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 24715 | . . . 4 ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) |
3 | nzrring 20034 | . . . 4 ⊢ (𝑃 ∈ NzRing → 𝑃 ∈ Ring) | |
4 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | mon1pid.o | . . . . 5 ⊢ 1 = (1r‘𝑃) | |
6 | 4, 5 | ringidcl 19318 | . . . 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 20033 | . . . 4 ⊢ (𝑃 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
10 | 2, 9 | syl 17 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
11 | nzrring 20034 | . . . . . . . 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 20460 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
15 | 11, 14 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
16 | 15 | fveq2d 6674 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (coe1‘ 1 )) |
17 | eqid 2821 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
18 | 17, 13 | ringidcl 19318 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
19 | eqid 2821 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
20 | 1, 12, 17, 19 | coe1scl 20455 | . . . . . . 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 6674 | . . . . . 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 20033 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
26 | mon1pid.d | . . . . . . . 8 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
27 | 26, 1, 17, 12, 19 | deg1scl 24707 | . . . . . . 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 6677 | . . . 4 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0)) |
31 | 0nn0 11913 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
32 | iftrue 4473 | . . . . . 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 6683 | . . . . . 6 ⊢ (1r‘𝑅) ∈ V | |
35 | 32, 33, 34 | fvmpt 6768 | . . . . 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 24736 | . . 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 1537 ∈ wcel 2114 ≠ wne 3016 ifcif 4467 ↦ cmpt 5146 ‘cfv 6355 0cc0 10537 ℕ0cn0 11898 Basecbs 16483 0gc0g 16713 1rcur 19251 Ringcrg 19297 NzRingcnzr 20030 algSccascl 20084 Poly1cpl1 20345 coe1cco1 20346 deg1 cdg1 24648 Monic1pcmn1 24719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 df-lmod 19636 df-lss 19704 df-nzr 20031 df-ascl 20087 df-psr 20136 df-mvr 20137 df-mpl 20138 df-opsr 20140 df-psr1 20348 df-vr1 20349 df-ply1 20350 df-coe1 20351 df-cnfld 20546 df-mdeg 24649 df-deg1 24650 df-mon1 24724 |
This theorem is referenced by: mon1psubm 39826 deg1mhm 39827 |
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