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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 2771 | . . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
2 | eqid 2771 | . . 3 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
3 | eqid 2771 | . . 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 24120 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝐹 ≠ (0g‘(Poly1‘𝑅)) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) |
8 | 7 | simp3bi 1141 | 1 ⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ‘cfv 6031 Basecbs 16064 0gc0g 16308 Unitcui 18847 Poly1cpl1 19762 coe1cco1 19763 deg1 cdg1 24034 Unic1pcuc1p 24106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5994 df-fun 6033 df-fv 6039 df-slot 16068 df-base 16070 df-uc1p 24111 |
This theorem is referenced by: uc1pmon1p 24131 q1peqb 24134 fta1glem1 24145 ig1peu 24151 |
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