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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 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 | mndrid 18768 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
| 6 | 1, 5 | sylan 580 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 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: grpridd 18988 grpinvid1 19009 grpinvid2 19010 grpidinv2 19015 grpasscan2 19020 grpidrcan 19021 grpraddf1o 19032 grpsubid1 19043 grpsubadd 19046 grppncan 19049 mulgaddcom 19116 mulgdirlem 19123 mulgmodid 19131 nmzsubg 19183 0nsg 19187 ghmquskerlem1 19301 cntzsubg 19357 cayleylem2 19431 odbezout 19576 lsmdisj2 19700 pj1lid 19719 frgpuplem 19790 abladdsub4 19829 odadd2 19867 gex2abl 19869 rnglz 20162 isabvd 20813 lmod0vrid 20891 lmodfopne 20898 islmhm2 21037 rnglidl0 21239 lsmcss 21710 mplcoe1 22055 mdetero 22616 mdetunilem6 22623 opnsubg 24116 tgpconncompeqg 24120 snclseqg 24124 clmvz 25144 deg1add 26142 gsumsubg 33049 ogrpaddltbi 33095 ogrpinvlt 33100 archiabllem2a 33201 archiabllem2c 33202 lindsunlem 33675 lflmul 39069 cdlemn4 41200 mapdh6cN 41740 hdmap1l6c 41814 hdmapinvlem3 41922 hdmapinvlem4 41923 hdmapglem7b 41930 fsuppind 42600 |
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