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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 18049 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
4 | grpinv.u | . . 3 ⊢ 0 = (0g‘𝐺) | |
5 | 1, 4 | grpidcl 18069 | . 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
6 | 1, 2, 4 | grplid 18071 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
7 | 1, 2 | grpass 18050 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 1, 2, 4 | grpinvex 18051 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
9 | simpr 485 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | grpinv.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
11 | 1, 10 | grpinvcl 18089 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
12 | 1, 2, 4, 10 | grplinv 18090 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) |
13 | 3, 5, 6, 7, 8, 9, 11, 12 | grprinvd 17872 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Grpcgrp 18041 invgcminusg 18042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 |
This theorem is referenced by: grpinvid1 18092 grpinvid2 18093 grplrinv 18095 grpasscan1 18100 grpinvinv 18104 grplmulf1o 18111 grpinvadd 18115 grpsubid 18121 dfgrp3 18136 mulgdirlem 18196 subginv 18224 nmzsubg 18255 eqger 18268 qusinv 18277 ghminv 18303 conjnmz 18330 gacan 18373 cntzsubg 18405 oppggrp 18423 oppginv 18425 psgnuni 18556 sylow2blem3 18676 frgpuplem 18827 ringnegl 19273 unitrinv 19357 isdrng2 19441 lmodvnegid 19605 lmodvsinv2 19738 lspsolvlem 19843 evpmodpmf1o 20668 grpvrinv 20935 mdetralt 21145 ghmcnp 22650 qustgpopn 22655 isngp4 23148 clmvsrinv 23638 ogrpinv0le 30643 ogrpaddltbi 30646 ogrpinv0lt 30650 ogrpinvlt 30651 archiabllem1b 30748 orngsqr 30804 lbsdiflsp0 30921 ldepsprlem 44455 |
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