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Theorem cntzel 16488
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 16487 . 2  |-  ( S 
C_  B  ->  ( X  e.  ( Z `  S )  <->  ( X  e.  B  /\  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) ) )
54baibd 909 1  |-  ( ( S  C_  B  /\  X  e.  B )  ->  ( X  e.  ( Z `  S )  <->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712  Cntzccntz 16480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-cntz 16482
This theorem is referenced by:  cntzsubg  16501  cntzcmn  16975  cntzsubr  17588  cntzsdrg  31355
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