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Theorem 0pval 25688
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 25687 . . 3 0𝑝 = (ℂ × {0})
21fveq1i 6894 . 2 (0𝑝𝐴) = ((ℂ × {0})‘𝐴)
3 c0ex 11249 . . 3 0 ∈ V
43fvconst2 7213 . 2 (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0)
52, 4eqtrid 2778 1 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {csn 4623   × cxp 5672  cfv 6546  cc 11147  0cc0 11149  0𝑝c0p 25686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-mulcl 11211  ax-i2m1 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-0p 25687
This theorem is referenced by:  0plef  25689  0pledm  25690  itg1ge0  25703  mbfi1fseqlem5  25737  itg2addlem  25776  ne0p  26231  plyeq0lem  26234  plydivlem3  26320  plymul02  34405  dgraa0p  42847
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