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Theorem elcntzsn 14894
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 14893 . . 3  |-  ( Y  e.  B  ->  ( Z `  { Y } )  =  {
x  e.  B  | 
( x  .+  Y
)  =  ( Y 
.+  x ) } )
54eleq2d 2425 . 2  |-  ( Y  e.  B  ->  ( X  e.  ( Z `  { Y } )  <-> 
X  e.  { x  e.  B  |  (
x  .+  Y )  =  ( Y  .+  x ) } ) )
6 oveq1 5949 . . . 4  |-  ( x  =  X  ->  (
x  .+  Y )  =  ( X  .+  Y ) )
7 oveq2 5950 . . . 4  |-  ( x  =  X  ->  ( Y  .+  x )  =  ( Y  .+  X
) )
86, 7eqeq12d 2372 . . 3  |-  ( x  =  X  ->  (
( x  .+  Y
)  =  ( Y 
.+  x )  <->  ( X  .+  Y )  =  ( Y  .+  X ) ) )
98elrab 2999 . 2  |-  ( X  e.  { x  e.  B  |  ( x 
.+  Y )  =  ( Y  .+  x
) }  <->  ( X  e.  B  /\  ( X  .+  Y )  =  ( Y  .+  X
) ) )
105, 9syl6bb 252 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 176    /\ wa 358    = wceq 1642    e. wcel 1710   {crab 2623   {csn 3716   ` cfv 5334  (class class class)co 5942   Basecbs 13239   +g cplusg 13299  Cntzccntz 14884
This theorem is referenced by:  gsumconst  15302  gsumpt  15315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-cntz 14886
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