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Theorem 0pval 25051
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 25050 . . 3 0𝑝 = (ℂ × {0})
21fveq1i 6848 . 2 (0𝑝𝐴) = ((ℂ × {0})‘𝐴)
3 c0ex 11156 . . 3 0 ∈ V
43fvconst2 7158 . 2 (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0)
52, 4eqtrid 2789 1 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {csn 4591   × cxp 5636  cfv 6501  cc 11056  0cc0 11058  0𝑝c0p 25049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-mulcl 11120  ax-i2m1 11126
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-0p 25050
This theorem is referenced by:  0plef  25052  0pledm  25053  itg1ge0  25066  mbfi1fseqlem5  25100  itg2addlem  25139  ne0p  25584  plyeq0lem  25587  plydivlem3  25671  plymul02  33198  dgraa0p  41505
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