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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 18375 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | rngz.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rngz.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 17273 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2610 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 2, 5, 3 | grplid 17275 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
7 | 4, 6 | mpdan 699 | . . . . . 6 ⊢ (𝑅 ∈ Grp → ( 0 (+g‘𝑅) 0 ) = 0 ) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ Ring → ( 0 (+g‘𝑅) 0 ) = 0 ) |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
10 | 9 | oveq2d 6565 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = (𝑋 · 0 )) |
11 | simpr 476 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
12 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
14 | 11, 13, 13 | 3jca 1235 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 0 ∈ 𝐵)) |
15 | rngz.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
16 | 2, 5, 15 | ringdi 18389 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 0 ∈ 𝐵)) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 ))) |
17 | 14, 16 | syldan 486 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 ))) |
18 | 1 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp) |
19 | 2, 15 | ringcl 18384 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 · 0 ) ∈ 𝐵) |
20 | 13, 19 | mpd3an3 1417 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) ∈ 𝐵) |
21 | 2, 5, 3 | grplid 17275 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → ( 0 (+g‘𝑅)(𝑋 · 0 )) = (𝑋 · 0 )) |
22 | 21 | eqcomd 2616 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → (𝑋 · 0 ) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
23 | 18, 20, 22 | syl2anc 691 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
24 | 10, 17, 23 | 3eqtr3d 2652 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
25 | 2, 5 | grprcan 17278 | . . 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 1320 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 )) ↔ (𝑋 · 0 ) = 0 )) |
27 | 24, 26 | mpbid 221 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 0gc0g 15923 Grpcgrp 17245 Ringcrg 18370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mgp 18313 df-ring 18372 |
This theorem is referenced by: ringsrg 18412 ringinvnz1ne0 18415 ringinvnzdiv 18416 rngnegr 18418 gsummgp0 18431 gsumdixp 18432 dvdsr02 18479 isdrng2 18580 drngmul0or 18591 cntzsubr 18635 isabvd 18643 lmodvs0 18720 rrgeq0 19111 unitrrg 19114 domneq0 19118 psrridm 19225 mpllsslem 19256 mplsubrglem 19260 mplcoe1 19286 mplmon2 19314 evlslem4 19329 coe1tmmul2 19467 cply1mul 19485 ocvlss 19835 frlmphl 19939 uvcresum 19951 mamurid 20067 matsc 20075 dmatmul 20122 dmatscmcl 20128 scmatscmide 20132 mulmarep1el 20197 mdetdiaglem 20223 mdetero 20235 mdetunilem8 20244 mdetunilem9 20245 mdetuni0 20246 maducoeval2 20265 madugsum 20268 smadiadetlem1a 20288 smadiadetglem2 20297 chpdmatlem2 20463 chfacfpmmul0 20486 cayhamlem4 20512 mdegvscale 23639 mdegmullem 23642 coe1mul3 23663 deg1mul3le 23680 ply1divex 23700 ply1rem 23727 fta1blem 23732 kerunit 29154 matunitlindflem1 32575 lfl0f 33374 lfl0sc 33387 lkrlss 33400 lcfrlem33 35882 hdmapinvlem3 36230 hdmapglem7b 36238 cntzsdrg 36791 mgpsumz 41934 domnmsuppn0 41944 rmsuppss 41945 ply1mulgsumlem2 41969 lincresunit2 42061 |
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