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Theorem 0pval 19563
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 19562 . . 3  |-  0 p  =  ( CC  X.  { 0 } )
21fveq1i 5729 . 2  |-  ( 0 p `  A )  =  ( ( CC 
X.  { 0 } ) `  A )
3 c0ex 9085 . . 3  |-  0  e.  _V
43fvconst2 5947 . 2  |-  ( A  e.  CC  ->  (
( CC  X.  {
0 } ) `  A )  =  0 )
52, 4syl5eq 2480 1  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {csn 3814    X. cxp 4876   ` cfv 5454   CCcc 8988   0cc0 8990   0 pc0p 19561
This theorem is referenced by:  0plef  19564  0pledm  19565  itg1ge0  19578  mbfi1fseqlem5  19611  itg2addlem  19650  ne0p  20126  plyeq0lem  20129  plydivlem3  20212  dgraa0p  27331
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-mulcl 9052  ax-i2m1 9058
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-0p 19562
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