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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 | ⊢ 𝐵 = (Base‘𝑅) |
ringidm.t | ⊢ · = (.r‘𝑅) |
ringidm.u | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
ringridm | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringidm.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | ringidm.t | . . 3 ⊢ · = (.r‘𝑅) | |
3 | ringidm.u | . . 3 ⊢ 1 = (1r‘𝑅) | |
4 | 1, 2, 3 | ringidmlem 20216 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) |
5 | 4 | simprd 494 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 .rcmulr 17237 1rcur 20133 Ringcrg 20185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mgp 20087 df-ur 20134 df-ring 20187 |
This theorem is referenced by: ringridmd 20221 ringidss 20225 ringinvnz1ne0 20248 ringnegr 20251 imasring 20278 xpsring1d 20281 opprring 20298 unitmulcl 20331 unitgrp 20334 dvr1 20358 dvrcan1 20360 dvrcan3 20361 subrginv 20539 issubrg2 20543 lidl1el 21134 freshmansdream 21525 uvcresum 21744 frlmssuvc2 21746 asclmul2 21837 psrridm 21925 mplcoe1 21997 mplmon2 22027 evlslem1 22050 mamurid 22388 matsc 22396 scmatscmide 22453 mat1scmat 22485 mulmarep1el 22518 mdet0pr 22538 mdetunilem9 22566 mdetuni0 22567 maducoeval2 22586 madugsum 22589 smadiadetglem2 22618 cramerimplem1 22629 chpmat1dlem 22781 chpdmatlem3 22786 nrginvrcnlem 24652 lgseisenlem4 27356 ress1r 33034 lfl1sc 38686 eqlkr 38701 eqlkr3 38703 lkrlsp 38704 lcfl7lem 41102 lclkrlem2m 41122 hdmapinvlem3 41523 hdmapglem5 41525 hgmapvvlem1 41526 hdmapglem7b 41531 0prjspnrel 42186 mgpsumn 47613 |
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