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Mirrors > Home > MPE Home > Th. List > mon1puc1p | Structured version Visualization version GIF version |
Description: Monic polynomials are unitic. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
mon1puc1p.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
mon1puc1p.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
Ref | Expression |
---|---|
mon1puc1p | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → 𝑋 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . . . 4 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
2 | eqid 2740 | . . . 4 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
3 | mon1puc1p.m | . . . 4 ⊢ 𝑀 = (Monic1p‘𝑅) | |
4 | 1, 2, 3 | mon1pcl 25305 | . . 3 ⊢ (𝑋 ∈ 𝑀 → 𝑋 ∈ (Base‘(Poly1‘𝑅))) |
5 | 4 | adantl 482 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → 𝑋 ∈ (Base‘(Poly1‘𝑅))) |
6 | eqid 2740 | . . . 4 ⊢ (0g‘(Poly1‘𝑅)) = (0g‘(Poly1‘𝑅)) | |
7 | 1, 6, 3 | mon1pn0 25307 | . . 3 ⊢ (𝑋 ∈ 𝑀 → 𝑋 ≠ (0g‘(Poly1‘𝑅))) |
8 | 7 | adantl 482 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → 𝑋 ≠ (0g‘(Poly1‘𝑅))) |
9 | eqid 2740 | . . . . 5 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
10 | eqid 2740 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
11 | 9, 10, 3 | mon1pldg 25310 | . . . 4 ⊢ (𝑋 ∈ 𝑀 → ((coe1‘𝑋)‘(( deg1 ‘𝑅)‘𝑋)) = (1r‘𝑅)) |
12 | 11 | adantl 482 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → ((coe1‘𝑋)‘(( deg1 ‘𝑅)‘𝑋)) = (1r‘𝑅)) |
13 | eqid 2740 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
14 | 13, 10 | 1unit 19896 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅)) |
15 | 14 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → (1r‘𝑅) ∈ (Unit‘𝑅)) |
16 | 12, 15 | eqeltrd 2841 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → ((coe1‘𝑋)‘(( deg1 ‘𝑅)‘𝑋)) ∈ (Unit‘𝑅)) |
17 | mon1puc1p.c | . . 3 ⊢ 𝐶 = (Unic1p‘𝑅) | |
18 | 1, 2, 6, 9, 17, 13 | isuc1p 25301 | . 2 ⊢ (𝑋 ∈ 𝐶 ↔ (𝑋 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝑋 ≠ (0g‘(Poly1‘𝑅)) ∧ ((coe1‘𝑋)‘(( deg1 ‘𝑅)‘𝑋)) ∈ (Unit‘𝑅))) |
19 | 5, 8, 16, 18 | syl3anbrc 1342 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → 𝑋 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ‘cfv 6431 Basecbs 16908 0gc0g 17146 1rcur 19733 Ringcrg 19779 Unitcui 19877 Poly1cpl1 21344 coe1cco1 21345 deg1 cdg1 25212 Monic1pcmn1 25286 Unic1pcuc1p 25287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-tpos 8031 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-plusg 16971 df-mulr 16972 df-0g 17148 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-grp 18576 df-mgp 19717 df-ur 19734 df-ring 19781 df-oppr 19858 df-dvdsr 19879 df-unit 19880 df-mon1 25291 df-uc1p 25292 |
This theorem is referenced by: ply1rem 25324 facth1 25325 fta1glem1 25326 fta1glem2 25327 ig1pdvds 25337 |
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