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Theorem cnmpt22f 20767
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 20334 . . . 4  |-  ( ( x  e.  X , 
y  e.  Y  |->  A )  e.  ( ( J  tX  K )  Cn  L )  ->  L  e.  Top )
63, 5syl 17 . . 3  |-  ( ph  ->  L  e.  Top )
7 eqid 2471 . . . 4  |-  U. L  =  U. L
87toptopon 20025 . . 3  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
96, 8sylib 201 . 2  |-  ( ph  ->  L  e.  (TopOn `  U. L ) )
10 cntop2 20334 . . . 4  |-  ( ( x  e.  X , 
y  e.  Y  |->  B )  e.  ( ( J  tX  K )  Cn  M )  ->  M  e.  Top )
114, 10syl 17 . . 3  |-  ( ph  ->  M  e.  Top )
12 eqid 2471 . . . 4  |-  U. M  =  U. M
1312toptopon 20025 . . 3  |-  ( M  e.  Top  <->  M  e.  (TopOn `  U. M ) )
1411, 13sylib 201 . 2  |-  ( ph  ->  M  e.  (TopOn `  U. M ) )
15 txtopon 20683 . . . . . . 7  |-  ( ( L  e.  (TopOn `  U. L )  /\  M  e.  (TopOn `  U. M ) )  ->  ( L  tX  M )  e.  (TopOn `  ( U. L  X.  U. M ) ) )
169, 14, 15syl2anc 673 . . . . . 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 20334 . . . . . . . 8  |-  ( F  e.  ( ( L 
tX  M )  Cn  N )  ->  N  e.  Top )
1917, 18syl 17 . . . . . . 7  |-  ( ph  ->  N  e.  Top )
20 eqid 2471 . . . . . . . 8  |-  U. N  =  U. N
2120toptopon 20025 . . . . . . 7  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
2219, 21sylib 201 . . . . . 6  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
23 cnf2 20342 . . . . . 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 1292 . . . . 5  |-  ( ph  ->  F : ( U. L  X.  U. M ) --> U. N )
25 ffn 5739 . . . . 5  |-  ( F : ( U. L  X.  U. M ) --> U. N  ->  F  Fn  ( U. L  X.  U. M ) )
2624, 25syl 17 . . . 4  |-  ( ph  ->  F  Fn  ( U. L  X.  U. M ) )
27 fnov 6423 . . . 4  |-  ( F  Fn  ( U. L  X.  U. M )  <->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2826, 27sylib 201 . . 3  |-  ( ph  ->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2928, 17eqeltrrd 2550 . 2  |-  ( ph  ->  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) )  e.  ( ( L  tX  M )  Cn  N
) )
30 oveq12 6317 . 2  |-  ( ( z  =  A  /\  w  =  B )  ->  ( z F w )  =  ( A F B ) )
311, 2, 3, 4, 9, 14, 29, 30cnmpt22 20766 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 1452    e. wcel 1904   U.cuni 4190    X. cxp 4837    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   Topctop 19994  TopOnctopon 19995    Cn ccn 20317    tX ctx 20652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320  df-tx 20654
This theorem is referenced by:  cnmptcom  20770  cnmpt2plusg  21181  istgp2  21184  cnmpt2vsca  21287  cnmpt2ds  21939  divcn  21978  cnrehmeo  22059  htpycom  22085  htpyco1  22087  htpycc  22089  reparphti  22106  pcohtpylem  22128  cnmpt2ip  22297  cxpcn  23764  vmcn  26416  dipcn  26440  mndpluscn  28806  cvxscon  30038
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