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Mirrors > Home > MPE Home > Th. List > ringrz | Structured version Visualization version GIF version |
Description: The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) |
Ref | Expression |
---|---|
rngz.b | ⊢ 𝐵 = (Base‘𝑅) |
rngz.t | ⊢ · = (.r‘𝑅) |
rngz.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
ringrz | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrp 18868 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | rngz.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rngz.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 17766 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2799 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 2, 5, 3 | grplid 17768 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
7 | 4, 6 | mpdan 679 | . . . . . 6 ⊢ (𝑅 ∈ Grp → ( 0 (+g‘𝑅) 0 ) = 0 ) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ Ring → ( 0 (+g‘𝑅) 0 ) = 0 ) |
9 | 8 | adantr 473 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
10 | 9 | oveq2d 6894 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = (𝑋 · 0 )) |
11 | simpr 478 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
12 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
13 | 12 | adantr 473 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
14 | 11, 13, 13 | 3jca 1159 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 0 ∈ 𝐵)) |
15 | rngz.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
16 | 2, 5, 15 | ringdi 18882 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 0 ∈ 𝐵)) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 ))) |
17 | 14, 16 | syldan 586 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 ))) |
18 | 1 | adantr 473 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp) |
19 | 2, 15 | ringcl 18877 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 · 0 ) ∈ 𝐵) |
20 | 13, 19 | mpd3an3 1587 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) ∈ 𝐵) |
21 | 2, 5, 3 | grplid 17768 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → ( 0 (+g‘𝑅)(𝑋 · 0 )) = (𝑋 · 0 )) |
22 | 21 | eqcomd 2805 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → (𝑋 · 0 ) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
23 | 18, 20, 22 | syl2anc 580 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
24 | 10, 17, 23 | 3eqtr3d 2841 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
25 | 2, 5 | grprcan 17771 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ ((𝑋 · 0 ) ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ (𝑋 · 0 ) ∈ 𝐵)) → (((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 )) ↔ (𝑋 · 0 ) = 0 )) |
26 | 18, 20, 13, 20, 25 | syl13anc 1492 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 )) ↔ (𝑋 · 0 ) = 0 )) |
27 | 24, 26 | mpbid 224 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 +gcplusg 16267 .rcmulr 16268 0gc0g 16415 Grpcgrp 17738 Ringcrg 18863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-plusg 16280 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-mgp 18806 df-ring 18865 |
This theorem is referenced by: ringsrg 18905 ringinvnz1ne0 18908 ringinvnzdiv 18909 rngnegr 18911 gsummgp0 18924 gsumdixp 18925 dvdsr02 18972 isdrng2 19075 drngmul0or 19086 cntzsubr 19130 isabvd 19138 lmodvs0 19215 rrgeq0 19613 unitrrg 19616 domneq0 19620 psrridm 19727 mpllsslem 19758 mplsubrglem 19762 mplcoe1 19788 mplmon2 19815 evlslem4 19830 coe1tmmul2 19968 cply1mul 19986 ocvlss 20341 frlmphl 20445 uvcresum 20457 mamurid 20573 matsc 20582 dmatmul 20629 dmatscmcl 20635 scmatscmide 20639 mulmarep1el 20704 mdetdiaglem 20730 mdetero 20742 mdetunilem8 20751 mdetunilem9 20752 mdetuni0 20753 maducoeval2 20772 madugsum 20775 smadiadetlem1a 20796 smadiadetglem2 20805 chpdmatlem2 20972 chfacfpmmul0 20995 cayhamlem4 21021 mdegvscale 24176 mdegmullem 24179 coe1mul3 24200 deg1mul3le 24217 ply1divex 24237 ply1rem 24264 fta1blem 24269 kerunit 30339 matunitlindflem1 33894 lfl0f 35090 lfl0sc 35103 lkrlss 35116 lcfrlem33 37596 hdmapinvlem3 37941 hdmapglem7b 37949 cntzsdrg 38557 mgpsumz 42940 domnmsuppn0 42949 rmsuppss 42950 ply1mulgsumlem2 42974 lincresunit2 43066 |
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