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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 2728 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
12 | eqid 2728 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | eqid 2728 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
14 | ply1divalg.g3 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) | |
15 | ply1divalg.u | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
16 | eqid 2728 | . . . . 5 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
17 | 15, 16, 12 | ringinvcl 20324 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) → ((invr‘𝑅)‘((coe1‘𝐺)‘(𝐷‘𝐺))) ∈ (Base‘𝑅)) |
18 | 7, 14, 17 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((invr‘𝑅)‘((coe1‘𝐺)‘(𝐷‘𝐺))) ∈ (Base‘𝑅)) |
19 | 15, 16, 13, 11 | unitrinv 20326 | . . . 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 583 | . . 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 26065 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
22 | eqid 2728 | . . . . . 6 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅) | |
23 | 22, 15 | unitrrg 21233 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (RLReg‘𝑅)) |
24 | 7, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ⊆ (RLReg‘𝑅)) |
25 | 24, 14 | sseldd 3979 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (RLReg‘𝑅)) |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 22 | ply1divmo 26064 | . 2 ⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
27 | reu5 3374 | . 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 582 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∃wrex 3066 ∃!wreu 3370 ∃*wrmo 3371 ⊆ wss 3945 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 < clt 11272 Basecbs 17173 .rcmulr 17227 0gc0g 17414 -gcsg 18885 1rcur 20114 Ringcrg 20166 Unitcui 20287 invrcinvr 20319 RLRegcrlreg 21219 Poly1cpl1 22089 coe1cco1 22090 deg1 cdg1 25980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-subrng 20476 df-subrg 20501 df-lmod 20738 df-lss 20809 df-rlreg 21223 df-cnfld 21273 df-psr 21835 df-mvr 21836 df-mpl 21837 df-opsr 21839 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-mdeg 25981 df-deg1 25982 |
This theorem is referenced by: ply1divalg2 26067 |
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