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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 18103 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
4 | grpinv.u | . . 3 ⊢ 0 = (0g‘𝐺) | |
5 | 1, 4 | grpidcl 18123 | . 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
6 | 1, 2, 4 | grplid 18125 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
7 | 1, 2 | grpass 18104 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 1, 2, 4 | grpinvex 18105 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
9 | simpr 488 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | grpinv.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
11 | 1, 10 | grpinvcl 18143 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
12 | 1, 2, 4, 10 | grplinv 18144 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) |
13 | 3, 5, 6, 7, 8, 9, 11, 12 | grprinvd 17876 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 |
This theorem is referenced by: grpinvid1 18146 grpinvid2 18147 grplrinv 18149 grpasscan1 18154 grpinvinv 18158 grplmulf1o 18165 grpinvadd 18169 grpsubid 18175 dfgrp3 18190 mulgdirlem 18250 subginv 18278 nmzsubg 18309 eqger 18322 qusinv 18331 ghminv 18357 conjnmz 18384 gacan 18427 cntzsubg 18459 oppggrp 18477 oppginv 18479 psgnuni 18619 sylow2blem3 18739 frgpuplem 18890 ringnegl 19340 unitrinv 19424 isdrng2 19505 lmodvnegid 19669 lmodvsinv2 19802 lspsolvlem 19907 evpmodpmf1o 20285 grpvrinv 21003 mdetralt 21213 ghmcnp 22720 qustgpopn 22725 isngp4 23218 clmvsrinv 23712 ogrpinv0le 30766 ogrpaddltbi 30769 ogrpinv0lt 30773 ogrpinvlt 30774 archiabllem1b 30871 orngsqr 30928 lbsdiflsp0 31110 ldepsprlem 44881 |
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