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Theorem 0pval 23729
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 23728 . . 3 0𝑝 = (ℂ × {0})
21fveq1i 6376 . 2 (0𝑝𝐴) = ((ℂ × {0})‘𝐴)
3 c0ex 10287 . . 3 0 ∈ V
43fvconst2 6662 . 2 (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0)
52, 4syl5eq 2811 1 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  {csn 4334   × cxp 5275  cfv 6068  cc 10187  0cc0 10189  0𝑝c0p 23727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-mulcl 10251  ax-i2m1 10257
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-0p 23728
This theorem is referenced by:  0plef  23730  0pledm  23731  itg1ge0  23744  mbfi1fseqlem5  23777  itg2addlem  23816  ne0p  24254  plyeq0lem  24257  plydivlem3  24341  plymul02  31072  dgraa0p  38396
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