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Theorem grpinvinv 18158

Description: Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)

**Hypotheses**
Ref	Expression
grpinvinv.b	⊢ 𝐵 = (Base‘𝐺)
grpinvinv.n	⊢ 𝑁 = (inv_g‘𝐺)

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

**Proof of Theorem grpinvinv**
Step	Hyp	Ref	Expression
1		grpinvinv.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2		grpinvinv.n	. . . . 5 ⊢ 𝑁 = (inv_g‘𝐺)
3	1, 2	grpinvcl 18143	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
4		eqid 2798	. . . . 5 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
5		eqid 2798	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
6	1, 4, 5, 2	grprinv 18145	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0_g‘𝐺))
7	3, 6	syldan 594	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0_g‘𝐺))
8	1, 4, 5, 2	grplinv 18144	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)𝑋) = (0_g‘𝐺))
9	7, 8	eqtr4d 2836	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+_g‘𝐺)𝑋))
10		simpl 486	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ Grp)
11	1, 2	grpinvcl 18143	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵)
12	3, 11	syldan 594	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵)
13		simpr 488	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
14	1, 4	grplcan 18153	. . 3 ⊢ ((𝐺 ∈ Grp ∧ ((𝑁‘(𝑁‘𝑋)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵)) → (((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+_g‘𝐺)𝑋) ↔ (𝑁‘(𝑁‘𝑋)) = 𝑋))
15	10, 12, 13, 3, 14	syl13anc 1369	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+_g‘𝐺)𝑋) ↔ (𝑁‘(𝑁‘𝑋)) = 𝑋))
16	9, 15	mpbid 235	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +_gcplusg 16557 0_gc0g 16705 Grpcgrp 18095 inv_gcminusg 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: grpinv11 18160 grpinvnz 18162 grpsubinv 18164 grpinvsub 18173 grpsubeq0 18177 grpnpcan 18183 mulgneg 18238 mulgnegneg 18239 mulginvinv 18245 mulgdir 18251 mulgass 18256 eqger 18322 frgpuptinv 18889 ablsub2inv 18924 mulgdi 18940 invghm 18947 ringm2neg 19344 unitinvinv 19421 unitnegcl 19427 irrednegb 19457 abvneg 19598 lspsnneg 19771 islindf4 20527 tgpconncomp 22718 archirngz 30868 archiabllem1b 30871 baerlem5amN 39012 baerlem5bmN 39013 baerlem5abmN 39014 nelsubginvcld 39423

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