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Theorem ne0p 24304
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 23779 . . . 4 (𝐴 ∈ ℂ → (0𝑝𝐴) = 0)
2 fveq1 6410 . . . . 5 (𝐹 = 0𝑝 → (𝐹𝐴) = (0𝑝𝐴))
32eqeq1d 2801 . . . 4 (𝐹 = 0𝑝 → ((𝐹𝐴) = 0 ↔ (0𝑝𝐴) = 0))
41, 3syl5ibrcom 239 . . 3 (𝐴 ∈ ℂ → (𝐹 = 0𝑝 → (𝐹𝐴) = 0))
54necon3d 2992 . 2 (𝐴 ∈ ℂ → ((𝐹𝐴) ≠ 0 → 𝐹 ≠ 0𝑝))
65imp 396 1 ((𝐴 ∈ ℂ ∧ (𝐹𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2971  cfv 6101  cc 10222  0cc0 10224  0𝑝c0p 23777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-mulcl 10286  ax-i2m1 10292
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-0p 23778
This theorem is referenced by:  dgrmulc  24368  qaa  24419  iaa  24421  aareccl  24422  dchrfi  25332
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