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Theorem 0pval 19124
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 19123 . . 3  |-  0 p  =  ( CC  X.  { 0 } )
21fveq1i 5606 . 2  |-  ( 0 p `  A )  =  ( ( CC 
X.  { 0 } ) `  A )
3 c0ex 8919 . . 3  |-  0  e.  _V
43fvconst2 5810 . 2  |-  ( A  e.  CC  ->  (
( CC  X.  {
0 } ) `  A )  =  0 )
52, 4syl5eq 2402 1  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   {csn 3716    X. cxp 4766   ` cfv 5334   CCcc 8822   0cc0 8824   0 pc0p 19122
This theorem is referenced by:  0plef  19125  0pledm  19126  itg1ge0  19139  mbfi1fseqlem5  19172  itg2addlem  19211  ne0p  19687  plyeq0lem  19690  plydivlem3  19773  dgraa0p  26677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-mulcl 8886  ax-i2m1 8892
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-0p 19123
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