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Mirrors > Home > MPE Home > Th. List > dvdsr1p | Structured version Visualization version GIF version |
Description: Divisibility in a polynomial ring in terms of the remainder. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
dvdsq1p.p | ⊢ 𝑃 = (Poly1‘𝑅) |
dvdsq1p.d | ⊢ ∥ = (∥r‘𝑃) |
dvdsq1p.b | ⊢ 𝐵 = (Base‘𝑃) |
dvdsq1p.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
dvdsr1p.z | ⊢ 0 = (0g‘𝑃) |
dvdsr1p.e | ⊢ 𝐸 = (rem1p‘𝑅) |
Ref | Expression |
---|---|
dvdsr1p | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐺 ∥ 𝐹 ↔ (𝐹𝐸𝐺) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsq1p.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | 1 | ply1ring 21417 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
3 | 2 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Ring) |
4 | ringgrp 19786 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑃 ∈ Grp) |
6 | simp2 1136 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 ∈ 𝐵) | |
7 | eqid 2740 | . . . . 5 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅) | |
8 | dvdsq1p.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
9 | dvdsq1p.c | . . . . 5 ⊢ 𝐶 = (Unic1p‘𝑅) | |
10 | 7, 1, 8, 9 | q1pcl 25318 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵) |
11 | 1, 8, 9 | uc1pcl 25306 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵) |
12 | 11 | 3ad2ant3 1134 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ∈ 𝐵) |
13 | eqid 2740 | . . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
14 | 8, 13 | ringcl 19798 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ (𝐹(quot1p‘𝑅)𝐺) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺) ∈ 𝐵) |
15 | 3, 10, 12, 14 | syl3anc 1370 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺) ∈ 𝐵) |
16 | dvdsr1p.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
17 | eqid 2740 | . . . 4 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
18 | 8, 16, 17 | grpsubeq0 18659 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺) ∈ 𝐵) → ((𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)) = 0 ↔ 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
19 | 5, 6, 15, 18 | syl3anc 1370 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)) = 0 ↔ 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
20 | dvdsr1p.e | . . . . 5 ⊢ 𝐸 = (rem1p‘𝑅) | |
21 | 20, 1, 8, 7, 13, 17 | r1pval 25319 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
22 | 6, 12, 21 | syl2anc 584 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) = (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
23 | 22 | eqeq1d 2742 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹𝐸𝐺) = 0 ↔ (𝐹(-g‘𝑃)((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺)) = 0 )) |
24 | dvdsq1p.d | . . 3 ⊢ ∥ = (∥r‘𝑃) | |
25 | 1, 24, 8, 9, 13, 7 | dvdsq1p 25323 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐺 ∥ 𝐹 ↔ 𝐹 = ((𝐹(quot1p‘𝑅)𝐺)(.r‘𝑃)𝐺))) |
26 | 19, 23, 25 | 3bitr4rd 312 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐺 ∥ 𝐹 ↔ (𝐹𝐸𝐺) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 .rcmulr 16961 0gc0g 17148 Grpcgrp 18575 -gcsg 18577 Ringcrg 19781 ∥rcdsr 19878 Poly1cpl1 21346 Unic1pcuc1p 25289 quot1pcq1p 25290 rem1pcr1p 25291 |
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 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
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 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-ofr 7528 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-fzo 13382 df-seq 13720 df-hash 14043 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-0g 17150 df-gsum 17151 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-subrg 20020 df-lmod 20123 df-lss 20192 df-rlreg 20552 df-cnfld 20596 df-psr 21110 df-mvr 21111 df-mpl 21112 df-opsr 21114 df-psr1 21349 df-vr1 21350 df-ply1 21351 df-coe1 21352 df-mdeg 25215 df-deg1 25216 df-uc1p 25294 df-q1p 25295 df-r1p 25296 |
This theorem is referenced by: facth1 25327 ig1pdvds 25339 |
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