Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > r1pid | Structured version Visualization version GIF version |
Description: Express the original polynomial 𝐹 as 𝐹 = (𝑞 · 𝐺) + 𝑟 using the quotient and remainder functions for 𝑞 and 𝑟. (Contributed by Mario Carneiro, 5-Jun-2015.) |
Ref | Expression |
---|---|
r1pid.p | ⊢ 𝑃 = (Poly1‘𝑅) |
r1pid.b | ⊢ 𝐵 = (Base‘𝑃) |
r1pid.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
r1pid.q | ⊢ 𝑄 = (quot1p‘𝑅) |
r1pid.e | ⊢ 𝐸 = (rem1p‘𝑅) |
r1pid.t | ⊢ · = (.r‘𝑃) |
r1pid.m | ⊢ + = (+g‘𝑃) |
Ref | Expression |
---|---|
r1pid | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1pid.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | r1pid.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
3 | r1pid.c | . . . . . 6 ⊢ 𝐶 = (Unic1p‘𝑅) | |
4 | 1, 2, 3 | uc1pcl 24731 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵) |
5 | r1pid.e | . . . . . 6 ⊢ 𝐸 = (rem1p‘𝑅) | |
6 | r1pid.q | . . . . . 6 ⊢ 𝑄 = (quot1p‘𝑅) | |
7 | r1pid.t | . . . . . 6 ⊢ · = (.r‘𝑃) | |
8 | eqid 2821 | . . . . . 6 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
9 | 5, 1, 2, 6, 7, 8 | r1pval 24744 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
10 | 4, 9 | sylan2 594 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
11 | 10 | 3adant1 1126 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
12 | 11 | oveq2d 7166 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺)) = (((𝐹𝑄𝐺) · 𝐺) + (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)))) |
13 | 1 | ply1ring 20410 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
14 | 13 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Ring) |
15 | ringabl 19324 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Abel) |
17 | 6, 1, 2, 3 | q1pcl 24743 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) ∈ 𝐵) |
18 | 4 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ∈ 𝐵) |
19 | 2, 7 | ringcl 19305 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ (𝐹𝑄𝐺) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) |
20 | 14, 17, 18, 19 | syl3anc 1367 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) |
21 | ringgrp 19296 | . . . . 5 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp) | |
22 | 14, 21 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Grp) |
23 | simp2 1133 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 ∈ 𝐵) | |
24 | 2, 8 | grpsubcl 18173 | . . . 4 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) → (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) ∈ 𝐵) |
25 | 22, 23, 20, 24 | syl3anc 1367 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) ∈ 𝐵) |
26 | r1pid.m | . . . 4 ⊢ + = (+g‘𝑃) | |
27 | 2, 26 | ablcom 18918 | . . 3 ⊢ ((𝑃 ∈ Abel ∧ ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵 ∧ (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) ∈ 𝐵) → (((𝐹𝑄𝐺) · 𝐺) + (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) = ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺))) |
28 | 16, 20, 25, 27 | syl3anc 1367 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (((𝐹𝑄𝐺) · 𝐺) + (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) = ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺))) |
29 | 2, 26, 8 | grpnpcan 18185 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) → ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺)) = 𝐹) |
30 | 22, 23, 20, 29 | syl3anc 1367 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺)) = 𝐹) |
31 | 12, 28, 30 | 3eqtrrd 2861 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 Grpcgrp 18097 -gcsg 18099 Abelcabl 18901 Ringcrg 19291 Poly1cpl1 20339 Unic1pcuc1p 24714 quot1pcq1p 24715 rem1pcr1p 24716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-subrg 19527 df-lmod 19630 df-lss 19698 df-rlreg 20050 df-psr 20130 df-mvr 20131 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-vr1 20343 df-ply1 20344 df-coe1 20345 df-cnfld 20540 df-mdeg 24643 df-deg1 24644 df-uc1p 24719 df-q1p 24720 df-r1p 24721 |
This theorem is referenced by: ply1rem 24751 |
Copyright terms: Public domain | W3C validator |