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Theorem elcntzsn 15155
Description: Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
cntzfval.b  |-  B  =  ( Base `  M
)
cntzfval.p  |-  .+  =  ( +g  `  M )
cntzfval.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
elcntzsn  |-  ( Y  e.  B  ->  ( X  e.  ( Z `  { Y } )  <-> 
( X  e.  B  /\  ( X  .+  Y
)  =  ( Y 
.+  X ) ) ) )

Proof of Theorem elcntzsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cntzfval.b . . . 4  |-  B  =  ( Base `  M
)
2 cntzfval.p . . . 4  |-  .+  =  ( +g  `  M )
3 cntzfval.z . . . 4  |-  Z  =  (Cntz `  M )
41, 2, 3cntzsnval 15154 . . 3  |-  ( Y  e.  B  ->  ( Z `  { Y } )  =  {
x  e.  B  | 
( x  .+  Y
)  =  ( Y 
.+  x ) } )
54eleq2d 2509 . 2  |-  ( Y  e.  B  ->  ( X  e.  ( Z `  { Y } )  <-> 
X  e.  { x  e.  B  |  (
x  .+  Y )  =  ( Y  .+  x ) } ) )
6 oveq1 6117 . . . 4  |-  ( x  =  X  ->  (
x  .+  Y )  =  ( X  .+  Y ) )
7 oveq2 6118 . . . 4  |-  ( x  =  X  ->  ( Y  .+  x )  =  ( Y  .+  X
) )
86, 7eqeq12d 2456 . . 3  |-  ( x  =  X  ->  (
( x  .+  Y
)  =  ( Y 
.+  x )  <->  ( X  .+  Y )  =  ( Y  .+  X ) ) )
98elrab 3098 . 2  |-  ( X  e.  { x  e.  B  |  ( x 
.+  Y )  =  ( Y  .+  x
) }  <->  ( X  e.  B  /\  ( X  .+  Y )  =  ( Y  .+  X
) ) )
105, 9syl6bb 254 1  |-  ( Y  e.  B  ->  ( X  e.  ( Z `  { Y } )  <-> 
( X  e.  B  /\  ( X  .+  Y
)  =  ( Y 
.+  X ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   {crab 2715   {csn 3838   ` cfv 5483  (class class class)co 6110   Basecbs 13500   +g cplusg 13560  Cntzccntz 15145
This theorem is referenced by:  gsumconst  15563  gsumpt  15576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-cntz 15147
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