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Mirrors > Home > MPE Home > Th. List > uc1pn0 | Structured version Visualization version GIF version |
Description: Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pn0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
uc1pn0.z | ⊢ 0 = (0g‘𝑃) |
uc1pn0.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
Ref | Expression |
---|---|
uc1pn0 | ⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uc1pn0.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | eqid 2798 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
3 | uc1pn0.z | . . 3 ⊢ 0 = (0g‘𝑃) | |
4 | eqid 2798 | . . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
5 | uc1pn0.c | . . 3 ⊢ 𝐶 = (Unic1p‘𝑅) | |
6 | eqid 2798 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | isuc1p 24741 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅))) |
8 | 7 | simp2bi 1143 | 1 ⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 Basecbs 16475 0gc0g 16705 Unitcui 19385 Poly1cpl1 20806 coe1cco1 20807 deg1 cdg1 24655 Unic1pcuc1p 24727 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-slot 16479 df-base 16481 df-uc1p 24732 |
This theorem is referenced by: uc1pdeg 24748 q1peqb 24755 |
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