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 25298 | . . . . 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 2740 | . . . . . 6 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
9 | 5, 1, 2, 6, 7, 8 | r1pval 25311 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
10 | 4, 9 | sylan2 593 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
11 | 10 | 3adant1 1129 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺))) |
12 | 11 | oveq2d 7285 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺)) = (((𝐹𝑄𝐺) · 𝐺) + (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)))) |
13 | 1 | ply1ring 21409 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
14 | 13 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Ring) |
15 | ringabl 19809 | . . . 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 25310 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) ∈ 𝐵) |
18 | 4 | 3ad2ant3 1134 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ∈ 𝐵) |
19 | 2, 7 | ringcl 19790 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ (𝐹𝑄𝐺) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) |
20 | 14, 17, 18, 19 | syl3anc 1370 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) |
21 | ringgrp 19778 | . . . . 5 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp) | |
22 | 14, 21 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Grp) |
23 | simp2 1136 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 ∈ 𝐵) | |
24 | 2, 8 | grpsubcl 18645 | . . . 4 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) → (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) ∈ 𝐵) |
25 | 22, 23, 20, 24 | syl3anc 1370 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) ∈ 𝐵) |
26 | r1pid.m | . . . 4 ⊢ + = (+g‘𝑃) | |
27 | 2, 26 | ablcom 19394 | . . 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 1370 | . 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 18657 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ((𝐹𝑄𝐺) · 𝐺) ∈ 𝐵) → ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺)) = 𝐹) |
30 | 22, 23, 20, 29 | syl3anc 1370 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹(-g‘𝑃)((𝐹𝑄𝐺) · 𝐺)) + ((𝐹𝑄𝐺) · 𝐺)) = 𝐹) |
31 | 12, 28, 30 | 3eqtrrd 2785 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ‘cfv 6431 (class class class)co 7269 Basecbs 16902 +gcplusg 16952 .rcmulr 16953 Grpcgrp 18567 -gcsg 18569 Abelcabl 19377 Ringcrg 19773 Poly1cpl1 21338 Unic1pcuc1p 25281 quot1pcq1p 25282 rem1pcr1p 25283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 ax-pre-sup 10942 ax-addf 10943 ax-mulf 10944 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-ofr 7526 df-om 7702 df-1st 7818 df-2nd 7819 df-supp 7963 df-tpos 8027 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-1o 8282 df-er 8473 df-map 8592 df-pm 8593 df-ixp 8661 df-en 8709 df-dom 8710 df-sdom 8711 df-fin 8712 df-fsupp 9099 df-sup 9171 df-oi 9239 df-card 9690 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-z 12312 df-dec 12429 df-uz 12574 df-fz 13231 df-fzo 13374 df-seq 13712 df-hash 14035 df-struct 16838 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-ress 16932 df-plusg 16965 df-mulr 16966 df-starv 16967 df-sca 16968 df-vsca 16969 df-tset 16971 df-ple 16972 df-ds 16974 df-unif 16975 df-0g 17142 df-gsum 17143 df-mre 17285 df-mrc 17286 df-acs 17288 df-mgm 18316 df-sgrp 18365 df-mnd 18376 df-mhm 18420 df-submnd 18421 df-grp 18570 df-minusg 18571 df-sbg 18572 df-mulg 18691 df-subg 18742 df-ghm 18822 df-cntz 18913 df-cmn 19378 df-abl 19379 df-mgp 19711 df-ur 19728 df-ring 19775 df-cring 19776 df-oppr 19852 df-dvdsr 19873 df-unit 19874 df-invr 19904 df-subrg 20012 df-lmod 20115 df-lss 20184 df-rlreg 20544 df-cnfld 20588 df-psr 21102 df-mvr 21103 df-mpl 21104 df-opsr 21106 df-psr1 21341 df-vr1 21342 df-ply1 21343 df-coe1 21344 df-mdeg 25207 df-deg1 25208 df-uc1p 25286 df-q1p 25287 df-r1p 25288 |
This theorem is referenced by: ply1rem 25318 |
Copyright terms: Public domain | W3C validator |