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Theorem grpo2inv 28317
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 28310 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
4 eqid 2801 . . . . 5 (GId‘𝐺) = (GId‘𝐺)
51, 4, 2grporinv 28313 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = (GId‘𝐺))
63, 5syldan 594 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = (GId‘𝐺))
71, 4, 2grpolinv 28312 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = (GId‘𝐺))
86, 7eqtr4d 2839 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴))
91, 2grpoinvcl 28310 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑁𝐴) ∈ 𝑋) → (𝑁‘(𝑁𝐴)) ∈ 𝑋)
103, 9syldan 594 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) ∈ 𝑋)
11 simpr 488 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → 𝐴𝑋)
1210, 11, 33jca 1125 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁‘(𝑁𝐴)) ∈ 𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋))
131grpolcan 28316 . . 3 ((𝐺 ∈ GrpOp ∧ ((𝑁‘(𝑁𝐴)) ∈ 𝑋𝐴𝑋 ∧ (𝑁𝐴) ∈ 𝑋)) → (((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁𝐴)) = 𝐴))
1412, 13syldan 594 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺(𝑁‘(𝑁𝐴))) = ((𝑁𝐴)𝐺𝐴) ↔ (𝑁‘(𝑁𝐴)) = 𝐴))
158, 14mpbid 235 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  ran crn 5524  cfv 6328  (class class class)co 7139  GrpOpcgr 28275  GIdcgi 28276  invcgn 28277
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-grpo 28279  df-gid 28280  df-ginv 28281
This theorem is referenced by:  grpoinvf  28318  grpodivinv  28322  grpoinvdiv  28323  nvnegneg  28435
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