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Theorem htpyco1 20509
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 18829 . . 3  |-  ( ( P  e.  ( J  Cn  K )  /\  F  e.  ( K  Cn  L ) )  -> 
( F  o.  P
)  e.  ( J  Cn  L ) )
52, 3, 4syl2anc 656 . 2  |-  ( ph  ->  ( F  o.  P
)  e.  ( J  Cn  L ) )
6 htpyco1.g . . 3  |-  ( ph  ->  G  e.  ( K  Cn  L ) )
7 cnco 18829 . . 3  |-  ( ( P  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  P
)  e.  ( J  Cn  L ) )
82, 6, 7syl2anc 656 . 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 20414 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1110a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
121, 11cnmpt1st 19200 . . . . 5  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  x )  e.  ( ( J  tX  II )  Cn  J ) )
131, 11, 12, 2cnmpt21f 19204 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( P `  x
) )  e.  ( ( J  tX  II )  Cn  K ) )
141, 11cnmpt2nd 19201 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  y )  e.  ( ( J  tX  II )  Cn  II ) )
15 cntop2 18804 . . . . . . . 8  |-  ( P  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 16 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
17 eqid 2441 . . . . . . . 8  |-  U. K  =  U. K
1817toptopon 18497 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
1916, 18sylib 196 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
2019, 3, 6htpycn 20504 . . . . 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 3354 . . . 4  |-  ( ph  ->  H  e.  ( ( K  tX  II )  Cn  L ) )
231, 11, 13, 14, 22cnmpt22f 19207 . . 3  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( ( P `  x ) H y ) )  e.  ( ( J  tX  II )  Cn  L ) )
249, 23syl5eqel 2525 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  L ) )
25 cnf2 18812 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  P  e.  ( J  Cn  K ) )  ->  P : X
--> U. K )
261, 19, 2, 25syl3anc 1213 . . . . . 6  |-  ( ph  ->  P : X --> U. K
)
2726ffvelrnda 5840 . . . . 5  |-  ( (
ph  /\  s  e.  X )  ->  ( P `  s )  e.  U. K )
2819, 3, 6, 21htpyi 20505 . . . . 5  |-  ( (
ph  /\  ( P `  s )  e.  U. K )  ->  (
( ( P `  s ) H 0 )  =  ( F `
 ( P `  s ) )  /\  ( ( P `  s ) H 1 )  =  ( G `
 ( P `  s ) ) ) )
2927, 28syldan 467 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  (
( ( P `  s ) H 0 )  =  ( F `
 ( P `  s ) )  /\  ( ( P `  s ) H 1 )  =  ( G `
 ( P `  s ) ) ) )
3029simpld 456 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( P `  s
) H 0 )  =  ( F `  ( P `  s ) ) )
31 simpr 458 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
32 0elunit 11399 . . . 4  |-  0  e.  ( 0 [,] 1
)
33 fveq2 5688 . . . . . 6  |-  ( x  =  s  ->  ( P `  x )  =  ( P `  s ) )
34 id 22 . . . . . 6  |-  ( y  =  0  ->  y  =  0 )
3533, 34oveqan12d 6109 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 0 ) )
36 ovex 6115 . . . . 5  |-  ( ( P `  s ) H 0 )  e. 
_V
3735, 9, 36ovmpt2a 6220 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( ( P `  s
) H 0 ) )
3831, 32, 37sylancl 657 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( ( P `
 s ) H 0 ) )
39 fvco3 5765 . . . 4  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( ( F  o.  P ) `  s )  =  ( F `  ( P `
 s ) ) )
4026, 39sylan 468 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( F  o.  P
) `  s )  =  ( F `  ( P `  s ) ) )
4130, 38, 403eqtr4d 2483 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( ( F  o.  P ) `  s ) )
4229simprd 460 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( P `  s
) H 1 )  =  ( G `  ( P `  s ) ) )
43 1elunit 11400 . . . 4  |-  1  e.  ( 0 [,] 1
)
44 id 22 . . . . . 6  |-  ( y  =  1  ->  y  =  1 )
4533, 44oveqan12d 6109 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 1 ) )
46 ovex 6115 . . . . 5  |-  ( ( P `  s ) H 1 )  e. 
_V
4745, 9, 46ovmpt2a 6220 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( ( P `  s
) H 1 ) )
4831, 43, 47sylancl 657 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( P `
 s ) H 1 ) )
49 fvco3 5765 . . . 4  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( ( G  o.  P ) `  s )  =  ( G `  ( P `
 s ) ) )
5026, 49sylan 468 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( G  o.  P
) `  s )  =  ( G `  ( P `  s ) ) )
5142, 48, 503eqtr4d 2483 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( G  o.  P ) `  s ) )
521, 5, 8, 24, 41, 51ishtpyd 20506 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 1364    e. wcel 1761   U.cuni 4088    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   0cc0 9278   1c1 9279   [,]cicc 11299   Topctop 18457  TopOnctopon 18458    Cn ccn 18787    tX ctx 19092   IIcii 20410   Htpy chtpy 20498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-icc 11303  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-cn 18790  df-tx 19094  df-ii 20412  df-htpy 20501
This theorem is referenced by: (None)
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