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Theorem 0pval 21841
Description: The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
0pval  |-  ( A  e.  CC  ->  (
0p `  A
)  =  0 )

Proof of Theorem 0pval
StepHypRef Expression
1 df-0p 21840 . . 3  |-  0p  =  ( CC  X.  { 0 } )
21fveq1i 5867 . 2  |-  ( 0p `  A )  =  ( ( CC 
X.  { 0 } ) `  A )
3 c0ex 9590 . . 3  |-  0  e.  _V
43fvconst2 6116 . 2  |-  ( A  e.  CC  ->  (
( CC  X.  {
0 } ) `  A )  =  0 )
52, 4syl5eq 2520 1  |-  ( A  e.  CC  ->  (
0p `  A
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {csn 4027    X. cxp 4997   ` cfv 5588   CCcc 9490   0cc0 9492   0pc0p 21839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-mulcl 9554  ax-i2m1 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-0p 21840
This theorem is referenced by:  0plef  21842  0pledm  21843  itg1ge0  21856  mbfi1fseqlem5  21889  itg2addlem  21928  ne0p  22367  plyeq0lem  22370  plydivlem3  22453  dgraa0p  30731
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