![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ringlidm | Structured version Visualization version GIF version |
Description: The unity element of a ring is a left 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 |
---|---|
ringlidm | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 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 20282 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) |
5 | 4 | simpld 494 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 𝑋) |
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 1rcur 20199 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-mgp 20153 df-ur 20200 df-ring 20253 |
This theorem is referenced by: ringlidmd 20286 ringo2times 20289 ringidss 20291 ringcomlem 20293 ring1eq0 20312 ringinvnzdiv 20315 ringnegl 20316 imasring 20344 xpsring1d 20347 opprring 20364 dvdsrid 20384 unitmulcl 20397 unitgrp 20400 1rinv 20412 dvreq1 20428 ringinvdv 20431 subrginv 20605 issubrg2 20609 unitrrg 20720 isdrng2 20760 drngmul0orOLD 20778 isdrngd 20782 isdrngdOLD 20784 abv1z 20842 issrngd 20873 sralmod 21212 rngqiprngfulem5 21343 mulgrhm 21506 dvdschrmulg 21561 freshmansdream 21611 asclmul1 21924 psrlmod 21998 psrlidm 22000 mplmonmul 22072 evlslem1 22124 coe1pwmul 22298 mamulid 22463 madetsumid 22483 1mavmul 22570 m1detdiag 22619 mdetralt 22630 mdetunilem7 22640 mdetuni 22644 mdetmul 22645 m2detleib 22653 chfacfpmmulgsum 22886 cpmadugsumlemB 22896 nrginvrcnlem 24728 cphsubrglem 25225 ply1divex 26191 ress1r 33224 dvrcan5 33226 ornglmullt 33317 orng0le1 33322 isarchiofld 33327 elrspunidl 33436 mxidlprm 33478 madjusmdetlem1 33788 matunitlindflem1 37603 lfl0 39047 lfladd 39048 eqlkr3 39083 lcfrlem1 41525 hdmapinvlem4 41904 hdmapglem5 41905 mon1psubm 43188 lidldomn1 48075 invginvrid 48212 ply1sclrmsm 48229 ldepsprlem 48318 |
Copyright terms: Public domain | W3C validator |