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Theorem grpinvinv 18623

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 18608	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
4		eqid 2739	. . . . 5 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
5		eqid 2739	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
6	1, 4, 5, 2	grprinv 18610	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0_g‘𝐺))
7	3, 6	syldan 590	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0_g‘𝐺))
8	1, 4, 5, 2	grplinv 18609	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)𝑋) = (0_g‘𝐺))
9	7, 8	eqtr4d 2782	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+_g‘𝐺)𝑋))
10		simpl 482	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ Grp)
11	1, 2	grpinvcl 18608	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵)
12	3, 11	syldan 590	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵)
13		simpr 484	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
14	1, 4	grplcan 18618	. . 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 1370	. 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 231	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 +_gcplusg 16943 0_gc0g 17131 Grpcgrp 18558 inv_gcminusg 18559

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-fv 6438 df-riota 7225 df-ov 7271 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562

This theorem is referenced by: grpinv11 18625 grpinvnz 18627 grpsubinv 18629 grpinvsub 18638 grpsubeq0 18642 grpnpcan 18648 mulgneg 18703 mulgnegneg 18704 mulginvinv 18710 mulgdir 18716 mulgass 18721 eqger 18787 frgpuptinv 19358 ablsub2inv 19393 mulgdi 19409 invghm 19416 ringm2neg 19818 unitinvinv 19898 unitnegcl 19904 irrednegb 19934 abvneg 20075 lspsnneg 20249 islindf4 21026 tgpconncomp 23245 archirngz 31422 archiabllem1b 31425 baerlem5amN 39709 baerlem5bmN 39710 baerlem5abmN 39711 nelsubginvcld 40200

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