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Theorem oppgcntz 16967
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 2429 . . . . . . 7  |-  ( ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x )  <-> 
( y ( +g  `  G ) x )  =  ( x ( +g  `  G ) y ) )
2 eqid 2420 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
3 oppggic.o . . . . . . . . 9  |-  O  =  (oppg
`  G )
4 eqid 2420 . . . . . . . . 9  |-  ( +g  `  O )  =  ( +g  `  O )
52, 3, 4oppgplus 16952 . . . . . . . 8  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  G ) x )
62, 3, 4oppgplus 16952 . . . . . . . 8  |-  ( y ( +g  `  O
) x )  =  ( x ( +g  `  G ) y )
75, 6eqeq12i 2440 . . . . . . 7  |-  ( ( x ( +g  `  O
) y )  =  ( y ( +g  `  O ) x )  <-> 
( y ( +g  `  G ) x )  =  ( x ( +g  `  G ) y ) )
81, 7bitr4i 255 . . . . . 6  |-  ( ( x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x )  <-> 
( x ( +g  `  O ) y )  =  ( y ( +g  `  O ) x ) )
98ralbii 2854 . . . . 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 698 . . . 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 698 . . 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 2420 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
13 oppgcntz.z . . . . . 6  |-  Z  =  (Cntz `  G )
1412, 13cntzrcl 16933 . . . . 5  |-  ( x  e.  ( Z `  A )  ->  ( G  e.  _V  /\  A  C_  ( Base `  G
) ) )
1514simprd 464 . . . 4  |-  ( x  e.  ( Z `  A )  ->  A  C_  ( Base `  G
) )
1612, 2, 13elcntz 16928 . . . 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 646 . . 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 16954 . . . . . 6  |-  ( Base `  G )  =  (
Base `  O )
19 eqid 2420 . . . . . 6  |-  (Cntz `  O )  =  (Cntz `  O )
2018, 19cntzrcl 16933 . . . . 5  |-  ( x  e.  ( (Cntz `  O ) `  A
)  ->  ( O  e.  _V  /\  A  C_  ( Base `  G )
) )
2120simprd 464 . . . 4  |-  ( x  e.  ( (Cntz `  O ) `  A
)  ->  A  C_  ( Base `  G ) )
2218, 4, 19elcntz 16928 . . . 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 646 . . 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 280 . 2  |-  ( x  e.  ( Z `  A )  <->  x  e.  ( (Cntz `  O ) `  A ) )
2524eqriv 2416 1  |-  ( Z `
 A )  =  ( (Cntz `  O
) `  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078    C_ wss 3433   ` cfv 5592  (class class class)co 6296   Basecbs 15081   +g cplusg 15150  Cntzccntz 16921  oppgcoppg 16948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-plusg 15163  df-cntz 16923  df-oppg 16949
This theorem is referenced by:  oppgcntr  16968  gsumzoppg  17518  gsumzinv  17519
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