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Theorem grpoinvop 20869
Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) 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
grpoinvop  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  B ) G ( N `  A ) ) )

Proof of Theorem grpoinvop
StepHypRef Expression
1 simp1 960 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
2 simp2 961 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
3 simp3 962 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
4 grpasscan1.1 . . . . . . 7  |-  X  =  ran  G
5 grpasscan1.2 . . . . . . 7  |-  N  =  ( inv `  G
)
64, 5grpoinvcl 20854 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( N `  B )  e.  X )
763adant2 979 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  e.  X )
84, 5grpoinvcl 20854 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
983adant3 980 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  e.  X )
104grpocl 20828 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( N `  B )  e.  X  /\  ( N `  A )  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  e.  X )
111, 7, 9, 10syl3anc 1187 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  e.  X )
124grpoass 20831 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  B ) G ( N `  A ) )  e.  X ) )  ->  ( ( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  ( A G ( B G ( ( N `
 B ) G ( N `  A
) ) ) ) )
131, 2, 3, 11, 12syl13anc 1189 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  ( A G ( B G ( ( N `  B
) G ( N `
 A ) ) ) ) )
14 eqid 2258 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
154, 14, 5grporinv 20857 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G ( N `  B ) )  =  (GId `  G )
)
16153adant2 979 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( N `  B ) )  =  (GId `  G )
)
1716oveq1d 5807 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( B G ( N `  B ) ) G ( N `
 A ) )  =  ( (GId `  G ) G ( N `  A ) ) )
184grpoass 20831 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  ( N `  B )  e.  X  /\  ( N `  A )  e.  X ) )  -> 
( ( B G ( N `  B
) ) G ( N `  A ) )  =  ( B G ( ( N `
 B ) G ( N `  A
) ) ) )
191, 3, 7, 9, 18syl13anc 1189 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( B G ( N `  B ) ) G ( N `
 A ) )  =  ( B G ( ( N `  B ) G ( N `  A ) ) ) )
204, 14grpolid 20847 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
218, 20syldan 458 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
22213adant3 980 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G ( N `  A ) )  =  ( N `  A
) )
2317, 19, 223eqtr3d 2298 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( ( N `
 B ) G ( N `  A
) ) )  =  ( N `  A
) )
2423oveq2d 5808 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( B G ( ( N `  B ) G ( N `  A ) ) ) )  =  ( A G ( N `  A ) ) )
254, 14, 5grporinv 20857 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
26253adant3 980 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
2713, 24, 263eqtrd 2294 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  (GId `  G
) )
284grpocl 20828 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
294, 14, 5grpoinvid1 20858 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A G B )  e.  X  /\  ( ( N `  B ) G ( N `  A ) )  e.  X )  ->  (
( N `  ( A G B ) )  =  ( ( N `
 B ) G ( N `  A
) )  <->  ( ( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  (GId
`  G ) ) )
301, 28, 11, 29syl3anc 1187 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  ( A G B ) )  =  ( ( N `
 B ) G ( N `  A
) )  <->  ( ( A G B ) G ( ( N `  B ) G ( N `  A ) ) )  =  (GId
`  G ) ) )
3127, 30mpbird 225 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  B ) G ( N `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   ran crn 4662   ` cfv 4673  (class class class)co 5792   GrpOpcgr 20814  GIdcgi 20815   invcgn 20816
This theorem is referenced by:  grpoinvdiv  20873  grpopnpcan2  20881  gxcom  20897  gxinv  20898  gxsuc  20900  gxdi  20924  abloinvop  24721  fprodneg  24746  invaddvec  24835
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-iota 6225  df-riota 6272  df-grpo 20819  df-gid 20820  df-ginv 20821
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