Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > uc1pldg | Structured version Visualization version GIF version |
Description: Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pldg.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
uc1pldg.u | ⊢ 𝑈 = (Unit‘𝑅) |
uc1pldg.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
Ref | Expression |
---|---|
uc1pldg | ⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
2 | eqid 2821 | . . 3 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
3 | eqid 2821 | . . 3 ⊢ (0g‘(Poly1‘𝑅)) = (0g‘(Poly1‘𝑅)) | |
4 | uc1pldg.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
5 | uc1pldg.c | . . 3 ⊢ 𝐶 = (Unic1p‘𝑅) | |
6 | uc1pldg.u | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | isuc1p 24733 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝐹 ≠ (0g‘(Poly1‘𝑅)) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
8 | 7 | simp3bi 1143 | 1 ⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6354 Basecbs 16482 0gc0g 16712 Unitcui 19388 Poly1cpl1 20344 coe1cco1 20345 deg1 cdg1 24647 Unic1pcuc1p 24719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-slot 16486 df-base 16488 df-uc1p 24724 |
This theorem is referenced by: uc1pmon1p 24744 q1peqb 24747 fta1glem1 24758 ig1peu 24764 |
Copyright terms: Public domain | W3C validator |