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Mirrors > Home > MPE Home > Th. List > ne0p | Structured version Visualization version GIF version |
Description: A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.) |
Ref | Expression |
---|---|
ne0p | ⊢ ((𝐴 ∈ ℂ ∧ (𝐹‘𝐴) ≠ 0) → 𝐹 ≠ 0𝑝) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0pval 23779 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) | |
2 | fveq1 6410 | . . . . 5 ⊢ (𝐹 = 0𝑝 → (𝐹‘𝐴) = (0𝑝‘𝐴)) | |
3 | 2 | eqeq1d 2801 | . . . 4 ⊢ (𝐹 = 0𝑝 → ((𝐹‘𝐴) = 0 ↔ (0𝑝‘𝐴) = 0)) |
4 | 1, 3 | syl5ibrcom 239 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐹 = 0𝑝 → (𝐹‘𝐴) = 0)) |
5 | 4 | necon3d 2992 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐹‘𝐴) ≠ 0 → 𝐹 ≠ 0𝑝)) |
6 | 5 | imp 396 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (𝐹‘𝐴) ≠ 0) → 𝐹 ≠ 0𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ‘cfv 6101 ℂcc 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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