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Mirrors > Home > MPE Home > Th. List > grprid | Structured version Visualization version GIF version |
Description: The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.) |
Ref | Expression |
---|---|
grpbn0.b | ⊢ 𝐵 = (Base‘𝐺) |
grplid.p | ⊢ + = (+g‘𝐺) |
grplid.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grprid | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18372 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpbn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grplid.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | grplid.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
5 | 2, 3, 4 | mndrid 18194 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
6 | 1, 5 | sylan 583 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 0gc0g 16944 Mndcmnd 18173 Grpcgrp 18365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-riota 7170 df-ov 7216 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 |
This theorem is referenced by: grprcan 18401 grpinvid1 18418 grpinvid2 18419 grpidinv2 18422 grpasscan2 18427 grpidrcan 18428 grpsubid1 18448 grpsubadd 18451 grppncan 18454 mulgaddcom 18515 mulgdirlem 18522 mulgmodid 18530 nmzsubg 18581 0nsg 18585 cntzsubg 18731 cayleylem2 18805 odbezout 18949 lsmdisj2 19072 pj1lid 19091 frgpuplem 19162 abladdsub4 19199 odadd2 19234 gex2abl 19236 ringlz 19605 isabvd 19856 lmod0vrid 19930 lmodfopne 19937 islmhm2 20075 lsmcss 20654 mplcoe1 20994 mdetero 21507 mdetunilem6 21514 opnsubg 23005 tgpconncompeqg 23009 snclseqg 23013 clmvz 24008 deg1add 25001 gsumsubg 31025 ogrpaddltbi 31063 ogrpinvlt 31068 archiabllem2a 31167 archiabllem2c 31168 lindsunlem 31419 lflmul 36819 cdlemn4 38949 mapdh6cN 39489 hdmap1l6c 39563 hdmapinvlem3 39671 hdmapinvlem4 39672 hdmapglem7b 39679 fsuppind 39989 rnglz 45115 |
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