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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 18958 | . 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 18767 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
| 6 | 1, 5 | sylan 580 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 Grpcgrp 18951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 |
| This theorem is referenced by: grplidd 18987 grprcan 18991 grpid 18993 isgrpid2 18994 grprinv 19008 grpinvid1 19009 grpinvid2 19010 grpidinv2 19015 grpinvid 19017 grplcan 19018 grpasscan1 19019 grpidlcan 19022 grplmulf1o 19031 grpidssd 19034 grpinvadd 19036 grpinvval2 19041 grplactcnv 19061 imasgrp 19074 mulgaddcom 19116 mulgdirlem 19123 subg0 19150 issubg2 19159 issubg4 19163 0subgOLD 19170 isnsg3 19178 nmzsubg 19183 ssnmz 19184 eqgid 19198 qusgrp 19204 qus0 19207 ghmid 19240 conjghm 19267 subgga 19318 cntzsubg 19357 sylow1lem2 19617 sylow2blem2 19639 sylow2blem3 19640 sylow3lem1 19645 lsmmod 19693 lsmdisj2 19700 pj1rid 19720 abladdsub4 19829 ablpncan2 19833 ablpnpcan 19837 ablnncan 19838 odadd1 19866 odadd2 19867 oddvdssubg 19873 dprdfadd 20040 pgpfac1lem3a 20096 rnglz 20162 rngrz 20163 isabvd 20813 lmod0vlid 20890 lmod0vs 20893 freshmansdream 21593 evpmodpmf1o 21614 ocvlss 21690 lsmcss 21710 psr0lid 21973 mplsubglem 22019 mplcoe1 22055 mdetunilem6 22623 mdetunilem9 22626 ghmcnp 24123 tgpt0 24127 qustgpopn 24128 mdegaddle 26113 ply1rem 26205 gsumsubg 33049 ogrpinv0le 33092 ogrpaddltrbid 33097 ogrpinv0lt 33099 ogrpinvlt 33100 cyc3genpmlem 33171 isarchi3 33194 archirngz 33196 archiabllem1b 33199 orngsqr 33334 ornglmulle 33335 orngrmulle 33336 qusker 33377 grplsm0l 33431 quslsm 33433 mxidlprm 33498 matunitlindflem1 37623 lfl0f 39070 lfladd0l 39075 lkrlss 39096 lkrin 39165 dvhgrp 41109 baerlem3lem1 41709 mapdh6bN 41739 hdmap1l6b 41813 hdmapinvlem3 41922 hdmapinvlem4 41923 hdmapglem7b 41930 fsuppind 42600 fsuppssind 42603 |
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