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Theorem cnmpt2c 20297
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 2458 . . 3  |-  ( z  =  <. x ,  y
>.  ->  P  =  P )
21mpt2mpt 6393 . 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 20218 . . . 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 20289 . 2  |-  ( ph  ->  ( z  e.  ( X  X.  Y ) 
|->  P )  e.  ( ( J  tX  K
)  Cn  L ) )
102, 9syl5eqelr 2550 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 1395    e. wcel 1819   <.cop 4038    |-> cmpt 4515    X. cxp 5006   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298  TopOnctopon 19522    Cn ccn 19852    tX ctx 20187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-topgen 14861  df-top 19526  df-bases 19528  df-topon 19529  df-cn 19855  df-cnp 19856  df-tx 20189
This theorem is referenced by:  cnrehmeo  21579  pcopt  21648  pcopt2  21649  vmcn  25736  dipcn  25760  cvxscon  28885
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