Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mon1pid | Structured version Visualization version GIF version |
Description: Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
mon1pid.p | ⊢ 𝑃 = (Poly1‘𝑅) |
mon1pid.o | ⊢ 1 = (1r‘𝑃) |
mon1pid.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
mon1pid.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
Ref | Expression |
---|---|
mon1pid | ⊢ (𝑅 ∈ NzRing → ( 1 ∈ 𝑀 ∧ (𝐷‘ 1 ) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mon1pid.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | 1 | ply1nz 24714 | . . . 4 ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) |
3 | nzrring 20033 | . . . 4 ⊢ (𝑃 ∈ NzRing → 𝑃 ∈ Ring) | |
4 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | mon1pid.o | . . . . 5 ⊢ 1 = (1r‘𝑃) | |
6 | 4, 5 | ringidcl 19317 | . . . 4 ⊢ (𝑃 ∈ Ring → 1 ∈ (Base‘𝑃)) |
7 | 2, 3, 6 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ∈ (Base‘𝑃)) |
8 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
9 | 5, 8 | nzrnz 20032 | . . . 4 ⊢ (𝑃 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
10 | 2, 9 | syl 17 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
11 | nzrring 20033 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
12 | eqid 2821 | . . . . . . . . 9 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
13 | eqid 2821 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
14 | 1, 12, 13, 5 | ply1scl1 20459 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
15 | 11, 14 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
16 | 15 | fveq2d 6673 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (coe1‘ 1 )) |
17 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
18 | 17, 13 | ringidcl 19317 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
19 | eqid 2821 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
20 | 1, 12, 17, 19 | coe1scl 20454 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
21 | 11, 18, 20 | syl2anc2 587 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
22 | 16, 21 | eqtr3d 2858 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (coe1‘ 1 ) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
23 | 15 | fveq2d 6673 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝐷‘ 1 )) |
24 | 11, 18 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ∈ (Base‘𝑅)) |
25 | 13, 19 | nzrnz 20032 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
26 | mon1pid.d | . . . . . . . 8 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
27 | 26, 1, 17, 12, 19 | deg1scl 24706 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = 0) |
28 | 11, 24, 25, 27 | syl3anc 1367 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = 0) |
29 | 23, 28 | eqtr3d 2858 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (𝐷‘ 1 ) = 0) |
30 | 22, 29 | fveq12d 6676 | . . . 4 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0)) |
31 | 0nn0 11911 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
32 | iftrue 4472 | . . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅)) | |
33 | eqid 2821 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅))) | |
34 | fvex 6682 | . . . . . 6 ⊢ (1r‘𝑅) ∈ V | |
35 | 32, 33, 34 | fvmpt 6767 | . . . . 5 ⊢ (0 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0) = (1r‘𝑅)) |
36 | 31, 35 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0) = (1r‘𝑅) |
37 | 30, 36 | syl6eq 2872 | . . 3 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = (1r‘𝑅)) |
38 | mon1pid.m | . . . 4 ⊢ 𝑀 = (Monic1p‘𝑅) | |
39 | 1, 4, 8, 26, 38, 13 | ismon1p 24735 | . . 3 ⊢ ( 1 ∈ 𝑀 ↔ ( 1 ∈ (Base‘𝑃) ∧ 1 ≠ (0g‘𝑃) ∧ ((coe1‘ 1 )‘(𝐷‘ 1 )) = (1r‘𝑅))) |
40 | 7, 10, 37, 39 | syl3anbrc 1339 | . 2 ⊢ (𝑅 ∈ NzRing → 1 ∈ 𝑀) |
41 | 40, 29 | jca 514 | 1 ⊢ (𝑅 ∈ NzRing → ( 1 ∈ 𝑀 ∧ (𝐷‘ 1 ) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ifcif 4466 ↦ cmpt 5145 ‘cfv 6354 0cc0 10536 ℕ0cn0 11896 Basecbs 16482 0gc0g 16712 1rcur 19250 Ringcrg 19296 NzRingcnzr 20029 algSccascl 20083 Poly1cpl1 20344 coe1cco1 20345 deg1 cdg1 24647 Monic1pcmn1 24718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-0g 16714 df-gsum 16715 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-subrg 19532 df-lmod 19635 df-lss 19703 df-nzr 20030 df-ascl 20086 df-psr 20135 df-mvr 20136 df-mpl 20137 df-opsr 20139 df-psr1 20347 df-vr1 20348 df-ply1 20349 df-coe1 20350 df-cnfld 20545 df-mdeg 24648 df-deg1 24649 df-mon1 24723 |
This theorem is referenced by: mon1psubm 39804 deg1mhm 39805 |
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