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Theorem oppgcntz 17027
Description: A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
oppggic.o  |-  O  =  (oppg
`  G )
oppgcntz.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
oppgcntz  |-  ( Z `
 A )  =  ( (Cntz `  O
) `  A )

Proof of Theorem oppgcntz
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2460 . . . . . . 7  |-  ( ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x )  <-> 
( y ( +g  `  G ) x )  =  ( x ( +g  `  G ) y ) )
2 eqid 2453 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
3 oppggic.o . . . . . . . . 9  |-  O  =  (oppg
`  G )
4 eqid 2453 . . . . . . . . 9  |-  ( +g  `  O )  =  ( +g  `  O )
52, 3, 4oppgplus 17012 . . . . . . . 8  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  G ) x )
62, 3, 4oppgplus 17012 . . . . . . . 8  |-  ( y ( +g  `  O
) x )  =  ( x ( +g  `  G ) y )
75, 6eqeq12i 2467 . . . . . . 7  |-  ( ( x ( +g  `  O
) y )  =  ( y ( +g  `  O ) x )  <-> 
( y ( +g  `  G ) x )  =  ( x ( +g  `  G ) y ) )
81, 7bitr4i 256 . . . . . 6  |-  ( ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x )  <-> 
( x ( +g  `  O ) y )  =  ( y ( +g  `  O ) x ) )
98ralbii 2821 . . . . 5  |-  ( A. y  e.  A  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x )  <->  A. y  e.  A  ( x ( +g  `  O ) y )  =  ( y ( +g  `  O ) x ) )
109anbi2i 701 . . . 4  |-  ( ( x  e.  ( Base `  G )  /\  A. y  e.  A  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )  <->  ( x  e.  ( Base `  G
)  /\  A. y  e.  A  ( x
( +g  `  O ) y )  =  ( y ( +g  `  O
) x ) ) )
1110anbi2i 701 . . 3  |-  ( ( A  C_  ( Base `  G )  /\  (
x  e.  ( Base `  G )  /\  A. y  e.  A  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) ) )  <->  ( A  C_  ( Base `  G
)  /\  ( x  e.  ( Base `  G
)  /\  A. y  e.  A  ( x
( +g  `  O ) y )  =  ( y ( +g  `  O
) x ) ) ) )
12 eqid 2453 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
13 oppgcntz.z . . . . . 6  |-  Z  =  (Cntz `  G )
1412, 13cntzrcl 16993 . . . . 5  |-  ( x  e.  ( Z `  A )  ->  ( G  e.  _V  /\  A  C_  ( Base `  G
) ) )
1514simprd 465 . . . 4  |-  ( x  e.  ( Z `  A )  ->  A  C_  ( Base `  G
) )
1612, 2, 13elcntz 16988 . . . 4  |-  ( A 
C_  ( Base `  G
)  ->  ( x  e.  ( Z `  A
)  <->  ( x  e.  ( Base `  G
)  /\  A. y  e.  A  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) ) ) )
1715, 16biadan2 648 . . 3  |-  ( x  e.  ( Z `  A )  <->  ( A  C_  ( Base `  G
)  /\  ( x  e.  ( Base `  G
)  /\  A. y  e.  A  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) ) ) )
183, 12oppgbas 17014 . . . . . 6  |-  ( Base `  G )  =  (
Base `  O )
19 eqid 2453 . . . . . 6  |-  (Cntz `  O )  =  (Cntz `  O )
2018, 19cntzrcl 16993 . . . . 5  |-  ( x  e.  ( (Cntz `  O ) `  A
)  ->  ( O  e.  _V  /\  A  C_  ( Base `  G )
) )
2120simprd 465 . . . 4  |-  ( x  e.  ( (Cntz `  O ) `  A
)  ->  A  C_  ( Base `  G ) )
2218, 4, 19elcntz 16988 . . . 4  |-  ( A 
C_  ( Base `  G
)  ->  ( x  e.  ( (Cntz `  O
) `  A )  <->  ( x  e.  ( Base `  G )  /\  A. y  e.  A  (
x ( +g  `  O
) y )  =  ( y ( +g  `  O ) x ) ) ) )
2321, 22biadan2 648 . . 3  |-  ( x  e.  ( (Cntz `  O ) `  A
)  <->  ( A  C_  ( Base `  G )  /\  ( x  e.  (
Base `  G )  /\  A. y  e.  A  ( x ( +g  `  O ) y )  =  ( y ( +g  `  O ) x ) ) ) )
2411, 17, 233bitr4i 281 . 2  |-  ( x  e.  ( Z `  A )  <->  x  e.  ( (Cntz `  O ) `  A ) )
2524eqriv 2450 1  |-  ( Z `
 A )  =  ( (Cntz `  O
) `  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   _Vcvv 3047    C_ wss 3406   ` cfv 5585  (class class class)co 6295   Basecbs 15133   +g cplusg 15202  Cntzccntz 16981  oppgcoppg 17008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-plusg 15215  df-cntz 16983  df-oppg 17009
This theorem is referenced by:  oppgcntr  17028  gsumzoppg  17589  gsumzinv  17590
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