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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 18906 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | rngz.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
3 | rngz.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | grpidcl 17804 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
5 | eqid 2825 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 2, 5, 3 | grplid 17806 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
7 | 4, 6 | mpdan 680 | . . . . . 6 ⊢ (𝑅 ∈ Grp → ( 0 (+g‘𝑅) 0 ) = 0 ) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ Ring → ( 0 (+g‘𝑅) 0 ) = 0 ) |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 (+g‘𝑅) 0 ) = 0 ) |
10 | 9 | oveq2d 6921 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = (𝑋 · 0 )) |
11 | simpr 479 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
12 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
13 | 12 | adantr 474 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
14 | 11, 13, 13 | 3jca 1164 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 0 ∈ 𝐵)) |
15 | rngz.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
16 | 2, 5, 15 | ringdi 18920 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 0 ∈ 𝐵)) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 ))) |
17 | 14, 16 | syldan 587 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · ( 0 (+g‘𝑅) 0 )) = ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 ))) |
18 | 1 | adantr 474 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Grp) |
19 | 2, 15 | ringcl 18915 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 · 0 ) ∈ 𝐵) |
20 | 13, 19 | mpd3an3 1592 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) ∈ 𝐵) |
21 | 2, 5, 3 | grplid 17806 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → ( 0 (+g‘𝑅)(𝑋 · 0 )) = (𝑋 · 0 )) |
22 | 21 | eqcomd 2831 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 0 ) ∈ 𝐵) → (𝑋 · 0 ) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
23 | 18, 20, 22 | syl2anc 581 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
24 | 10, 17, 23 | 3eqtr3d 2869 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 0 )(+g‘𝑅)(𝑋 · 0 )) = ( 0 (+g‘𝑅)(𝑋 · 0 ))) |
25 | 2, 5 | grprcan 17809 | . . 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 1497 | . 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 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 .rcmulr 16306 0gc0g 16453 Grpcgrp 17776 Ringcrg 18901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-mgp 18844 df-ring 18903 |
This theorem is referenced by: ringsrg 18943 ringinvnz1ne0 18946 ringinvnzdiv 18947 rngnegr 18949 gsummgp0 18962 gsumdixp 18963 dvdsr02 19010 isdrng2 19113 drngmul0or 19124 cntzsubr 19168 isabvd 19176 lmodvs0 19253 rrgeq0 19651 unitrrg 19654 domneq0 19658 psrridm 19765 mpllsslem 19796 mplsubrglem 19800 mplcoe1 19826 mplmon2 19853 evlslem4 19868 coe1tmmul2 20006 cply1mul 20024 ocvlss 20379 frlmphl 20487 uvcresum 20499 mamurid 20615 matsc 20624 dmatmul 20671 dmatscmcl 20677 scmatscmide 20681 mulmarep1el 20746 mdetdiaglem 20772 mdetero 20784 mdetunilem8 20793 mdetunilem9 20794 mdetuni0 20795 maducoeval2 20814 madugsum 20817 smadiadetlem1a 20838 smadiadetglem2 20847 chpdmatlem2 21014 chfacfpmmul0 21037 cayhamlem4 21063 mdegvscale 24234 mdegmullem 24237 coe1mul3 24258 deg1mul3le 24275 ply1divex 24295 ply1rem 24322 fta1blem 24327 kerunit 30368 matunitlindflem1 33949 lfl0f 35144 lfl0sc 35157 lkrlss 35170 lcfrlem33 37650 hdmapinvlem3 37995 hdmapglem7b 38003 cntzsdrg 38615 mgpsumz 42988 domnmsuppn0 42997 rmsuppss 42998 ply1mulgsumlem2 43022 lincresunit2 43114 |
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