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Theorem ne0p 24807
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 24278 . . . 4 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
2 fveq1 6648 . . . . 5 (𝐹 = 0𝑝 → (𝐹𝐴) = (0𝑝𝐴))
32eqeq1d 2803 . . . 4 (𝐹 = 0𝑝 → ((𝐹𝐴) = 0 ↔ (0𝑝𝐴) = 0))
41, 3syl5ibrcom 250 . . 3 (𝐴 ∈ ℂ → (𝐹 = 0𝑝 → (𝐹𝐴) = 0))
54necon3d 3011 . 2 (𝐴 ∈ ℂ → ((𝐹𝐴) ≠ 0 → 𝐹 ≠ 0𝑝))
65imp 410 1 ((𝐴 ∈ ℂ ∧ (𝐹𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  cfv 6328  cc 10528  0cc0 10530  0𝑝c0p 24276
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-mulcl 10592  ax-i2m1 10598
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-0p 24277
This theorem is referenced by:  dgrmulc  24871  qaa  24922  iaa  24924  aareccl  24925  dchrfi  25842
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