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Theorem 0pval 24281
 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 24280 . . 3 0𝑝 = (ℂ × {0})
21fveq1i 6662 . 2 (0𝑝𝐴) = ((ℂ × {0})‘𝐴)
3 c0ex 10633 . . 3 0 ∈ V
43fvconst2 6957 . 2 (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0)
52, 4syl5eq 2871 1 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  {csn 4550   × cxp 5540  ‘cfv 6343  ℂcc 10533  0cc0 10535  0𝑝c0p 24279 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-mulcl 10597  ax-i2m1 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-0p 24280 This theorem is referenced by:  0plef  24282  0pledm  24283  itg1ge0  24296  mbfi1fseqlem5  24329  itg2addlem  24368  ne0p  24810  plyeq0lem  24813  plydivlem3  24897  plymul02  31876  dgraa0p  40013
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