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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 20299 | . 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 20184 | . 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 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 0gc0g 17486 Rngcrng 20170 Ringcrg 20251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 |
This theorem is referenced by: ringrzd 20310 ringsrg 20311 ringinvnz1ne0 20314 ringinvnzdiv 20315 ringnegr 20317 gsummgp0 20332 gsumdixp 20333 dvdsr02 20389 cntzsubr 20623 rrgeq0 20717 unitrrg 20720 domneq0 20725 isdomn4 20733 isdrng2 20760 drngmul0orOLD 20778 cntzsdrg 20820 isabvd 20830 lmodvs0 20911 rngqiprngimf1 21328 ocvlss 21708 frlmphl 21819 uvcresum 21831 psrridm 22001 mpllsslem 22038 mplsubrglem 22042 mplcoe1 22073 mplmon2 22103 evlslem4 22118 coe1tmmul2 22295 cply1mul 22316 mamurid 22464 matsc 22472 dmatmul 22519 dmatscmcl 22525 scmatscmide 22529 mulmarep1el 22594 mdetdiaglem 22620 mdetero 22632 mdetunilem8 22641 mdetunilem9 22642 mdetuni0 22643 maducoeval2 22662 madugsum 22665 smadiadetlem1a 22685 smadiadetglem2 22694 chpdmatlem2 22861 chfacfpmmul0 22884 cayhamlem4 22910 mdegvscale 26129 mdegmullem 26132 coe1mul3 26153 deg1mul3le 26171 ply1divex 26191 ply1rem 26220 fta1blem 26225 kerunit 33329 dvdsruasso 33393 mxidlirredi 33479 irngnzply1lem 33705 matunitlindflem1 37603 lfl0f 39051 lfl0sc 39064 lkrlss 39077 lcfrlem33 41558 hdmapinvlem3 41903 hdmapglem7b 41911 ringexp0nn 42116 mnringmulrcld 44224 mgpsumz 48207 domnmsuppn0 48214 rmsuppss 48215 ply1mulgsumlem2 48233 lincresunit2 48324 |
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