Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mon1pn0 | Structured version Visualization version GIF version |
Description: Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pn0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
uc1pn0.z | ⊢ 0 = (0g‘𝑃) |
mon1pn0.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
Ref | Expression |
---|---|
mon1pn0 | ⊢ (𝐹 ∈ 𝑀 → 𝐹 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uc1pn0.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | eqid 2823 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
3 | uc1pn0.z | . . 3 ⊢ 0 = (0g‘𝑃) | |
4 | eqid 2823 | . . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
5 | mon1pn0.m | . . 3 ⊢ 𝑀 = (Monic1p‘𝑅) | |
6 | eqid 2823 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | ismon1p 24738 | . 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) = (1r‘𝑅))) |
8 | 7 | simp2bi 1142 | 1 ⊢ (𝐹 ∈ 𝑀 → 𝐹 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 Basecbs 16485 0gc0g 16715 1rcur 19253 Poly1cpl1 20347 coe1cco1 20348 deg1 cdg1 24650 Monic1pcmn1 24721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-slot 16489 df-base 16491 df-mon1 24726 |
This theorem is referenced by: mon1puc1p 24746 deg1submon1p 24748 mon1psubm 39813 |
Copyright terms: Public domain | W3C validator |