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Mirrors > Home > MPE Home > Th. List > grprinv | Structured version Visualization version GIF version |
Description: The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grpinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinv.p | ⊢ + = (+g‘𝐺) |
grpinv.u | ⊢ 0 = (0g‘𝐺) |
grpinv.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grprinv | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinv.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpinv.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | 1, 2 | grpcl 18110 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
4 | grpinv.u | . . 3 ⊢ 0 = (0g‘𝐺) | |
5 | 1, 4 | grpidcl 18130 | . 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
6 | 1, 2, 4 | grplid 18132 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
7 | 1, 2 | grpass 18111 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 1, 2, 4 | grpinvex 18112 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
9 | simpr 487 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | grpinv.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
11 | 1, 10 | grpinvcl 18150 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
12 | 1, 2, 4, 10 | grplinv 18151 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) |
13 | 3, 5, 6, 7, 8, 9, 11, 12 | grprinvd 17883 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 0gc0g 16712 Grpcgrp 18102 invgcminusg 18103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-riota 7113 df-ov 7158 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 |
This theorem is referenced by: grpinvid1 18153 grpinvid2 18154 grplrinv 18156 grpasscan1 18161 grpinvinv 18165 grplmulf1o 18172 grpinvadd 18176 grpsubid 18182 dfgrp3 18197 mulgdirlem 18257 subginv 18285 nmzsubg 18316 eqger 18329 qusinv 18338 ghminv 18364 conjnmz 18391 gacan 18434 cntzsubg 18466 oppggrp 18484 oppginv 18486 psgnuni 18626 sylow2blem3 18746 frgpuplem 18897 ringnegl 19343 unitrinv 19427 isdrng2 19511 lmodvnegid 19675 lmodvsinv2 19808 lspsolvlem 19913 evpmodpmf1o 20739 grpvrinv 21006 mdetralt 21216 ghmcnp 22722 qustgpopn 22727 isngp4 23220 clmvsrinv 23710 ogrpinv0le 30716 ogrpaddltbi 30719 ogrpinv0lt 30723 ogrpinvlt 30724 archiabllem1b 30821 orngsqr 30877 lbsdiflsp0 31022 ldepsprlem 44526 |
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