Theorem grpo2inv 29779

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 29772	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
4		eqid 2732	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 4, 2	grporinv 29775	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
6	3, 5	syldan 591	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
7	1, 4, 2	grpolinv 29774	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = (GId‘𝐺))
8	6, 7	eqtr4d 2775	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴))
9	1, 2	grpoinvcl 29772	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
10	3, 9	syldan 591	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
11		simpr 485	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
12	10, 11, 3	3jca 1128	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐴)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋))
13	1	grpolcan 29778	. . 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 591	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁‘𝐴)) = 𝐴))
15	8, 14	mpbid 231	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ran crn 5677  ‘cfv 6543  (class class class)co 7408  GrpOpcgr 29737  GIdcgi 29738  invcgn 29739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-grpo 29741  df-gid 29742  df-ginv 29743

This theorem is referenced by:  grpoinvf  29780  grpodivinv  29784  grpoinvdiv  29785  nvnegneg  29897