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Theorem cnmpt2c 19202
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 2442 . . 3  |-  ( z  =  <. x ,  y
>.  ->  P  =  P )
21mpt2mpt 6181 . 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 19123 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
63, 4, 5syl2anc 656 . . 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 19194 . 2  |-  ( ph  ->  ( z  e.  ( X  X.  Y ) 
|->  P )  e.  ( ( J  tX  K
)  Cn  L ) )
102, 9syl5eqelr 2526 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 1364    e. wcel 1761   <.cop 3880    e. cmpt 4347    X. cxp 4834   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092  TopOnctopon 18458    Cn ccn 18787    tX ctx 19092
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212  df-topgen 14378  df-top 18462  df-bases 18464  df-topon 18465  df-cn 18790  df-cnp 18791  df-tx 19094
This theorem is referenced by:  cnrehmeo  20484  pcopt  20553  pcopt2  20554  vmcn  24029  dipcn  24053  cvxscon  27062
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