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Theorem cntziinsn 15975
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 2454 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
3 cntzrec.z . . 3  |-  Z  =  (Cntz `  M )
41, 2, 3cntzval 15962 . 2  |-  ( S 
C_  B  ->  ( Z `  S )  =  { y  e.  B  |  A. x  e.  S  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) } )
5 ssel2 3462 . . . . . 6  |-  ( ( S  C_  B  /\  x  e.  S )  ->  x  e.  B )
61, 2, 3cntzsnval 15965 . . . . . 6  |-  ( x  e.  B  ->  ( Z `  { x } )  =  {
y  e.  B  | 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) } )
75, 6syl 16 . . . . 5  |-  ( ( S  C_  B  /\  x  e.  S )  ->  ( Z `  {
x } )  =  { y  e.  B  |  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) } )
87iineq2dv 4304 . . . 4  |-  ( S 
C_  B  ->  |^|_ x  e.  S  ( Z `  { x } )  =  |^|_ x  e.  S  { y  e.  B  |  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) } )
98ineq2d 3663 . . 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 4357 . . 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 2511 . 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 2498 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 369    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803    i^i cin 3438    C_ wss 3439   {csn 3988   |^|_ciin 4283   ` cfv 5529  (class class class)co 6203   Basecbs 14296   +g cplusg 14361  Cntzccntz 15956
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-cntz 15958
This theorem is referenced by: (None)
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