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Theorem 0pval 19592
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 19591 . . 3  |-  0 p  =  ( CC  X.  { 0 } )
21fveq1i 5758 . 2  |-  ( 0 p `  A )  =  ( ( CC 
X.  { 0 } ) `  A )
3 c0ex 9116 . . 3  |-  0  e.  _V
43fvconst2 5976 . 2  |-  ( A  e.  CC  ->  (
( CC  X.  {
0 } ) `  A )  =  0 )
52, 4syl5eq 2486 1  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   {csn 3838    X. cxp 4905   ` cfv 5483   CCcc 9019   0cc0 9021   0 pc0p 19590
This theorem is referenced by:  0plef  19593  0pledm  19594  itg1ge0  19607  mbfi1fseqlem5  19640  itg2addlem  19679  ne0p  20157  plyeq0lem  20160  plydivlem3  20243  dgraa0p  27369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-mulcl 9083  ax-i2m1 9089
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-0p 19591
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