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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 17429 | . 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 17312 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
6 | 1, 5 | sylan 488 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 Mndcmnd 17294 Grpcgrp 17422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 |
This theorem is referenced by: grprcan 17455 grpinvid1 17470 grpinvid2 17471 grpidinv2 17474 grpasscan2 17479 grpidrcan 17480 grpsubid1 17500 grpsubadd 17503 grppncan 17506 mulgaddcom 17564 mulgdirlem 17572 mulgmodid 17581 nmzsubg 17635 0nsg 17639 cntzsubg 17769 cayleylem2 17833 odbezout 17975 lsmdisj2 18095 pj1lid 18114 frgpuplem 18185 abladdsub4 18219 odadd2 18252 gex2abl 18254 ringlz 18587 isabvd 18820 lmod0vrid 18894 lmodfopne 18901 islmhm2 19038 mplcoe1 19465 lsmcss 20036 mdetero 20416 mdetunilem6 20423 opnsubg 21911 tgpconncompeqg 21915 snclseqg 21919 clmvz 22911 deg1add 23863 ogrpaddltbi 29719 ogrpinvlt 29724 archiabllem2a 29748 archiabllem2c 29749 lflmul 34355 cdlemn4 36487 mapdh6cN 37027 hdmap1l6c 37102 hdmapinvlem3 37212 hdmapinvlem4 37213 hdmapglem7b 37220 rnglz 41884 |
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