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Theorem cntziinsn 16696
Description: Express any centralizer as an intersection of singleton 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
cntziinsn  |-  ( S 
C_  B  ->  ( Z `  S )  =  ( B  i^i  |^|_
x  e.  S  ( Z `  { x } ) ) )
Distinct variable groups:    x, B    x, M    x, S    x, Z

Proof of Theorem cntziinsn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cntzrec.b . . 3  |-  B  =  ( Base `  M
)
2 eqid 2402 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
3 cntzrec.z . . 3  |-  Z  =  (Cntz `  M )
41, 2, 3cntzval 16683 . 2  |-  ( S 
C_  B  ->  ( Z `  S )  =  { y  e.  B  |  A. x  e.  S  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) } )
5 ssel2 3437 . . . . . 6  |-  ( ( S  C_  B  /\  x  e.  S )  ->  x  e.  B )
61, 2, 3cntzsnval 16686 . . . . . 6  |-  ( x  e.  B  ->  ( Z `  { x } )  =  {
y  e.  B  | 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) } )
75, 6syl 17 . . . . 5  |-  ( ( S  C_  B  /\  x  e.  S )  ->  ( Z `  {
x } )  =  { y  e.  B  |  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) } )
87iineq2dv 4294 . . . 4  |-  ( S 
C_  B  ->  |^|_ x  e.  S  ( Z `  { x } )  =  |^|_ x  e.  S  { y  e.  B  |  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) } )
98ineq2d 3641 . . 3  |-  ( S 
C_  B  ->  ( B  i^i  |^|_ x  e.  S  ( Z `  { x } ) )  =  ( B  i^i  |^|_ x  e.  S  { y  e.  B  |  ( y ( +g  `  M
) x )  =  ( x ( +g  `  M ) y ) } ) )
10 riinrab 4347 . . 3  |-  ( B  i^i  |^|_ x  e.  S  { y  e.  B  |  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) } )  =  { y  e.  B  |  A. x  e.  S  (
y ( +g  `  M
) x )  =  ( x ( +g  `  M ) y ) }
119, 10syl6eq 2459 . 2  |-  ( S 
C_  B  ->  ( B  i^i  |^|_ x  e.  S  ( Z `  { x } ) )  =  { y  e.  B  |  A. x  e.  S  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) } )
124, 11eqtr4d 2446 1  |-  ( S 
C_  B  ->  ( Z `  S )  =  ( B  i^i  |^|_
x  e.  S  ( Z `  { x } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   {crab 2758    i^i cin 3413    C_ wss 3414   {csn 3972   |^|_ciin 4272   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909  Cntzccntz 16677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-cntz 16679
This theorem is referenced by: (None)
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