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Theorem sscntz 15054
Description: A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b  |-  B  =  ( Base `  M
)
cntzfval.p  |-  .+  =  ( +g  `  M )
cntzfval.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
sscntz  |-  ( ( S  C_  B  /\  T  C_  B )  -> 
( S  C_  ( Z `  T )  <->  A. x  e.  S  A. y  e.  T  (
x  .+  y )  =  ( y  .+  x ) ) )
Distinct variable groups:    x, y,  .+    x, B    x, M, y    x, T, y    x, S, y
Allowed substitution hints:    B( y)    Z( x, y)

Proof of Theorem sscntz
StepHypRef Expression
1 cntzfval.b . . . . 5  |-  B  =  ( Base `  M
)
2 cntzfval.p . . . . 5  |-  .+  =  ( +g  `  M )
3 cntzfval.z . . . . 5  |-  Z  =  (Cntz `  M )
41, 2, 3cntzval 15049 . . . 4  |-  ( T 
C_  B  ->  ( Z `  T )  =  { x  e.  B  |  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) } )
54sseq2d 3321 . . 3  |-  ( T 
C_  B  ->  ( S  C_  ( Z `  T )  <->  S  C_  { x  e.  B  |  A. y  e.  T  (
x  .+  y )  =  ( y  .+  x ) } ) )
6 ssrab 3366 . . 3  |-  ( S 
C_  { x  e.  B  |  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) }  <->  ( S  C_  B  /\  A. x  e.  S  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) ) )
75, 6syl6bb 253 . 2  |-  ( T 
C_  B  ->  ( S  C_  ( Z `  T )  <->  ( S  C_  B  /\  A. x  e.  S  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) ) ) )
8 ibar 491 . . 3  |-  ( S 
C_  B  ->  ( A. x  e.  S  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x )  <->  ( S  C_  B  /\  A. x  e.  S  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) ) ) )
98bicomd 193 . 2  |-  ( S 
C_  B  ->  (
( S  C_  B  /\  A. x  e.  S  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) )  <->  A. x  e.  S  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) ) )
107, 9sylan9bbr 682 1  |-  ( ( S  C_  B  /\  T  C_  B )  -> 
( S  C_  ( Z `  T )  <->  A. x  e.  S  A. y  e.  T  (
x  .+  y )  =  ( y  .+  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   A.wral 2651   {crab 2655    C_ wss 3265   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458  Cntzccntz 15043
This theorem is referenced by:  cntz2ss  15060  cntzrec  15061  submcmn2  15387
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-cntz 15045
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