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 23881 | . . . 4 ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) |
3 | nzrring 19261 | . . . 4 ⊢ (𝑃 ∈ NzRing → 𝑃 ∈ Ring) | |
4 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | mon1pid.o | . . . . 5 ⊢ 1 = (1r‘𝑃) | |
6 | 4, 5 | ringidcl 18568 | . . . 4 ⊢ (𝑃 ∈ Ring → 1 ∈ (Base‘𝑃)) |
7 | 2, 3, 6 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ∈ (Base‘𝑃)) |
8 | eqid 2622 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
9 | 5, 8 | nzrnz 19260 | . . . 4 ⊢ (𝑃 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
10 | 2, 9 | syl 17 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
11 | nzrring 19261 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
12 | eqid 2622 | . . . . . . . . 9 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
13 | eqid 2622 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
14 | 1, 12, 13, 5 | ply1scl1 19662 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
15 | 11, 14 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
16 | 15 | fveq2d 6195 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (coe1‘ 1 )) |
17 | eqid 2622 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
18 | 17, 13 | ringidcl 18568 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
19 | 11, 18 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ∈ (Base‘𝑅)) |
20 | eqid 2622 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
21 | 1, 12, 17, 20 | coe1scl 19657 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
22 | 11, 19, 21 | syl2anc 693 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
23 | 16, 22 | eqtr3d 2658 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (coe1‘ 1 ) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
24 | 15 | fveq2d 6195 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝐷‘ 1 )) |
25 | 13, 20 | nzrnz 19260 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
26 | mon1pid.d | . . . . . . . 8 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
27 | 26, 1, 17, 12, 20 | deg1scl 23873 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = 0) |
28 | 11, 19, 25, 27 | syl3anc 1326 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = 0) |
29 | 24, 28 | eqtr3d 2658 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (𝐷‘ 1 ) = 0) |
30 | 23, 29 | fveq12d 6197 | . . . 4 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0)) |
31 | 0nn0 11307 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
32 | iftrue 4092 | . . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅)) | |
33 | eqid 2622 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅))) | |
34 | fvex 6201 | . . . . . 6 ⊢ (1r‘𝑅) ∈ V | |
35 | 32, 33, 34 | fvmpt 6282 | . . . . 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 2672 | . . 3 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = (1r‘𝑅)) |
38 | mon1pid.m | . . . 4 ⊢ 𝑀 = (Monic1p‘𝑅) | |
39 | 1, 4, 8, 26, 38, 13 | ismon1p 23902 | . . 3 ⊢ ( 1 ∈ 𝑀 ↔ ( 1 ∈ (Base‘𝑃) ∧ 1 ≠ (0g‘𝑃) ∧ ((coe1‘ 1 )‘(𝐷‘ 1 )) = (1r‘𝑅))) |
40 | 7, 10, 37, 39 | syl3anbrc 1246 | . 2 ⊢ (𝑅 ∈ NzRing → 1 ∈ 𝑀) |
41 | 40, 29 | jca 554 | 1 ⊢ (𝑅 ∈ NzRing → ( 1 ∈ 𝑀 ∧ (𝐷‘ 1 ) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ifcif 4086 ↦ cmpt 4729 ‘cfv 5888 0cc0 9936 ℕ0cn0 11292 Basecbs 15857 0gc0g 16100 1rcur 18501 Ringcrg 18547 NzRingcnzr 19257 algSccascl 19311 Poly1cpl1 19547 coe1cco1 19548 deg1 cdg1 23814 Monic1pcmn1 23885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-lmod 18865 df-lss 18933 df-nzr 19258 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-cnfld 19747 df-mdeg 23815 df-deg1 23816 df-mon1 23890 |
This theorem is referenced by: mon1psubm 37784 deg1mhm 37785 |
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