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Theorem cntzel 15042
Description: Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b  |-  B  =  ( Base `  M
)
cntzfval.p  |-  .+  =  ( +g  `  M )
cntzfval.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
cntzel  |-  ( ( S  C_  B  /\  X  e.  B )  ->  ( X  e.  ( Z `  S )  <->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) )
Distinct variable groups:    y,  .+    y, M    y, S    y, X
Allowed substitution hints:    B( y)    Z( y)

Proof of Theorem cntzel
StepHypRef Expression
1 cntzfval.b . . 3  |-  B  =  ( Base `  M
)
2 cntzfval.p . . 3  |-  .+  =  ( +g  `  M )
3 cntzfval.z . . 3  |-  Z  =  (Cntz `  M )
41, 2, 3elcntz 15041 . 2  |-  ( S 
C_  B  ->  ( X  e.  ( Z `  S )  <->  ( X  e.  B  /\  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) ) )
54baibd 876 1  |-  ( ( S  C_  B  /\  X  e.  B )  ->  ( X  e.  ( Z `  S )  <->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642    C_ wss 3256   ` cfv 5387  (class class class)co 6013   Basecbs 13389   +g cplusg 13449  Cntzccntz 15034
This theorem is referenced by:  cntzsubg  15055  cntzcmn  15379  cntzsubr  15820  cntzsdrg  27172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-cntz 15036
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