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Theorem grpinvinv 18972

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 18954	. . . 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 18957	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0_g‘𝐺))
7	3, 6	syldan 597	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0_g‘𝐺))
8	1, 4, 5, 2	grplinv 18956	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)𝑋) = (0_g‘𝐺))
9	7, 8	eqtr4d 2777	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+_g‘𝐺)𝑋))
10		simpl 483	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ Grp)
11	1, 2	grpinvcl 18954	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵)
12	3, 11	syldan 597	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵)
13		simpr 485	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
14	1, 4	grplcan 18967	. . 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 1380	. 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 233	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 +_gcplusg 17211 0_gc0g 17393 Grpcgrp 18900 inv_gcminusg 18901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-fv 6493 df-riota 7313 df-ov 7359 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904

This theorem is referenced by: grpinv11 18974 grpinv11OLD 18975 grpinvnz 18977 grpsubinv 18979 grpinvsub 18989 grpsubeq0 18993 grpnpcan 18999 mulgneg 19059 mulgnegneg 19060 mulginvinv 19067 mulgdir 19073 mulgass 19078 eqger 19144 frgpuptinv 19737 ablsub2inv 19774 mulgdi 19792 invghm 19799 rngm2neg 20141 unitinvinv 20362 unitnegcl 20368 irrednegb 20402 abvneg 20798 lspsnneg 20996 islindf4 21813 tgpconncomp 24096 grpinvinvd 33120 archirngz 33270 archiabllem1b 33273 ply1divalg3 35870 baerlem5amN 42208 baerlem5bmN 42209 baerlem5abmN 42210 fldhmf1 42575 nelsubginvcld 42986

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