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Mirrors > Home > MPE Home > Th. List > ply1divalg | Structured version Visualization version GIF version |
Description: The division algorithm for univariate polynomials over a ring. For polynomials 𝐹, 𝐺 such that 𝐺 ≠ 0 and the leading coefficient of 𝐺 is a unit, there are unique polynomials 𝑞 and 𝑟 = 𝐹 − (𝐺 · 𝑞) such that the degree of 𝑟 is less than the degree of 𝐺. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
ply1divalg.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1divalg.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
ply1divalg.b | ⊢ 𝐵 = (Base‘𝑃) |
ply1divalg.m | ⊢ − = (-g‘𝑃) |
ply1divalg.z | ⊢ 0 = (0g‘𝑃) |
ply1divalg.t | ⊢ ∙ = (.r‘𝑃) |
ply1divalg.r1 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ply1divalg.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
ply1divalg.g1 | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
ply1divalg.g2 | ⊢ (𝜑 → 𝐺 ≠ 0 ) |
ply1divalg.g3 | ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) |
ply1divalg.u | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
ply1divalg | ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1divalg.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | ply1divalg.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | ply1divalg.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
4 | ply1divalg.m | . . 3 ⊢ − = (-g‘𝑃) | |
5 | ply1divalg.z | . . 3 ⊢ 0 = (0g‘𝑃) | |
6 | ply1divalg.t | . . 3 ⊢ ∙ = (.r‘𝑃) | |
7 | ply1divalg.r1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | ply1divalg.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
9 | ply1divalg.g1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
10 | ply1divalg.g2 | . . 3 ⊢ (𝜑 → 𝐺 ≠ 0 ) | |
11 | eqid 2798 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
12 | eqid 2798 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | eqid 2798 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
14 | ply1divalg.g3 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) | |
15 | ply1divalg.u | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
16 | eqid 2798 | . . . . 5 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
17 | 15, 16, 12 | ringinvcl 19422 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) → ((invr‘𝑅)‘((coe1‘𝐺)‘(𝐷‘𝐺))) ∈ (Base‘𝑅)) |
18 | 7, 14, 17 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((invr‘𝑅)‘((coe1‘𝐺)‘(𝐷‘𝐺))) ∈ (Base‘𝑅)) |
19 | 15, 16, 13, 11 | unitrinv 19424 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) → (((coe1‘𝐺)‘(𝐷‘𝐺))(.r‘𝑅)((invr‘𝑅)‘((coe1‘𝐺)‘(𝐷‘𝐺)))) = (1r‘𝑅)) |
20 | 7, 14, 19 | syl2anc 587 | . . 3 ⊢ (𝜑 → (((coe1‘𝐺)‘(𝐷‘𝐺))(.r‘𝑅)((invr‘𝑅)‘((coe1‘𝐺)‘(𝐷‘𝐺)))) = (1r‘𝑅)) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 20 | ply1divex 24737 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
22 | eqid 2798 | . . . . . 6 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅) | |
23 | 22, 15 | unitrrg 20059 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (RLReg‘𝑅)) |
24 | 7, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ⊆ (RLReg‘𝑅)) |
25 | 24, 14 | sseldd 3916 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (RLReg‘𝑅)) |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 22 | ply1divmo 24736 | . 2 ⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
27 | reu5 3375 | . 2 ⊢ (∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ↔ (∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺) ∧ ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))) | |
28 | 21, 26, 27 | sylanbrc 586 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ∃!wreu 3108 ∃*wrmo 3109 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 < clt 10664 Basecbs 16475 .rcmulr 16558 0gc0g 16705 -gcsg 18097 1rcur 19244 Ringcrg 19290 Unitcui 19385 invrcinvr 19417 RLRegcrlreg 20045 Poly1cpl1 20806 coe1cco1 20807 deg1 cdg1 24655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-subrg 19526 df-lmod 19629 df-lss 19697 df-rlreg 20049 df-cnfld 20092 df-psr 20594 df-mvr 20595 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-vr1 20810 df-ply1 20811 df-coe1 20812 df-mdeg 24656 df-deg1 24657 |
This theorem is referenced by: ply1divalg2 24739 |
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