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Theorem cntzrec 15052
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 2804 . . . 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 2382 . . . . 5  |-  ( ( x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x )  <-> 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
322ralbii 2668 . . . 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 241 . . 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 2380 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
8 cntzrec.z . . 3  |-  Z  =  (Cntz `  M )
96, 7, 8sscntz 15045 . 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 15045 . . 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 440 . 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 277 1  |-  ( ( S  C_  B  /\  T  C_  B )  -> 
( S  C_  ( Z `  T )  <->  T 
C_  ( Z `  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   A.wral 2642    C_ wss 3256   ` cfv 5387  (class class class)co 6013   Basecbs 13389   +g cplusg 13449  Cntzccntz 15034
This theorem is referenced by:  cntzrecd  15230  lsmcntzr  15232  cntzspan  15380  dprdfadd  15498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-cntz 15036
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