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Theorem cnmpt22f 19906
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 19503 . . . 4  |-  ( ( x  e.  X , 
y  e.  Y  |->  A )  e.  ( ( J  tX  K )  Cn  L )  ->  L  e.  Top )
63, 5syl 16 . . 3  |-  ( ph  ->  L  e.  Top )
7 eqid 2462 . . . 4  |-  U. L  =  U. L
87toptopon 19196 . . 3  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
96, 8sylib 196 . 2  |-  ( ph  ->  L  e.  (TopOn `  U. L ) )
10 cntop2 19503 . . . 4  |-  ( ( x  e.  X , 
y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M )  ->  M  e.  Top )
114, 10syl 16 . . 3  |-  ( ph  ->  M  e.  Top )
12 eqid 2462 . . . 4  |-  U. M  =  U. M
1312toptopon 19196 . . 3  |-  ( M  e.  Top  <->  M  e.  (TopOn `  U. M ) )
1411, 13sylib 196 . 2  |-  ( ph  ->  M  e.  (TopOn `  U. M ) )
15 txtopon 19822 . . . . . . 7  |-  ( ( L  e.  (TopOn `  U. L )  /\  M  e.  (TopOn `  U. M ) )  ->  ( L  tX  M )  e.  (TopOn `  ( U. L  X.  U. M ) ) )
169, 14, 15syl2anc 661 . . . . . 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 19503 . . . . . . . 8  |-  ( F  e.  ( ( L 
tX  M )  Cn  N )  ->  N  e.  Top )
1917, 18syl 16 . . . . . . 7  |-  ( ph  ->  N  e.  Top )
20 eqid 2462 . . . . . . . 8  |-  U. N  =  U. N
2120toptopon 19196 . . . . . . 7  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
2219, 21sylib 196 . . . . . 6  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
23 cnf2 19511 . . . . . 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 1223 . . . . 5  |-  ( ph  ->  F : ( U. L  X.  U. M ) --> U. N )
25 ffn 5724 . . . . 5  |-  ( F : ( U. L  X.  U. M ) --> U. N  ->  F  Fn  ( U. L  X.  U. M ) )
2624, 25syl 16 . . . 4  |-  ( ph  ->  F  Fn  ( U. L  X.  U. M ) )
27 fnov 6387 . . . 4  |-  ( F  Fn  ( U. L  X.  U. M )  <->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2826, 27sylib 196 . . 3  |-  ( ph  ->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2928, 17eqeltrrd 2551 . 2  |-  ( ph  ->  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) )  e.  ( ( L  tX  M )  Cn  N
) )
30 oveq12 6286 . 2  |-  ( ( z  =  A  /\  w  =  B )  ->  ( z F w )  =  ( A F B ) )
311, 2, 3, 4, 9, 14, 29, 30cnmpt22 19905 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A F B ) )  e.  ( ( J  tX  K
)  Cn  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   U.cuni 4240    X. cxp 4992    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   Topctop 19156  TopOnctopon 19157    Cn ccn 19486    tX ctx 19791
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-map 7414  df-topgen 14690  df-top 19161  df-bases 19163  df-topon 19164  df-cn 19489  df-tx 19793
This theorem is referenced by:  cnmptcom  19909  cnmpt2plusg  20317  istgp2  20320  cnmpt2vsca  20427  cnmpt2ds  21078  divcn  21102  cnrehmeo  21183  htpycom  21206  htpyco1  21208  htpycc  21210  reparphti  21227  pcohtpylem  21249  cnmpt2ip  21418  cxpcn  22842  vmcn  25273  dipcn  25297  mndpluscn  27532  cvxscon  28316
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