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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.) (Proof shortened by AV, 30-Mar-2025.) |
| Ref | Expression |
|---|---|
| ringz.b | ⊢ 𝐵 = (Base‘𝑅) |
| ringz.t | ⊢ · = (.r‘𝑅) |
| ringz.z | ⊢ 0 = (0g‘𝑅) |
| Ref | Expression |
|---|---|
| ringrz | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ringrng 20282 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Rng) | |
| 2 | ringz.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | ringz.t | . . 3 ⊢ · = (.r‘𝑅) | |
| 4 | ringz.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
| 5 | 2, 3, 4 | rngrz 20163 | . 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
| 6 | 1, 5 | sylan 580 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 0gc0g 17484 Rngcrng 20149 Ringcrg 20230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 |
| This theorem is referenced by: ringrzd 20293 ringsrg 20294 ringinvnz1ne0 20297 ringinvnzdiv 20298 ringnegr 20300 gsummgp0 20315 gsumdixp 20316 dvdsr02 20372 cntzsubr 20606 rrgeq0 20700 unitrrg 20703 domneq0 20708 isdomn4 20716 isdrng2 20743 drngmul0orOLD 20761 cntzsdrg 20803 isabvd 20813 lmodvs0 20894 rngqiprngimf1 21310 ocvlss 21690 frlmphl 21801 uvcresum 21813 psrridm 21983 mpllsslem 22020 mplsubrglem 22024 mplcoe1 22055 mplmon2 22085 evlslem4 22100 coe1tmmul2 22279 cply1mul 22300 mamurid 22448 matsc 22456 dmatmul 22503 dmatscmcl 22509 scmatscmide 22513 mulmarep1el 22578 mdetdiaglem 22604 mdetero 22616 mdetunilem8 22625 mdetunilem9 22626 mdetuni0 22627 maducoeval2 22646 madugsum 22649 smadiadetlem1a 22669 smadiadetglem2 22678 chpdmatlem2 22845 chfacfpmmul0 22868 cayhamlem4 22894 mdegvscale 26114 mdegmullem 26117 coe1mul3 26138 deg1mul3le 26156 ply1divex 26176 ply1rem 26205 fta1blem 26210 kerunit 33349 dvdsruasso 33413 mxidlirredi 33499 irngnzply1lem 33740 matunitlindflem1 37623 lfl0f 39070 lfl0sc 39083 lkrlss 39096 lcfrlem33 41577 hdmapinvlem3 41922 hdmapglem7b 41930 ringexp0nn 42135 mnringmulrcld 44247 mgpsumz 48278 domnmsuppn0 48285 rmsuppss 48286 ply1mulgsumlem2 48304 lincresunit2 48395 |
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