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| 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 2737 | . . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
| 2 | eqid 2737 | . . 3 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
| 3 | eqid 2737 | . . 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 26180 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝐹 ≠ (0g‘(Poly1‘𝑅)) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) | 
| 8 | 7 | simp3bi 1148 | 1 ⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 Basecbs 17247 0gc0g 17484 Unitcui 20355 Poly1cpl1 22178 coe1cco1 22179 deg1cdg1 26093 Unic1pcuc1p 26166 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-uc1p 26171 | 
| This theorem is referenced by: uc1pmon1p 26191 q1peqb 26195 fta1glem1 26207 ig1peu 26214 ply1divalg3 35647 | 
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