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Theorem grpo2inv 30293
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 = (invβ€˜πΊ)
Assertion
Ref Expression
grpo2inv ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜(π‘β€˜π΄)) = 𝐴)

Proof of Theorem grpo2inv
StepHypRef Expression
1 grpasscan1.1 . . . . 5 𝑋 = ran 𝐺
2 grpasscan1.2 . . . . 5 𝑁 = (invβ€˜πΊ)
31, 2grpoinvcl 30286 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ 𝑋)
4 eqid 2726 . . . . 5 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
51, 4, 2grporinv 30289 . . . 4 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺(π‘β€˜(π‘β€˜π΄))) = (GIdβ€˜πΊ))
63, 5syldan 590 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺(π‘β€˜(π‘β€˜π΄))) = (GIdβ€˜πΊ))
71, 4, 2grpolinv 30288 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺𝐴) = (GIdβ€˜πΊ))
86, 7eqtr4d 2769 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄)𝐺(π‘β€˜(π‘β€˜π΄))) = ((π‘β€˜π΄)𝐺𝐴))
91, 2grpoinvcl 30286 . . . . 5 ((𝐺 ∈ GrpOp ∧ (π‘β€˜π΄) ∈ 𝑋) β†’ (π‘β€˜(π‘β€˜π΄)) ∈ 𝑋)
103, 9syldan 590 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜(π‘β€˜π΄)) ∈ 𝑋)
11 simpr 484 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
1210, 11, 33jca 1125 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜(π‘β€˜π΄)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋))
131grpolcan 30292 . . 3 ((𝐺 ∈ GrpOp ∧ ((π‘β€˜(π‘β€˜π΄)) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (π‘β€˜π΄) ∈ 𝑋)) β†’ (((π‘β€˜π΄)𝐺(π‘β€˜(π‘β€˜π΄))) = ((π‘β€˜π΄)𝐺𝐴) ↔ (π‘β€˜(π‘β€˜π΄)) = 𝐴))
1412, 13syldan 590 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (((π‘β€˜π΄)𝐺(π‘β€˜(π‘β€˜π΄))) = ((π‘β€˜π΄)𝐺𝐴) ↔ (π‘β€˜(π‘β€˜π΄)) = 𝐴))
158, 14mpbid 231 1 ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜(π‘β€˜π΄)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5670  β€˜cfv 6537  (class class class)co 7405  GrpOpcgr 30251  GIdcgi 30252  invcgn 30253
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-grpo 30255  df-gid 30256  df-ginv 30257
This theorem is referenced by:  grpoinvf  30294  grpodivinv  30298  grpoinvdiv  30299  nvnegneg  30411
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