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Theorem cntzrec 15125
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 2861 . . . 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 2438 . . . . 5  |-  ( ( x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x )  <-> 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) )
322ralbii 2724 . . . 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 2436 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
8 cntzrec.z . . 3  |-  Z  =  (Cntz `  M )
96, 7, 8sscntz 15118 . 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 15118 . . 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 1652   A.wral 2698    C_ wss 3313   ` cfv 5447  (class class class)co 6074   Basecbs 13462   +g cplusg 13522  Cntzccntz 15107
This theorem is referenced by:  cntzrecd  15303  lsmcntzr  15305  cntzspan  15453  dprdfadd  15571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-cntz 15109
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