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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 2736 | . . 3 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
4 | eqid 2736 | . . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
5 | uc1pcl.c | . . 3 ⊢ 𝐶 = (Unic1p‘𝑅) | |
6 | eqid 2736 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | isuc1p 24992 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅))) |
8 | 7 | simp1bi 1147 | 1 ⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ‘cfv 6358 Basecbs 16666 0gc0g 16898 Unitcui 19611 Poly1cpl1 21052 coe1cco1 21053 deg1 cdg1 24903 Unic1pcuc1p 24978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-addcl 10754 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-nn 11796 df-ndx 16669 df-slot 16670 df-base 16672 df-uc1p 24983 |
This theorem is referenced by: uc1pdeg 24999 uc1pmon1p 25003 q1peqb 25006 r1pcl 25009 r1pdeglt 25010 r1pid 25011 dvdsq1p 25012 dvdsr1p 25013 |
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