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Theorem grpo2inv 27937
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 27930 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
4 eqid 2825 . . . . 5 (GId‘𝐺) = (GId‘𝐺)
51, 4, 2grporinv 27933 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = (GId‘𝐺))
63, 5syldan 585 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = (GId‘𝐺))
71, 4, 2grpolinv 27932 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = (GId‘𝐺))
86, 7eqtr4d 2864 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴))
91, 2grpoinvcl 27930 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑁‘(𝑁𝐴)) ∈ 𝑋)
103, 9syldan 585 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) ∈ 𝑋)
11 simpr 479 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → 𝐴𝑋)
1210, 11, 33jca 1162 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁‘(𝑁𝐴)) ∈ 𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
131grpolcan 27936 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝑁‘(𝑁𝐴)) ∈ 𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → (((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁𝐴)) = 𝐴))
1412, 13syldan 585 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁𝐴)) = 𝐴))
158, 14mpbid 224 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  ran crn 5347  cfv 6127  (class class class)co 6910  GrpOpcgr 27895  GIdcgi 27896  invcgn 27897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-grpo 27899  df-gid 27900  df-ginv 27901
This theorem is referenced by:  grpoinvf  27938  grpodivinv  27942  grpoinvdiv  27943  nvnegneg  28055
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