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Theorem 0pval 23483
Description: The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
0pval (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)

Proof of Theorem 0pval
StepHypRef Expression
1 df-0p 23482 . . 3 0𝑝 = (ℂ × {0})
21fveq1i 6230 . 2 (0𝑝𝐴) = ((ℂ × {0})‘𝐴)
3 c0ex 10072 . . 3 0 ∈ V
43fvconst2 6510 . 2 (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0)
52, 4syl5eq 2697 1 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {csn 4210   × cxp 5141  cfv 5926  cc 9972  0cc0 9974  0𝑝c0p 23481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-mulcl 10036  ax-i2m1 10042
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-0p 23482
This theorem is referenced by:  0plef  23484  0pledm  23485  itg1ge0  23498  mbfi1fseqlem5  23531  itg2addlem  23570  ne0p  24008  plyeq0lem  24011  plydivlem3  24095  plymul02  30751  dgraa0p  38036
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