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Theorem cntzrec 14825
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 2713 . . . 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 2298 . . . . 5  |-  ( ( x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x )  <-> 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
322ralbii 2582 . . . 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 240 . . 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 10 . 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 2296 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
8 cntzrec.z . . 3  |-  Z  =  (Cntz `  M )
96, 7, 8sscntz 14818 . 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 14818 . . 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 439 . 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 276 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 176    /\ wa 358    = wceq 1632   A.wral 2556    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Cntzccntz 14807
This theorem is referenced by:  cntzrecd  15003  lsmcntzr  15005  cntzspan  15153  dprdfadd  15271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-cntz 14809
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