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Theorem oppgcntz 15162
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 2440 . . . . . . 7  |-  ( ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x )  <-> 
( y ( +g  `  G ) x )  =  ( x ( +g  `  G ) y ) )
2 eqid 2438 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
3 oppggic.o . . . . . . . . 9  |-  O  =  (oppg
`  G )
4 eqid 2438 . . . . . . . . 9  |-  ( +g  `  O )  =  ( +g  `  O )
52, 3, 4oppgplus 15147 . . . . . . . 8  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  G ) x )
62, 3, 4oppgplus 15147 . . . . . . . 8  |-  ( y ( +g  `  O
) x )  =  ( x ( +g  `  G ) y )
75, 6eqeq12i 2451 . . . . . . 7  |-  ( ( x ( +g  `  O
) y )  =  ( y ( +g  `  O ) x )  <-> 
( y ( +g  `  G ) x )  =  ( x ( +g  `  G ) y ) )
81, 7bitr4i 245 . . . . . 6  |-  ( ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x )  <-> 
( x ( +g  `  O ) y )  =  ( y ( +g  `  O ) x ) )
98ralbii 2731 . . . . 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 677 . . . 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 677 . . 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 2438 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
13 oppgcntz.z . . . . . 6  |-  Z  =  (Cntz `  G )
1412, 13cntzrcl 15128 . . . . 5  |-  ( x  e.  ( Z `  A )  ->  ( G  e.  _V  /\  A  C_  ( Base `  G
) ) )
1514simprd 451 . . . 4  |-  ( x  e.  ( Z `  A )  ->  A  C_  ( Base `  G
) )
1612, 2, 13elcntz 15123 . . . 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 625 . . 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 15149 . . . . . 6  |-  ( Base `  G )  =  (
Base `  O )
19 eqid 2438 . . . . . 6  |-  (Cntz `  O )  =  (Cntz `  O )
2018, 19cntzrcl 15128 . . . . 5  |-  ( x  e.  ( (Cntz `  O ) `  A
)  ->  ( O  e.  _V  /\  A  C_  ( Base `  G )
) )
2120simprd 451 . . . 4  |-  ( x  e.  ( (Cntz `  O ) `  A
)  ->  A  C_  ( Base `  G ) )
2218, 4, 19elcntz 15123 . . . 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 625 . . 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 270 . 2  |-  ( x  e.  ( Z `  A )  <->  x  e.  ( (Cntz `  O ) `  A ) )
2524eqriv 2435 1  |-  ( Z `
 A )  =  ( (Cntz `  O
) `  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531  Cntzccntz 15116  oppgcoppg 15143
This theorem is referenced by:  oppgcntr  15163  gsumzoppg  15541  gsumzinv  15542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-cntz 15118  df-oppg 15144
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