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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 25019 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) | |
2 | fveq1 6838 | . . . . 5 ⊢ (𝐹 = 0𝑝 → (𝐹‘𝐴) = (0𝑝‘𝐴)) | |
3 | 2 | eqeq1d 2738 | . . . 4 ⊢ (𝐹 = 0𝑝 → ((𝐹‘𝐴) = 0 ↔ (0𝑝‘𝐴) = 0)) |
4 | 1, 3 | syl5ibrcom 246 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐹 = 0𝑝 → (𝐹‘𝐴) = 0)) |
5 | 4 | necon3d 2962 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐹‘𝐴) ≠ 0 → 𝐹 ≠ 0𝑝)) |
6 | 5 | imp 407 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (𝐹‘𝐴) ≠ 0) → 𝐹 ≠ 0𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ‘cfv 6493 ℂcc 11045 0cc0 11047 0𝑝c0p 25017 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-mulcl 11109 ax-i2m1 11115 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-0p 25018 |
This theorem is referenced by: dgrmulc 25616 qaa 25667 iaa 25669 aareccl 25670 dchrfi 26587 |
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