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Theorem 0pval 23189
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 23188 . . 3 0𝑝 = (ℂ × {0})
21fveq1i 6089 . 2 (0𝑝𝐴) = ((ℂ × {0})‘𝐴)
3 c0ex 9891 . . 3 0 ∈ V
43fvconst2 6352 . 2 (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0)
52, 4syl5eq 2655 1 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  {csn 4124   × cxp 5026  cfv 5790  cc 9791  0cc0 9793  0𝑝c0p 23187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-mulcl 9855  ax-i2m1 9861
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-0p 23188
This theorem is referenced by:  0plef  23190  0pledm  23191  itg1ge0  23204  mbfi1fseqlem5  23237  itg2addlem  23276  ne0p  23712  plyeq0lem  23715  plydivlem3  23799  plymul02  29783  dgraa0p  36562
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