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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 2622 | . . 3 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
3 | uc1pn0.z | . . 3 ⊢ 0 = (0g‘𝑃) | |
4 | eqid 2622 | . . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
5 | uc1pn0.c | . . 3 ⊢ 𝐶 = (Unic1p‘𝑅) | |
6 | eqid 2622 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | isuc1p 23900 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅))) |
8 | 7 | simp2bi 1077 | 1 ⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ‘cfv 5888 Basecbs 15857 0gc0g 16100 Unitcui 18639 Poly1cpl1 19547 coe1cco1 19548 deg1 cdg1 23814 Unic1pcuc1p 23886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-slot 15861 df-base 15863 df-uc1p 23891 |
This theorem is referenced by: uc1pdeg 23907 q1peqb 23914 |
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