MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0pval Unicode version

Theorem 0pval 19524
Description: The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
0pval  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )

Proof of Theorem 0pval
StepHypRef Expression
1 df-0p 19523 . . 3  |-  0 p  =  ( CC  X.  { 0 } )
21fveq1i 5696 . 2  |-  ( 0 p `  A )  =  ( ( CC 
X.  { 0 } ) `  A )
3 c0ex 9049 . . 3  |-  0  e.  _V
43fvconst2 5914 . 2  |-  ( A  e.  CC  ->  (
( CC  X.  {
0 } ) `  A )  =  0 )
52, 4syl5eq 2456 1  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {csn 3782    X. cxp 4843   ` cfv 5421   CCcc 8952   0cc0 8954   0 pc0p 19522
This theorem is referenced by:  0plef  19525  0pledm  19526  itg1ge0  19539  mbfi1fseqlem5  19572  itg2addlem  19611  ne0p  20087  plyeq0lem  20090  plydivlem3  20173  dgraa0p  27230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-mulcl 9016  ax-i2m1 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-0p 19523
  Copyright terms: Public domain W3C validator