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Theorem elcntzsn 15083
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 15082 . . 3  |-  ( Y  e.  B  ->  ( Z `  { Y } )  =  {
x  e.  B  | 
( x  .+  Y
)  =  ( Y 
.+  x ) } )
54eleq2d 2475 . 2  |-  ( Y  e.  B  ->  ( X  e.  ( Z `  { Y } )  <-> 
X  e.  { x  e.  B  |  (
x  .+  Y )  =  ( Y  .+  x ) } ) )
6 oveq1 6051 . . . 4  |-  ( x  =  X  ->  (
x  .+  Y )  =  ( X  .+  Y ) )
7 oveq2 6052 . . . 4  |-  ( x  =  X  ->  ( Y  .+  x )  =  ( Y  .+  X
) )
86, 7eqeq12d 2422 . . 3  |-  ( x  =  X  ->  (
( x  .+  Y
)  =  ( Y 
.+  x )  <->  ( X  .+  Y )  =  ( Y  .+  X ) ) )
98elrab 3056 . 2  |-  ( X  e.  { x  e.  B  |  ( x 
.+  Y )  =  ( Y  .+  x
) }  <->  ( X  e.  B  /\  ( X  .+  Y )  =  ( Y  .+  X
) ) )
105, 9syl6bb 253 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2674   {csn 3778   ` cfv 5417  (class class class)co 6044   Basecbs 13428   +g cplusg 13488  Cntzccntz 15073
This theorem is referenced by:  gsumconst  15491  gsumpt  15504
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-cntz 15075
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