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Theorem cnmpt2c 17364
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 2284 . . 3  |-  ( z  =  <. x ,  y
>.  ->  P  =  P )
21mpt2mpt 5939 . 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 17286 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
63, 4, 5syl2anc 642 . . 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 17356 . 2  |-  ( ph  ->  ( z  e.  ( X  X.  Y ) 
|->  P )  e.  ( ( J  tX  K
)  Cn  L ) )
102, 9syl5eqelr 2368 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 1623    e. wcel 1684   <.cop 3643    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255
This theorem is referenced by:  cnrehmeo  18451  pcopt  18520  pcopt2  18521  vmcn  21272  dipcn  21296  cvxscon  23774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-cnp 16958  df-tx 17257
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