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Theorem cnmpt2c 20039
Description: A constant function 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 ) )
cnmpt2c.l  |-  ( ph  ->  L  e.  (TopOn `  Z ) )
cnmpt2c.p  |-  ( ph  ->  P  e.  Z )
Assertion
Ref Expression
cnmpt2c  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  P )  e.  ( ( J  tX  K
)  Cn  L ) )
Distinct variable groups:    x, y, L    ph, x, y    x, X, y    x, P, y   
x, Y, y    x, Z, y
Allowed substitution hints:    J( x, y)    K( x, y)

Proof of Theorem cnmpt2c
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqidd 2468 . . 3  |-  ( z  =  <. x ,  y
>.  ->  P  =  P )
21mpt2mpt 6389 . 2  |-  ( z  e.  ( X  X.  Y )  |->  P )  =  ( x  e.  X ,  y  e.  Y  |->  P )
3 cnmpt21.j . . . 4  |-  ( ph  ->  J  e.  (TopOn `  X ) )
4 cnmpt21.k . . . 4  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
5 txtopon 19960 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
63, 4, 5syl2anc 661 . . 3  |-  ( ph  ->  ( J  tX  K
)  e.  (TopOn `  ( X  X.  Y
) ) )
7 cnmpt2c.l . . 3  |-  ( ph  ->  L  e.  (TopOn `  Z ) )
8 cnmpt2c.p . . 3  |-  ( ph  ->  P  e.  Z )
96, 7, 8cnmptc 20031 . 2  |-  ( ph  ->  ( z  e.  ( X  X.  Y ) 
|->  P )  e.  ( ( J  tX  K
)  Cn  L ) )
102, 9syl5eqelr 2560 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  P )  e.  ( ( J  tX  K
)  Cn  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   <.cop 4039    |-> cmpt 4511    X. cxp 5003   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297  TopOnctopon 19264    Cn ccn 19593    tX ctx 19929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-topgen 14716  df-top 19268  df-bases 19270  df-topon 19271  df-cn 19596  df-cnp 19597  df-tx 19931
This theorem is referenced by:  cnrehmeo  21321  pcopt  21390  pcopt2  21391  vmcn  25432  dipcn  25456  cvxscon  28513
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