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Theorem grpinvinv 19045

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 19027	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
4		eqid 2740	. . . . 5 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
5		eqid 2740	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
6	1, 4, 5, 2	grprinv 19030	. . . 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 19029	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)𝑋) = (0_g‘𝐺))
9	7, 8	eqtr4d 2783	. 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 19027	. . . 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 19040	. . 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 1372	. 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 232	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +_gcplusg 17311 0_gc0g 17499 Grpcgrp 18973 inv_gcminusg 18974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-riota 7404 df-ov 7451 df-0g 17501 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977

This theorem is referenced by: grpinv11 19047 grpinv11OLD 19048 grpinvnz 19050 grpsubinv 19052 grpinvsub 19062 grpsubeq0 19066 grpnpcan 19072 mulgneg 19132 mulgnegneg 19133 mulginvinv 19140 mulgdir 19146 mulgass 19151 eqger 19218 frgpuptinv 19813 ablsub2inv 19850 mulgdi 19868 invghm 19875 rngm2neg 20196 unitinvinv 20417 unitnegcl 20423 irrednegb 20457 abvneg 20849 lspsnneg 21027 islindf4 21881 tgpconncomp 24142 archirngz 33169 archiabllem1b 33172 ply1divalg3 35610 baerlem5amN 41673 baerlem5bmN 41674 baerlem5abmN 41675 fldhmf1 42047 nelsubginvcld 42451

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