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Theorem 0pval 23658
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 23657 . . 3 0𝑝 = (ℂ × {0})
21fveq1i 6334 . 2 (0𝑝𝐴) = ((ℂ × {0})‘𝐴)
3 c0ex 10240 . . 3 0 ∈ V
43fvconst2 6616 . 2 (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0)
52, 4syl5eq 2817 1 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {csn 4317   × cxp 5248  cfv 6030  cc 10140  0cc0 10142  0𝑝c0p 23656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-mulcl 10204  ax-i2m1 10210
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-0p 23657
This theorem is referenced by:  0plef  23659  0pledm  23660  itg1ge0  23673  mbfi1fseqlem5  23706  itg2addlem  23745  ne0p  24183  plyeq0lem  24186  plydivlem3  24270  plymul02  30963  dgraa0p  38245
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