Theorem grpo2inv 30823

Description: Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpasscan1.1	⊢ 𝑋 = ran 𝐺
grpasscan1.2	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
grpo2inv	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) = 𝐴)

Proof of Theorem grpo2inv

Step	Hyp	Ref	Expression
1		grpasscan1.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2		grpasscan1.2	. . . . 5 ⊢ 𝑁 = (inv‘𝐺)
3	1, 2	grpoinvcl 30816	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
4		eqid 2769	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 4, 2	grporinv 30819	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
6	3, 5	syldan 602	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
7	1, 4, 2	grpolinv 30818	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = (GId‘𝐺))
8	6, 7	eqtr4d 2807	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴))
9	1, 2	grpoinvcl 30816	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
10	3, 9	syldan 602	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
11		simpr 489	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
12	10, 11, 3	3jca 1144	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐴)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋))
13	1	grpolcan 30822	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ ((𝑁‘(𝑁‘𝐴)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋)) → (((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁‘𝐴)) = 𝐴))
14	12, 13	syldan 602	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁‘𝐴)) = 𝐴))
15	8, 14	mpbid 235	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ran crn 5663  ‘cfv 6537  (class class class)co 7411  GrpOpcgr 30781  GIdcgi 30782  invcgn 30783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-grpo 30785  df-gid 30786  df-ginv 30787

This theorem is referenced by:  grpoinvf  30824  grpodivinv  30828  grpoinvdiv  30829  nvnegneg  30941