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Theorem sscntz 14818
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 14813 . . . 4  |-  ( T 
C_  B  ->  ( Z `  T )  =  { x  e.  B  |  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) } )
54sseq2d 3219 . . 3  |-  ( T 
C_  B  ->  ( S  C_  ( Z `  T )  <->  S  C_  { x  e.  B  |  A. y  e.  T  (
x  .+  y )  =  ( y  .+  x ) } ) )
6 ssrab 3264 . . 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 252 . 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 490 . . 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 192 . 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 681 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 176    /\ wa 358    = wceq 1632   A.wral 2556   {crab 2560    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Cntzccntz 14807
This theorem is referenced by:  cntz2ss  14824  cntzrec  14825  submcmn2  15151
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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