
17 eqid 2443 . . . . . . . 8  |-  U. K  =  U. K
1817toptopon 18543 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
1916, 18sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
2019, 3, 6htpycn 20550 . . . . 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 3362 . . . 4  |-  ( ph  ->  H  e.  ( ( K  tX  II )  Cn  L ) )
231, 11, 13, 14, 22cnmpt22f 19253 . . 3  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( ( P `  x ) H y ) )  e.  ( ( J  tX  II )  Cn  L ) )
249, 23syl5eqel 2527 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  L ) )
25 cnf2 18858 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  P  e.  ( J  Cn  K ) )  ->  P : X
--> U. K )
261, 19, 2, 25syl3anc 1218 . . . . . 6  |-  ( ph  ->  P : X --> U. K
)
2726ffvelrnda 5848 . . . . 5  |-  ( (
ph  /\  s  e.  X )  ->  ( P `  s )  e.  U. K )
2819, 3, 6, 21htpyi 20551 . . . . 5  |-  ( (
ph  /\  ( P `  s )  e.  U. K )  ->  (
( ( P `  s ) H 0 )  =  ( F `
 ( P `  s ) )  /\  ( ( P `  s ) H 1 )  =  ( G `
 ( P `  s ) ) ) )
2927, 28syldan 470 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  (
( ( P `  s ) H 0 )  =  ( F `
 ( P `  s ) )  /\  ( ( P `  s ) H 1 )  =  ( G `
 ( P `  s ) ) ) )
3029simpld 459 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( P `  s
) H 0 )  =  ( F `  ( P `  s ) ) )
31 simpr 461 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
32 0elunit 11408 . . . 4  |-  0  e.  ( 0 [,] 1
)
33 fveq2 5696 . . . . . 6  |-  ( x  =  s  ->  ( P `  x )  =  ( P `  s ) )
34 id 22 . . . . . 6  |-  ( y  =  0  ->  y  =  0 )
3533, 34oveqan12d 6115 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 0 ) )
36 ovex 6121 . . . . 5  |-  ( ( P `  s ) H 0 )  e. 
_V
3735, 9, 36ovmpt2a 6226 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( ( P `  s
) H 0 ) )
3831, 32, 37sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( ( P `
 s ) H 0 ) )
39 fvco3 5773 . . . 4  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( ( F  o.  P ) `  s )  =  ( F `  ( P `
 s ) ) )
4026, 39sylan 471 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( F  o.  P
) `  s )  =  ( F `  ( P `  s ) ) )
4130, 38, 403eqtr4d 2485 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( ( F  o.  P ) `  s ) )
4229simprd 463 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( P `  s
) H 1 )  =  ( G `  ( P `  s ) ) )
43 1elunit 11409 . . . 4  |-  1  e.  ( 0 [,] 1
)
44 id 22 . . . . . 6  |-  ( y  =  1  ->  y  =  1 )
4533, 44oveqan12d 6115 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 1 ) )
46 ovex 6121 . . . . 5  |-  ( ( P `  s ) H 1 )  e. 
_V
4745, 9, 46ovmpt2a 6226 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( ( P `  s
) H 1 ) )
4831, 43, 47sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( P `
 s ) H 1 ) )
49 fvco3 5773 . . . 4  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( ( G  o.  P ) `  s )  =  ( G `  ( P `
 s ) ) )
5026, 49sylan 471 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( G  o.  P
) `  s )  =  ( G `  ( P `  s ) ) )
5142, 48, 503eqtr4d 2485 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( G  o.  P ) `  s ) )
521, 5, 8, 24, 41, 51ishtpyd 20552 1  |-  ( ph  ->  N  e.  ( ( F  o.  P ) ( J Htpy  L ) ( G  o.  P
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   U.cuni 4096    o. ccom 4849   -->wf 5419   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   0cc0 9287   1c1 9288   [,]cicc 11308   Topctop 18503  TopOnctopon 18504    Cn ccn 18833    tX ctx 19138   IIcii 20456   Htpy chtpy 20544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-icc 11312  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-top 18508  df-bases 18510  df-topon 18511  df-cn 18836  df-tx 19140  df-ii 20458  df-htpy 20547
This theorem is referenced by: (None)
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