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Theorem cnmpt22f 17699
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 17297 . . . 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 2435 . . . 4  |-  U. L  =  U. L
87toptopon 16990 . . 3  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
96, 8sylib 189 . 2  |-  ( ph  ->  L  e.  (TopOn `  U. L ) )
10 cntop2 17297 . . . 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 2435 . . . 4  |-  U. M  =  U. M
1312toptopon 16990 . . 3  |-  ( M  e.  Top  <->  M  e.  (TopOn `  U. M ) )
1411, 13sylib 189 . 2  |-  ( ph  ->  M  e.  (TopOn `  U. M ) )
15 txtopon 17615 . . . . . . 7  |-  ( ( L  e.  (TopOn `  U. L )  /\  M  e.  (TopOn `  U. M ) )  ->  ( L  tX  M )  e.  (TopOn `  ( U. L  X.  U. M ) ) )
169, 14, 15syl2anc 643 . . . . . 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 17297 . . . . . . . 8  |-  ( F  e.  ( ( L 
tX  M )  Cn  N )  ->  N  e.  Top )
1917, 18syl 16 . . . . . . 7  |-  ( ph  ->  N  e.  Top )
20 eqid 2435 . . . . . . . 8  |-  U. N  =  U. N
2120toptopon 16990 . . . . . . 7  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
2219, 21sylib 189 . . . . . 6  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
23 cnf2 17305 . . . . . 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 1184 . . . . 5  |-  ( ph  ->  F : ( U. L  X.  U. M ) --> U. N )
25 ffn 5583 . . . . 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 6170 . . . 4  |-  ( F  Fn  ( U. L  X.  U. M )  <->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2826, 27sylib 189 . . 3  |-  ( ph  ->  F  =  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) ) )
2928, 17eqeltrrd 2510 . 2  |-  ( ph  ->  ( z  e.  U. L ,  w  e.  U. M  |->  ( z F w ) )  e.  ( ( L  tX  M )  Cn  N
) )
30 oveq12 6082 . 2  |-  ( ( z  =  A  /\  w  =  B )  ->  ( z F w )  =  ( A F B ) )
311, 2, 3, 4, 9, 14, 29, 30cnmpt22 17698 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 1652    e. wcel 1725   U.cuni 4007    X. cxp 4868    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Topctop 16950  TopOnctopon 16951    Cn ccn 17280    tX ctx 17584
This theorem is referenced by:  cnmptcom  17702  cnmpt2plusg  18110  istgp2  18113  cnmpt2vsca  18216  cnmpt2ds  18866  divcn  18890  cnrehmeo  18970  htpycom  18993  htpyco1  18995  htpycc  18997  reparphti  19014  pcohtpylem  19036  cnmpt2ip  19194  cxpcn  20621  vmcn  22187  dipcn  22211  mndpluscn  24304  cvxscon  24922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-cn 17283  df-tx 17586
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