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Mirrors > Home > MPE Home > Th. List > mon1pcl | Structured version Visualization version GIF version |
Description: Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pcl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
uc1pcl.b | ⊢ 𝐵 = (Base‘𝑃) |
mon1pcl.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
Ref | Expression |
---|---|
mon1pcl | ⊢ (𝐹 ∈ 𝑀 → 𝐹 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uc1pcl.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | uc1pcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
3 | eqid 2798 | . . 3 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
4 | eqid 2798 | . . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
5 | mon1pcl.m | . . 3 ⊢ 𝑀 = (Monic1p‘𝑅) | |
6 | eqid 2798 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | ismon1p 24743 | . 2 ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) = (1r‘𝑅))) |
8 | 7 | simp1bi 1142 | 1 ⊢ (𝐹 ∈ 𝑀 → 𝐹 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 Basecbs 16475 0gc0g 16705 1rcur 19244 Poly1cpl1 20806 coe1cco1 20807 deg1 cdg1 24655 Monic1pcmn1 24726 |
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-mon1 24731 |
This theorem is referenced by: mon1puc1p 24751 deg1submon1p 24753 ply1rem 24764 fta1glem1 24766 fta1glem2 24767 mon1psubm 40150 |
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