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Mirrors > Home > MPE Home > Th. List > ringridm | Structured version Visualization version GIF version |
Description: The unity element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) |
Ref | Expression |
---|---|
ringidm.b | โข ๐ต = (Baseโ๐ ) |
ringidm.t | โข ยท = (.rโ๐ ) |
ringidm.u | โข 1 = (1rโ๐ ) |
Ref | Expression |
---|---|
ringridm | โข ((๐ โ Ring โง ๐ โ ๐ต) โ (๐ ยท 1 ) = ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringidm.b | . . 3 โข ๐ต = (Baseโ๐ ) | |
2 | ringidm.t | . . 3 โข ยท = (.rโ๐ ) | |
3 | ringidm.u | . . 3 โข 1 = (1rโ๐ ) | |
4 | 1, 2, 3 | ringidmlem 20084 | . 2 โข ((๐ โ Ring โง ๐ โ ๐ต) โ (( 1 ยท ๐) = ๐ โง (๐ ยท 1 ) = ๐)) |
5 | 4 | simprd 496 | 1 โข ((๐ โ Ring โง ๐ โ ๐ต) โ (๐ ยท 1 ) = ๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 396 = wceq 1541 โ wcel 2106 โcfv 6543 (class class class)co 7408 Basecbs 17143 .rcmulr 17197 1rcur 20003 Ringcrg 20055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mgp 19987 df-ur 20004 df-ring 20057 |
This theorem is referenced by: ringridmd 20089 ringidss 20093 ringinvnz1ne0 20111 ringnegr 20114 imasring 20142 xpsring1d 20145 opprring 20160 unitmulcl 20193 unitgrp 20196 dvr1 20220 dvrcan1 20222 dvrcan3 20223 subrginv 20334 issubrg2 20338 lidl1el 20840 uvcresum 21347 frlmssuvc2 21349 asclmul2 21440 psrridm 21523 mplcoe1 21591 mplmon2 21621 evlslem1 21644 mamurid 21943 matsc 21951 scmatscmide 22008 mat1scmat 22040 mulmarep1el 22073 mdet0pr 22093 mdetunilem9 22121 mdetuni0 22122 maducoeval2 22141 madugsum 22144 smadiadetglem2 22173 cramerimplem1 22184 chpmat1dlem 22336 chpdmatlem3 22341 nrginvrcnlem 24207 lgseisenlem4 26878 freshmansdream 32376 ress1r 32378 lfl1sc 37949 eqlkr 37964 eqlkr3 37966 lkrlsp 37967 lcfl7lem 40365 lclkrlem2m 40385 hdmapinvlem3 40786 hdmapglem5 40788 hgmapvvlem1 40789 hdmapglem7b 40794 0prjspnrel 41370 mgpsumn 47029 |
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