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Theorem cnmpt2c 17694
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 2436 . . 3  |-  ( z  =  <. x ,  y
>.  ->  P  =  P )
21mpt2mpt 6157 . 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 17615 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
63, 4, 5syl2anc 643 . . 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 17686 . 2  |-  ( ph  ->  ( z  e.  ( X  X.  Y ) 
|->  P )  e.  ( ( J  tX  K
)  Cn  L ) )
102, 9syl5eqelr 2520 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  P )  e.  ( ( J  tX  K
)  Cn  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   <.cop 3809    e. cmpt 4258    X. cxp 4868   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075  TopOnctopon 16951    Cn ccn 17280    tX ctx 17584
This theorem is referenced by:  cnrehmeo  18970  pcopt  19039  pcopt2  19040  vmcn  22187  dipcn  22211  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-cnp 17284  df-tx 17586
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