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Theorem cntzrec 16166
Description: Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cntzrec.b  |-  B  =  ( Base `  M
)
cntzrec.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
cntzrec  |-  ( ( S  C_  B  /\  T  C_  B )  -> 
( S  C_  ( Z `  T )  <->  T 
C_  ( Z `  S ) ) )

Proof of Theorem cntzrec
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ralcom 3022 . . . 4  |-  ( A. x  e.  S  A. y  e.  T  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x )  <->  A. y  e.  T  A. x  e.  S  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) )
2 eqcom 2476 . . . . 5  |-  ( ( x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x )  <-> 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
322ralbii 2896 . . . 4  |-  ( A. y  e.  T  A. x  e.  S  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x )  <->  A. y  e.  T  A. x  e.  S  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
41, 3bitri 249 . . 3  |-  ( A. x  e.  S  A. y  e.  T  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x )  <->  A. y  e.  T  A. x  e.  S  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
54a1i 11 . 2  |-  ( ( S  C_  B  /\  T  C_  B )  -> 
( A. x  e.  S  A. y  e.  T  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M
) x )  <->  A. y  e.  T  A. x  e.  S  ( y
( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) ) )
6 cntzrec.b . . 3  |-  B  =  ( Base `  M
)
7 eqid 2467 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
8 cntzrec.z . . 3  |-  Z  =  (Cntz `  M )
96, 7, 8sscntz 16159 . 2  |-  ( ( S  C_  B  /\  T  C_  B )  -> 
( S  C_  ( Z `  T )  <->  A. x  e.  S  A. y  e.  T  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) ) )
106, 7, 8sscntz 16159 . . 3  |-  ( ( T  C_  B  /\  S  C_  B )  -> 
( T  C_  ( Z `  S )  <->  A. y  e.  T  A. x  e.  S  (
y ( +g  `  M
) x )  =  ( x ( +g  `  M ) y ) ) )
1110ancoms 453 . 2  |-  ( ( S  C_  B  /\  T  C_  B )  -> 
( T  C_  ( Z `  S )  <->  A. y  e.  T  A. x  e.  S  (
y ( +g  `  M
) x )  =  ( x ( +g  `  M ) y ) ) )
125, 9, 113bitr4d 285 1  |-  ( ( S  C_  B  /\  T  C_  B )  -> 
( S  C_  ( Z `  T )  <->  T 
C_  ( Z `  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   A.wral 2814    C_ wss 3476   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551  Cntzccntz 16148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-cntz 16150
This theorem is referenced by:  cntzrecd  16492  lsmcntzr  16494  cntzspan  16643  dprdfadd  16850  dprdfaddOLD  16857
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