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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 18803 | . 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 18625 | . 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 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 +gcplusg 17181 0gc0g 17369 Mndcmnd 18604 Grpcgrp 18796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-iota 6485 df-fun 6535 df-fv 6541 df-riota 7350 df-ov 7397 df-0g 17371 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-grp 18799 |
This theorem is referenced by: grpridd 18832 grprcan 18835 grpinvid1 18853 grpinvid2 18854 grpidinv2 18858 grpasscan2 18863 grpidrcan 18864 grpsubid1 18884 grpsubadd 18887 grppncan 18890 mulgaddcom 18952 mulgdirlem 18959 mulgmodid 18967 nmzsubg 19019 0nsg 19023 cntzsubg 19169 cayleylem2 19247 odbezout 19392 lsmdisj2 19516 pj1lid 19535 frgpuplem 19606 abladdsub4 19644 odadd2 19679 gex2abl 19681 ringlz 20066 isabvd 20379 lmod0vrid 20454 lmodfopne 20461 islmhm2 20600 lsmcss 21180 mplcoe1 21522 mdetero 22043 mdetunilem6 22050 opnsubg 23543 tgpconncompeqg 23547 snclseqg 23551 clmvz 24558 deg1add 25552 gsumsubg 32133 ogrpaddltbi 32171 ogrpinvlt 32176 archiabllem2a 32275 archiabllem2c 32276 ghmquskerlem1 32448 lindsunlem 32611 lflmul 37807 cdlemn4 39938 mapdh6cN 40478 hdmap1l6c 40552 hdmapinvlem3 40660 hdmapinvlem4 40661 hdmapglem7b 40668 fsuppind 41015 rnglz 46494 |
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