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Theorem cnmpt22f 17369
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
cnmpt21.k  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
cnmpt21.a  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J  tX  K
)  Cn  L ) )
cnmpt2t.b  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K
)  Cn  M ) )
cnmpt22f.f  |-  ( ph  ->  F  e.  ( ( L  tX  M )  Cn  N ) )
Assertion
Ref Expression
cnmpt22f  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A F B ) )  e.  ( ( J  tX  K
)  Cn  N ) )
Distinct variable groups:    x, y, F    x, L, y    ph, x, y    x, X, y    x, M, y    x, N, y   
x, Y, y
Allowed substitution hints:    A( x, y)    B( x, y)    J( x, y)    K( x, y)

Proof of Theorem cnmpt22f
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnmpt21.j . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 cnmpt21.k . 2  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
3 cnmpt21.a . 2  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( J  tX  K
)  Cn  L ) )
4 cnmpt2t.b . 2  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( J  tX  K
)  Cn  M ) )
5 cntop2 16971 . . . 4  |-  ( ( x  e.  X , 
y  e.  Y  |->  A )  e.  ( ( J  tX  K )  Cn  L )  ->  L  e.  Top )
63, 5syl 15 . . 3  |-  ( ph  ->  L  e.  Top )
7 eqid 2283 . . . 4  |-  U. L  =  U. L
87toptopon 16671 . . 3  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
96, 8sylib 188 . 2  |-  ( ph  ->  L  e.  (TopOn `  U. L ) )
10 cntop2 16971 . . . 4  |-  ( ( x  e.  X , 
y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M )  ->  M  e.  Top )
114, 10syl 15 . . 3  |-  ( ph  ->  M  e.  Top )
12 eqid 2283 . . . 4  |-  U. M  =  U. M
1312toptopon 16671 . . 3  |-  ( M  e.  Top  <->  M  e.  (TopOn `  U. M ) )
1411, 13sylib 188 . 2  |-  ( ph  ->  M  e.  (TopOn `  U. M ) )
15 txtopon 17286 . . . . . . 7  |-  ( ( L  e.  (TopOn `  U. L )  /\  M  e.  (TopOn `  U. M ) )  ->  ( L  tX  M )  e.  (TopOn `  ( U. L  X.  U. M ) ) )
169, 14, 15syl2anc 642 . . . . . 6  |-  ( ph  ->  ( L  tX  M
)  e.  (TopOn `  ( U. L  X.  U. M ) ) )
17 cnmpt22f.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( ( L  tX  M )  Cn  N ) )
18 cntop2 16971 . . . . . . . 8  |-  ( F  e.  ( ( L 
tX  M )  Cn  N )  ->  N  e.  Top )
1917, 18syl 15 . . . . . . 7  |-  ( ph  ->  N  e.  Top )
20 eqid 2283 . . . . . . . 8  |-  U. N  =  U. N
2120toptopon 16671 . . . . . . 7  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
2219, 21sylib 188 . . . . . 6  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
23 cnf2 16979 . . . . . 6  |-  ( ( ( L  tX  M
)  e.  (TopOn `  ( U. L  X.  U. M ) )  /\  N  e.  (TopOn `  U. N )  /\  F  e.  ( ( L  tX  M )  Cn  N
) )  ->  F : ( U. L  X.  U. M ) --> U. N )
2416, 22, 17, 23syl3anc 1182 . . . . 5  |-  ( ph  ->  F : ( U. L  X.  U. M ) --> U. N )
25 ffn 5389 . . . . 5  |-  ( F : ( U. L  X.  U. M ) --> U. N  ->  F  Fn  ( U. L  X.  U. M ) )
2624, 25syl 15 . . . 4  |-  ( ph  ->  F  Fn  ( U. L  X.  U. M ) )
27 fnov 5952 . . . 4  |-  ( F  Fn  ( U. L  X.  U. M )  <->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2826, 27sylib 188 . . 3  |-  ( ph  ->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2928, 17eqeltrrd 2358 . 2  |-  ( ph  ->  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) )  e.  ( ( L  tX  M )  Cn  N
) )
30 oveq12 5867 . 2  |-  ( ( z  =  A  /\  w  =  B )  ->  ( z F w )  =  ( A F B ) )
311, 2, 3, 4, 9, 14, 29, 30cnmpt22 17368 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A F B ) )  e.  ( ( J  tX  K
)  Cn  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   U.cuni 3827    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Topctop 16631  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255
This theorem is referenced by:  cnmptcom  17372  cnmpt2plusg  17771  istgp2  17774  cnmpt2vsca  17877  cnmpt2ds  18348  divcn  18372  cnrehmeo  18451  htpycom  18474  htpyco1  18476  htpycc  18478  reparphti  18495  pcohtpylem  18517  cnmpt2ip  18675  cxpcn  20085  vmcn  21272  dipcn  21296  mndpluscn  23299  cvxscon  23774
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-tx 17257
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