Theorem grpo2inv 30369

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 30362	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
4		eqid 2728	. . . . 5 ⊢ (GId‘𝐺) = (GId‘𝐺)
5	1, 4, 2	grporinv 30365	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
6	3, 5	syldan 589	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = (GId‘𝐺))
7	1, 4, 2	grpolinv 30364	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = (GId‘𝐺))
8	6, 7	eqtr4d 2771	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺(𝑁‘(𝑁‘𝐴))) = ((𝑁‘𝐴)𝐺𝐴))
9	1, 2	grpoinvcl 30362	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑁‘𝐴) ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
10	3, 9	syldan 589	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝑁‘𝐴)) ∈ 𝑋)
11		simpr 483	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
12	10, 11, 3	3jca 1125	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐴)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋))
13	1	grpolcan 30368	. . 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 589	. 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5683  ‘cfv 6553  (class class class)co 7426  GrpOpcgr 30327  GIdcgi 30328  invcgn 30329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-grpo 30331  df-gid 30332  df-ginv 30333

This theorem is referenced by:  grpoinvf  30370  grpodivinv  30374  grpoinvdiv  30375  nvnegneg  30487