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Theorem grpinvinv 19130

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 19112	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
4		eqid 2760	. . . . 5 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
5		eqid 2760	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
6	1, 4, 5, 2	grprinv 19115	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0_g‘𝐺))
7	3, 6	syldan 603	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = (0_g‘𝐺))
8	1, 4, 5, 2	grplinv 19114	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)𝑋) = (0_g‘𝐺))
9	7, 8	eqtr4d 2798	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋)(+_g‘𝐺)(𝑁‘(𝑁‘𝑋))) = ((𝑁‘𝑋)(+_g‘𝐺)𝑋))
10		simpl 488	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ Grp)
11	1, 2	grpinvcl 19112	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑁‘𝑋) ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵)
12	3, 11	syldan 603	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) ∈ 𝐵)
13		simpr 490	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
14	1, 4	grplcan 19125	. . 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 1399	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ‘cfv 6533 (class class class)co 7414 Basecbs 17302 +_gcplusg 17343 0_gc0g 17525 Grpcgrp 19058 inv_gcminusg 19059

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-riota 7371 df-ov 7417 df-0g 17527 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-grp 19061 df-minusg 19062

This theorem is used by: grpinv11 19132 grpinvnz 19134 grpsubinv 19136 grpinvsub 19146 grpsubeq0 19150 grpnpcan 19156 mulgneg 19216 mulgnegneg 19217 mulginvinv 19224 mulgdir 19230 mulgass 19235 eqger 19304 frgpuptinv 19899 ablsub2inv 19936 mulgdi 19954 invghm 19961 rngm2neg 20305 unitinvinv 20533 unitnegcl 20539 irrednegb 20573 abvneg 20993 lspsnneg 21191 islindf4 22052 tgpconncomp 24340 grpinvinvd 33481 archirngz 33630 archiabllem1b 33633 ply1divalg3 36222 baerlem5amN 42590 baerlem5bmN 42591 baerlem5abmN 42592 fldhmf1 42957 nelsubginvcld 43385

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