![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 25694 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) | |
2 | fveq1 6902 | . . . . 5 ⊢ (𝐹 = 0𝑝 → (𝐹‘𝐴) = (0𝑝‘𝐴)) | |
3 | 2 | eqeq1d 2728 | . . . 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 2951 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐹‘𝐴) ≠ 0 → 𝐹 ≠ 0𝑝)) |
6 | 5 | imp 405 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (𝐹‘𝐴) ≠ 0) → 𝐹 ≠ 0𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ‘cfv 6556 ℂcc 11158 0cc0 11160 0𝑝c0p 25692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-mulcl 11222 ax-i2m1 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-fv 6564 df-0p 25693 |
This theorem is referenced by: dgrmulc 26302 qaa 26354 iaa 26356 aareccl 26357 dchrfi 27287 |
Copyright terms: Public domain | W3C validator |