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Mirrors > Home > MPE Home > Th. List > grplid | Structured version Visualization version GIF version |
Description: The identity element of a group is a left 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 |
---|---|
grplid | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 17252 | . 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 | mndlid 17134 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
6 | 1, 5 | sylan 487 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Mndcmnd 17117 Grpcgrp 17245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 |
This theorem is referenced by: grprcan 17278 grpid 17280 isgrpid2 17281 grprinv 17292 grpinvid1 17293 grpinvid2 17294 grpidinv2 17297 grpinvid 17299 grplcan 17300 grpasscan1 17301 grpidlcan 17304 grplmulf1o 17312 grpidssd 17314 grpinvadd 17316 grpinvval2 17321 grplactcnv 17341 imasgrp 17354 mulgaddcom 17387 mulgdirlem 17395 subg0 17423 issubg2 17432 issubg4 17436 0subg 17442 isnsg3 17451 nmzsubg 17458 ssnmz 17459 eqger 17467 eqgid 17469 qusgrp 17472 qus0 17475 ghmid 17489 conjghm 17514 conjnmz 17517 subgga 17556 cntzsubg 17592 sylow1lem2 17837 sylow2blem2 17859 sylow2blem3 17860 sylow3lem1 17865 lsmmod 17911 lsmdisj2 17918 pj1rid 17938 abladdsub4 18042 ablpncan2 18044 ablpnpcan 18048 ablnncan 18049 odadd1 18074 odadd2 18075 oddvdssubg 18081 dprdfadd 18242 pgpfac1lem3a 18298 ringlz 18410 ringrz 18411 isabvd 18643 lmod0vlid 18716 lmod0vs 18719 psr0lid 19216 mplsubglem 19255 mplcoe1 19286 evpmodpmf1o 19761 ocvlss 19835 lsmcss 19855 mdetunilem6 20242 mdetunilem9 20245 ghmcnp 21728 tgpt0 21732 qustgpopn 21733 mdegaddle 23638 ply1rem 23727 ogrpinvOLD 29046 ogrpinv0le 29047 ogrpaddltrbid 29052 ogrpinv0lt 29054 ogrpinvlt 29055 isarchi3 29072 archirngz 29074 archiabllem1b 29077 orngsqr 29135 ornglmulle 29136 orngrmulle 29137 ofldchr 29145 matunitlindflem1 32575 lfl0f 33374 lfladd0l 33379 lkrlss 33400 lkrin 33469 dvhgrp 35414 baerlem3lem1 36014 mapdh6bN 36044 hdmap1l6b 36119 hdmapinvlem3 36230 hdmapinvlem4 36231 hdmapglem7b 36238 rnglz 41674 |
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