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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 2826 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
12 | eqid 2826 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | eqid 2826 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
14 | ply1divalg.g3 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) | |
15 | ply1divalg.u | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
16 | eqid 2826 | . . . . 5 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
17 | 15, 16, 12 | ringinvcl 19031 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) → ((invr‘𝑅)‘((coe1‘𝐺)‘(𝐷‘𝐺))) ∈ (Base‘𝑅)) |
18 | 7, 14, 17 | syl2anc 581 | . . 3 ⊢ (𝜑 → ((invr‘𝑅)‘((coe1‘𝐺)‘(𝐷‘𝐺))) ∈ (Base‘𝑅)) |
19 | 15, 16, 13, 11 | unitrinv 19033 | . . . 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 581 | . . 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 24296 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
22 | eqid 2826 | . . . . . 6 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅) | |
23 | 22, 15 | unitrrg 19655 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ (RLReg‘𝑅)) |
24 | 7, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ⊆ (RLReg‘𝑅)) |
25 | 24, 14 | sseldd 3829 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (RLReg‘𝑅)) |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 22 | ply1divmo 24295 | . 2 ⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
27 | reu5 3372 | . 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 580 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∃wrex 3119 ∃!wreu 3120 ∃*wrmo 3121 ⊆ wss 3799 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 < clt 10392 Basecbs 16223 .rcmulr 16307 0gc0g 16454 -gcsg 17779 1rcur 18856 Ringcrg 18902 Unitcui 18994 invrcinvr 19026 RLRegcrlreg 19641 Poly1cpl1 19908 coe1cco1 19909 deg1 cdg1 24214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-ofr 7159 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-tpos 7618 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-2o 7828 df-oadd 7831 df-er 8010 df-map 8125 df-pm 8126 df-ixp 8177 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-sup 8618 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-fzo 12762 df-seq 13097 df-hash 13412 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-0g 16456 df-gsum 16457 df-mre 16600 df-mrc 16601 df-acs 16603 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-mhm 17689 df-submnd 17690 df-grp 17780 df-minusg 17781 df-sbg 17782 df-mulg 17896 df-subg 17943 df-ghm 18010 df-cntz 18101 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-cring 18905 df-oppr 18978 df-dvdsr 18996 df-unit 18997 df-invr 19027 df-subrg 19135 df-lmod 19222 df-lss 19290 df-rlreg 19645 df-psr 19718 df-mvr 19719 df-mpl 19720 df-opsr 19722 df-psr1 19911 df-vr1 19912 df-ply1 19913 df-coe1 19914 df-cnfld 20108 df-mdeg 24215 df-deg1 24216 |
This theorem is referenced by: ply1divalg2 24298 |
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