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Mirrors > Home > MPE Home > Th. List > uc1pcl | Structured version Visualization version GIF version |
Description: Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
uc1pcl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
uc1pcl.b | ⊢ 𝐵 = (Base‘𝑃) |
uc1pcl.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
Ref | Expression |
---|---|
uc1pcl | ⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uc1pcl.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | uc1pcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
3 | eqid 2610 | . . 3 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
4 | eqid 2610 | . . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
5 | uc1pcl.c | . . 3 ⊢ 𝐶 = (Unic1p‘𝑅) | |
6 | eqid 2610 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | isuc1p 23704 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅))) |
8 | 7 | simp1bi 1069 | 1 ⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 Basecbs 15695 0gc0g 15923 Unitcui 18462 Poly1cpl1 19368 coe1cco1 19369 deg1 cdg1 23618 Unic1pcuc1p 23690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-slot 15699 df-base 15700 df-uc1p 23695 |
This theorem is referenced by: uc1pdeg 23711 uc1pmon1p 23715 q1peqb 23718 r1pcl 23721 r1pdeglt 23722 r1pid 23723 dvdsq1p 23724 dvdsr1p 23725 |
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