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Theorem cntzrec 16570
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 3015 . . . 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 2463 . . . . 5  |-  ( ( x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x )  <-> 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
322ralbii 2886 . . . 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 2454 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
8 cntzrec.z . . 3  |-  Z  =  (Cntz `  M )
96, 7, 8sscntz 16563 . 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 16563 . . 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 451 . 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 367    = wceq 1398   A.wral 2804    C_ wss 3461   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784  Cntzccntz 16552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-cntz 16554
This theorem is referenced by:  cntzrecd  16895  lsmcntzr  16897  cntzspan  17049  dprdfadd  17255  dprdfaddOLD  17262
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