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Theorem grpo2inv 26769
 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 ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) = 𝐴)

Proof of Theorem grpo2inv
StepHypRef Expression
1 grpasscan1.1 . . . . 5 𝑋 = ran 𝐺
2 grpasscan1.2 . . . . 5 𝑁 = (inv‘𝐺)
31, 2grpoinvcl 26762 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
4 eqid 2610 . . . . 5 (GId‘𝐺) = (GId‘𝐺)
51, 4, 2grporinv 26765 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = (GId‘𝐺))
63, 5syldan 486 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = (GId‘𝐺))
71, 4, 2grpolinv 26764 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = (GId‘𝐺))
86, 7eqtr4d 2647 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴))
91, 2grpoinvcl 26762 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑁‘(𝑁𝐴)) ∈ 𝑋)
103, 9syldan 486 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) ∈ 𝑋)
11 simpr 476 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → 𝐴𝑋)
1210, 11, 33jca 1235 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁‘(𝑁𝐴)) ∈ 𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
131grpolcan 26768 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝑁‘(𝑁𝐴)) ∈ 𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → (((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁𝐴)) = 𝐴))
1412, 13syldan 486 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁𝐴)) = 𝐴))
158, 14mpbid 221 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ran crn 5039  ‘cfv 5804  (class class class)co 6549  GrpOpcgr 26727  GIdcgi 26728  invcgn 26729 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-grpo 26731  df-gid 26732  df-ginv 26733 This theorem is referenced by:  grpoinvf  26770  grpodivinv  26774  grpoinvdiv  26775  nvnegneg  26888
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