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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 18936 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
4 | grpinv.u | . . 3 ⊢ 0 = (0g‘𝐺) | |
5 | 1, 4 | grpidcl 18960 | . 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
6 | 1, 2, 4 | grplid 18962 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
7 | 1, 2 | grpass 18937 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 1, 2, 4 | grpinvex 18938 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
9 | simpr 483 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | grpinv.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
11 | 1, 10 | grpinvcl 18982 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
12 | 1, 2, 4, 10 | grplinv 18984 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) |
13 | 3, 5, 6, 7, 8, 9, 11, 12 | grpinva 18667 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 0gc0g 17454 Grpcgrp 18928 invgcminusg 18929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-riota 7380 df-ov 7427 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 |
This theorem is referenced by: grpinvid1 18986 grpinvid2 18987 grprinvd 18990 grplrinv 18991 grpasscan1 18996 grpinvinv 19000 grplmulf1o 19007 grpinvadd 19012 grpsubid 19018 dfgrp3 19033 mulgdirlem 19099 subginv 19127 nmzsubg 19159 eqger 19172 qusinv 19184 ghminv 19217 gacan 19299 cntzsubg 19333 oppggrp 19354 oppginv 19356 psgnuni 19497 sylow2blem3 19620 frgpuplem 19770 ringnegl 20281 unitrinv 20376 isdrng2 20721 lmodvnegid 20880 lmodvsinv2 21015 lspsolvlem 21123 evpmodpmf1o 21592 grpvrinv 22390 mdetralt 22601 ghmcnp 24110 qustgpopn 24115 isngp4 24612 clmvsrinv 25125 ogrpinv0le 32950 ogrpaddltbi 32953 ogrpinv0lt 32957 ogrpinvlt 32958 archiabllem1b 33057 orngsqr 33182 quslsm 33280 lbsdiflsp0 33521 fldhmf1 41789 ldepsprlem 47855 |
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