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Theorem grprinv 18947

Description: The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

**Hypotheses**
Ref	Expression
grpinv.b	⊢ 𝐵 = (Base‘𝐺)
grpinv.p	⊢ + = (+_g‘𝐺)
grpinv.u	⊢ 0 = (0_g‘𝐺)
grpinv.n	⊢ 𝑁 = (inv_g‘𝐺)

**Assertion**
Ref	Expression
grprinv	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 )

Proof of Theorem grprinv Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinv.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		grpinv.p	. . 3 ⊢ + = (+_g‘𝐺)
3	1, 2	grpcl 18898	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
4		grpinv.u	. . 3 ⊢ 0 = (0_g‘𝐺)
5	1, 4	grpidcl 18922	. 2 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
6	1, 2, 4	grplid 18924	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
7	1, 2	grpass 18899	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
8	1, 2, 4	grpinvex 18900	. 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 = (inv_g‘𝐺)
11	1, 10	grpinvcl 18944	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
12	1, 2, 4, 10	grplinv 18946	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
13	3, 5, 6, 7, 8, 9, 11, 12	grpinva 18634	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +_gcplusg 17233 0_gc0g 17421 Grpcgrp 18890 inv_gcminusg 18891

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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-riota 7376 df-ov 7423 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894

This theorem is referenced by: grpinvid1 18948 grpinvid2 18949 grprinvd 18952 grplrinv 18953 grpasscan1 18958 grpinvinv 18962 grplmulf1o 18969 grpinvadd 18974 grpsubid 18980 dfgrp3 18995 mulgdirlem 19060 subginv 19088 nmzsubg 19120 eqger 19133 qusinv 19145 ghminv 19177 gacan 19256 cntzsubg 19290 oppggrp 19311 oppginv 19313 psgnuni 19454 sylow2blem3 19577 frgpuplem 19727 ringnegl 20238 unitrinv 20333 isdrng2 20638 lmodvnegid 20787 lmodvsinv2 20922 lspsolvlem 21030 evpmodpmf1o 21528 grpvrinv 22311 mdetralt 22523 ghmcnp 24032 qustgpopn 24037 isngp4 24534 clmvsrinv 25047 ogrpinv0le 32808 ogrpaddltbi 32811 ogrpinv0lt 32815 ogrpinvlt 32816 archiabllem1b 32913 orngsqr 33032 quslsm 33128 lbsdiflsp0 33324 fldhmf1 41561 ldepsprlem 47540

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