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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 17253 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
4 | grpinv.u | . . 3 ⊢ 0 = (0g‘𝐺) | |
5 | 1, 4 | grpidcl 17273 | . 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
6 | 1, 2, 4 | grplid 17275 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
7 | 1, 2 | grpass 17254 | . 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 1, 2, 4 | grpinvex 17255 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
9 | simpr 476 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
10 | grpinv.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
11 | 1, 10 | grpinvcl 17290 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵) |
12 | 1, 2, 4, 10 | grplinv 17291 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 ) |
13 | 3, 5, 6, 7, 8, 9, 11, 12 | grprinvd 6771 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 |
This theorem is referenced by: grpinvid1 17293 grpinvid2 17294 grplrinv 17296 grpasscan1 17301 grpinvinv 17305 grplmulf1o 17312 grpinvadd 17316 grpsubid 17322 dfgrp3 17337 mulgdirlem 17395 subginv 17424 nmzsubg 17458 eqger 17467 qusinv 17476 ghminv 17490 conjnmz 17517 gacan 17561 cntzsubg 17592 oppggrp 17610 oppginv 17612 psgnuni 17742 sylow2blem3 17860 frgpuplem 18008 ringnegl 18417 unitrinv 18501 isdrng2 18580 lmodvnegid 18728 lmodvsinv2 18858 lspsolvlem 18963 evpmodpmf1o 19761 grpvrinv 20021 mdetralt 20233 ghmcnp 21728 qustgpopn 21733 isngp4 22226 clmvsrinv 22715 ogrpinvOLD 29046 ogrpinv0le 29047 ogrpaddltbi 29050 ogrpinv0lt 29054 ogrpinvlt 29055 archiabllem1b 29077 orngsqr 29135 ldepsprlem 42055 |
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