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Theorem sscntz 15948
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 15943 . . . 4  |-  ( T 
C_  B  ->  ( Z `  T )  =  { x  e.  B  |  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) } )
54sseq2d 3484 . . 3  |-  ( T 
C_  B  ->  ( S  C_  ( Z `  T )  <->  S  C_  { x  e.  B  |  A. y  e.  T  (
x  .+  y )  =  ( y  .+  x ) } ) )
6 ssrab 3530 . . 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 261 . 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 504 . . 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 201 . 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 700 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   A.wral 2795   {crab 2799    C_ wss 3428   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342  Cntzccntz 15937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-cntz 15939
This theorem is referenced by:  cntz2ss  15954  cntzrec  15955  submcmn2  16429  mplcoe5lem  17656
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