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Theorem ne0p 23767
Description: A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
ne0p ((𝐴 ∈ ℂ ∧ (𝐹𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)

Proof of Theorem ne0p
StepHypRef Expression
1 0pval 23244 . . . 4 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
2 fveq1 6102 . . . . 5 (𝐹 = 0𝑝 → (𝐹𝐴) = (0𝑝𝐴))
32eqeq1d 2612 . . . 4 (𝐹 = 0𝑝 → ((𝐹𝐴) = 0 ↔ (0𝑝𝐴) = 0))
41, 3syl5ibrcom 236 . . 3 (𝐴 ∈ ℂ → (𝐹 = 0𝑝 → (𝐹𝐴) = 0))
54necon3d 2803 . 2 (𝐴 ∈ ℂ → ((𝐹𝐴) ≠ 0 → 𝐹 ≠ 0𝑝))
65imp 444 1 ((𝐴 ∈ ℂ ∧ (𝐹𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  cfv 5804  cc 9813  0cc0 9815  0𝑝c0p 23242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-0p 23243
This theorem is referenced by:  dgrmulc  23831  qaa  23882  iaa  23884  aareccl  23885  dchrfi  24780
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