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Theorem cntziinsn 14810
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 2283 . . 3  |-  ( +g  `  M )  =  ( +g  `  M )
3 cntzrec.z . . 3  |-  Z  =  (Cntz `  M )
41, 2, 3cntzval 14797 . 2  |-  ( S 
C_  B  ->  ( Z `  S )  =  { y  e.  B  |  A. x  e.  S  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) } )
5 ssel2 3175 . . . . . 6  |-  ( ( S  C_  B  /\  x  e.  S )  ->  x  e.  B )
61, 2, 3cntzsnval 14800 . . . . . 6  |-  ( x  e.  B  ->  ( Z `  { x } )  =  {
y  e.  B  | 
( y ( +g  `  M ) x )  =  ( x ( +g  `  M ) y ) } )
75, 6syl 15 . . . . 5  |-  ( ( S  C_  B  /\  x  e.  S )  ->  ( Z `  {
x } )  =  { y  e.  B  |  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) } )
87iineq2dv 3927 . . . 4  |-  ( S 
C_  B  ->  |^|_ x  e.  S  ( Z `  { x } )  =  |^|_ x  e.  S  { y  e.  B  |  ( y ( +g  `  M ) x )  =  ( x ( +g  `  M
) y ) } )
98ineq2d 3370 . . 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 3977 . . 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 2331 . 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 2318 1  |-  ( S 
C_  B  ->  ( Z `  S )  =  ( B  i^i  |^|_
x  e.  S  ( Z `  { x } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    i^i cin 3151    C_ wss 3152   {csn 3640   |^|_ciin 3906   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-cntz 14793
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