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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 18960 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | 
| 4 | grpinv.u | . . 3 ⊢ 0 = (0g‘𝐺) | |
| 5 | 1, 4 | grpidcl 18984 | . 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) | 
| 6 | 1, 2, 4 | grplid 18986 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | 
| 7 | 1, 2 | grpass 18961 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | 
| 8 | 1, 2, 4 | grpinvex 18962 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) | 
| 9 | simpr 484 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
| 10 | grpinv.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
| 11 | 1, 10 | grpinvcl 19006 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) | 
| 12 | 1, 2, 4, 10 | grplinv 19008 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) | 
| 13 | 3, 5, 6, 7, 8, 9, 11, 12 | grpinva 18688 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Grpcgrp 18952 invgcminusg 18953 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 | 
| This theorem is referenced by: grpinvid1 19010 grpinvid2 19011 grprinvd 19014 grplrinv 19015 grpasscan1 19020 grpinvinv 19024 grplmulf1o 19032 grpinvadd 19037 grpsubid 19043 dfgrp3 19058 mulgdirlem 19124 subginv 19152 nmzsubg 19184 eqger 19197 qusinv 19209 ghminv 19242 gacan 19324 cntzsubg 19358 oppggrp 19377 oppginv 19379 psgnuni 19518 sylow2blem3 19641 frgpuplem 19791 ringnegl 20300 unitrinv 20395 isdrng2 20744 lmodvnegid 20903 lmodvsinv2 21037 lspsolvlem 21145 evpmodpmf1o 21615 grpvrinv 22404 mdetralt 22615 ghmcnp 24124 qustgpopn 24129 isngp4 24626 clmvsrinv 25141 ogrpinv0le 33093 ogrpaddltbi 33096 ogrpinv0lt 33100 ogrpinvlt 33101 archiabllem1b 33200 orngsqr 33335 quslsm 33434 lbsdiflsp0 33678 fldhmf1 42092 ldepsprlem 48394 | 
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