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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 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 | mndrid 17135 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
6 | 1, 5 | sylan 487 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 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 grpinvid1 17293 grpinvid2 17294 grpidinv2 17297 grpasscan2 17302 grpidrcan 17303 grpsubid1 17323 grpsubadd 17326 grppncan 17329 mulgaddcom 17387 mulgdirlem 17395 mulgmodid 17404 nmzsubg 17458 0nsg 17462 cntzsubg 17592 cayleylem2 17656 odbezout 17798 lsmdisj2 17918 pj1lid 17937 frgpuplem 18008 abladdsub4 18042 odadd2 18075 gex2abl 18077 ringlz 18410 isabvd 18643 lmod0vrid 18717 lmodfopne 18724 islmhm2 18859 mplcoe1 19286 lsmcss 19855 mdetero 20235 mdetunilem6 20242 opnsubg 21721 tgpconcompeqg 21725 snclseqg 21729 clmvz 22719 deg1add 23667 ogrpaddltbi 29050 ogrpinvlt 29055 archiabllem2a 29079 archiabllem2c 29080 lflmul 33373 cdlemn4 35505 mapdh6cN 36045 hdmap1l6c 36120 hdmapinvlem3 36230 hdmapinvlem4 36231 hdmapglem7b 36238 rnglz 41674 |
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