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Theorem grpo2inv 30560
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 30553 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
4 eqid 2735 . . . . 5 (GId‘𝐺) = (GId‘𝐺)
51, 4, 2grporinv 30556 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = (GId‘𝐺))
63, 5syldan 591 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = (GId‘𝐺))
71, 4, 2grpolinv 30555 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = (GId‘𝐺))
86, 7eqtr4d 2778 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴))
91, 2grpoinvcl 30553 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑁‘(𝑁𝐴)) ∈ 𝑋)
103, 9syldan 591 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) ∈ 𝑋)
11 simpr 484 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → 𝐴𝑋)
1210, 11, 33jca 1127 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁‘(𝑁𝐴)) ∈ 𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
131grpolcan 30559 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝑁‘(𝑁𝐴)) ∈ 𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → (((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁𝐴)) = 𝐴))
1412, 13syldan 591 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁𝐴)) = 𝐴))
158, 14mpbid 232 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  ran crn 5690  cfv 6563  (class class class)co 7431  GrpOpcgr 30518  GIdcgi 30519  invcgn 30520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-grpo 30522  df-gid 30523  df-ginv 30524
This theorem is referenced by:  grpoinvf  30561  grpodivinv  30565  grpoinvdiv  30566  nvnegneg  30678
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