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 23244 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) | |
2 | fveq1 6102 | . . . . 5 ⊢ (𝐹 = 0𝑝 → (𝐹‘𝐴) = (0𝑝‘𝐴)) | |
3 | 2 | eqeq1d 2612 | . . . 4 ⊢ (𝐹 = 0𝑝 → ((𝐹‘𝐴) = 0 ↔ (0𝑝‘𝐴) = 0)) |
4 | 1, 3 | syl5ibrcom 236 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐹 = 0𝑝 → (𝐹‘𝐴) = 0)) |
5 | 4 | necon3d 2803 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐹‘𝐴) ≠ 0 → 𝐹 ≠ 0𝑝)) |
6 | 5 | imp 444 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (𝐹‘𝐴) ≠ 0) → 𝐹 ≠ 0𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 ℂcc 9813 0cc0 9815 0𝑝c0p 23242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-0p 23243 |
This theorem is referenced by: dgrmulc 23831 qaa 23882 iaa 23884 aareccl 23885 dchrfi 24780 |
Copyright terms: Public domain | W3C validator |