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Theorem htpyco1 18476
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 16995 . . 3  |-  ( ( P  e.  ( J  Cn  K )  /\  F  e.  ( K  Cn  L ) )  -> 
( F  o.  P
)  e.  ( J  Cn  L ) )
52, 3, 4syl2anc 642 . 2  |-  ( ph  ->  ( F  o.  P
)  e.  ( J  Cn  L ) )
6 htpyco1.g . . 3  |-  ( ph  ->  G  e.  ( K  Cn  L ) )
7 cnco 16995 . . 3  |-  ( ( P  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  P
)  e.  ( J  Cn  L ) )
82, 6, 7syl2anc 642 . 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 18383 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1110a1i 10 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
121, 11cnmpt1st 17362 . . . . 5  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  x )  e.  ( ( J  tX  II )  Cn  J ) )
131, 11, 12, 2cnmpt21f 17366 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( P `  x
) )  e.  ( ( J  tX  II )  Cn  K ) )
141, 11cnmpt2nd 17363 . . . 4  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  y )  e.  ( ( J  tX  II )  Cn  II ) )
15 cntop2 16971 . . . . . . . 8  |-  ( P  e.  ( J  Cn  K )  ->  K  e.  Top )
162, 15syl 15 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
17 eqid 2283 . . . . . . . 8  |-  U. K  =  U. K
1817toptopon 16671 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
1916, 18sylib 188 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  U. K ) )
2019, 3, 6htpycn 18471 . . . . 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 3181 . . . 4  |-  ( ph  ->  H  e.  ( ( K  tX  II )  Cn  L ) )
231, 11, 13, 14, 22cnmpt22f 17369 . . 3  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  ( ( P `  x ) H y ) )  e.  ( ( J  tX  II )  Cn  L ) )
249, 23syl5eqel 2367 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  L ) )
25 cnf2 16979 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  U. K )  /\  P  e.  ( J  Cn  K ) )  ->  P : X
--> U. K )
261, 19, 2, 25syl3anc 1182 . . . . . 6  |-  ( ph  ->  P : X --> U. K
)
27 ffvelrn 5663 . . . . . 6  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( P `  s )  e.  U. K )
2826, 27sylan 457 . . . . 5  |-  ( (
ph  /\  s  e.  X )  ->  ( P `  s )  e.  U. K )
2919, 3, 6, 21htpyi 18472 . . . . 5  |-  ( (
ph  /\  ( P `  s )  e.  U. K )  ->  (
( ( P `  s ) H 0 )  =  ( F `
 ( P `  s ) )  /\  ( ( P `  s ) H 1 )  =  ( G `
 ( P `  s ) ) ) )
3028, 29syldan 456 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  (
( ( P `  s ) H 0 )  =  ( F `
 ( P `  s ) )  /\  ( ( P `  s ) H 1 )  =  ( G `
 ( P `  s ) ) ) )
3130simpld 445 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( P `  s
) H 0 )  =  ( F `  ( P `  s ) ) )
32 simpr 447 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
33 0elunit 10754 . . . 4  |-  0  e.  ( 0 [,] 1
)
34 fveq2 5525 . . . . . 6  |-  ( x  =  s  ->  ( P `  x )  =  ( P `  s ) )
35 id 19 . . . . . 6  |-  ( y  =  0  ->  y  =  0 )
3634, 35oveqan12d 5877 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 0 ) )
37 ovex 5883 . . . . 5  |-  ( ( P `  s ) H 0 )  e. 
_V
3836, 9, 37ovmpt2a 5978 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( ( P `  s
) H 0 ) )
3932, 33, 38sylancl 643 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( ( P `
 s ) H 0 ) )
40 fvco3 5596 . . . 4  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( ( F  o.  P ) `  s )  =  ( F `  ( P `
 s ) ) )
4126, 40sylan 457 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( F  o.  P
) `  s )  =  ( F `  ( P `  s ) ) )
4231, 39, 413eqtr4d 2325 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( ( F  o.  P ) `  s ) )
4330simprd 449 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( P `  s
) H 1 )  =  ( G `  ( P `  s ) ) )
44 1elunit 10755 . . . 4  |-  1  e.  ( 0 [,] 1
)
45 id 19 . . . . . 6  |-  ( y  =  1  ->  y  =  1 )
4634, 45oveqan12d 5877 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( P `
 x ) H y )  =  ( ( P `  s
) H 1 ) )
47 ovex 5883 . . . . 5  |-  ( ( P `  s ) H 1 )  e. 
_V
4846, 9, 47ovmpt2a 5978 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( ( P `  s
) H 1 ) )
4932, 44, 48sylancl 643 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( P `
 s ) H 1 ) )
50 fvco3 5596 . . . 4  |-  ( ( P : X --> U. K  /\  s  e.  X
)  ->  ( ( G  o.  P ) `  s )  =  ( G `  ( P `
 s ) ) )
5126, 50sylan 457 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( G  o.  P
) `  s )  =  ( G `  ( P `  s ) ) )
5243, 49, 513eqtr4d 2325 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( ( G  o.  P ) `  s ) )
531, 5, 8, 24, 42, 52ishtpyd 18473 1  |-  ( ph  ->  N  e.  ( ( F  o.  P ) ( J Htpy  L ) ( G  o.  P
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   U.cuni 3827    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   0cc0 8737   1c1 8738   [,]cicc 10659   Topctop 16631  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255   IIcii 18379   Htpy chtpy 18465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-tx 17257  df-ii 18381  df-htpy 18468
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