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Mirrors > Home > MPE Home > Th. List > ringridm | Structured version Visualization version GIF version |
Description: The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) |
Ref | Expression |
---|---|
rngidm.b | ⊢ 𝐵 = (Base‘𝑅) |
rngidm.t | ⊢ · = (.r‘𝑅) |
rngidm.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ringridm | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngidm.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | rngidm.t | . . 3 ⊢ · = (.r‘𝑅) | |
3 | rngidm.u | . . 3 ⊢ 1 = (1r‘𝑅) | |
4 | 1, 2, 3 | ringidmlem 19319 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) |
5 | 4 | simprd 498 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 .rcmulr 16565 1rcur 19250 Ringcrg 19296 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mgp 19239 df-ur 19251 df-ring 19298 |
This theorem is referenced by: ringidss 19326 ringinvnz1ne0 19341 rngnegr 19344 imasring 19368 opprring 19380 unitmulcl 19413 unitgrp 19416 dvr1 19438 dvrcan1 19440 dvrcan3 19441 subrginv 19550 issubrg2 19554 lidl1el 19990 asclmul2 20114 psrridm 20183 mplcoe1 20245 mplmon2 20272 evlslem1 20294 uvcresum 20936 frlmssuvc2 20938 mamurid 21050 matsc 21058 scmatscmide 21115 mat1scmat 21147 mulmarep1el 21180 mdet0pr 21200 mdetunilem9 21228 mdetuni0 21229 maducoeval2 21248 madugsum 21251 smadiadetglem2 21280 cramerimplem1 21291 chpmat1dlem 21442 chpdmatlem3 21447 nrginvrcnlem 23299 lgseisenlem4 25953 freshmansdream 30859 ress1r 30860 lfl1sc 36219 eqlkr 36234 eqlkr3 36236 lkrlsp 36237 lcfl7lem 38634 lclkrlem2m 38654 hdmapinvlem3 39055 hdmapglem5 39057 hgmapvvlem1 39058 hdmapglem7b 39063 0prjspnrel 39267 mgpsumn 44410 |
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