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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 18110 | . 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 17932 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
6 | 1, 5 | sylan 582 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Mndcmnd 17911 Grpcgrp 18103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 |
This theorem is referenced by: grprcan 18137 grpinvid1 18154 grpinvid2 18155 grpidinv2 18158 grpasscan2 18163 grpidrcan 18164 grpsubid1 18184 grpsubadd 18187 grppncan 18190 mulgaddcom 18251 mulgdirlem 18258 mulgmodid 18266 nmzsubg 18317 0nsg 18321 cntzsubg 18467 cayleylem2 18541 odbezout 18685 lsmdisj2 18808 pj1lid 18827 frgpuplem 18898 abladdsub4 18934 odadd2 18969 gex2abl 18971 ringlz 19337 isabvd 19591 lmod0vrid 19665 lmodfopne 19672 islmhm2 19810 mplcoe1 20246 mhpinvcl 20339 lsmcss 20836 mdetero 21219 mdetunilem6 21226 opnsubg 22716 tgpconncompeqg 22720 snclseqg 22724 clmvz 23715 deg1add 24697 gsumsubg 30684 ogrpaddltbi 30719 ogrpinvlt 30724 archiabllem2a 30823 archiabllem2c 30824 lindsunlem 31020 lflmul 36219 cdlemn4 38349 mapdh6cN 38889 hdmap1l6c 38963 hdmapinvlem3 39071 hdmapinvlem4 39072 hdmapglem7b 39079 rnglz 44175 |
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