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Theorem elcntz 14814
Description: Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b  |-  B  =  ( Base `  M
)
cntzfval.p  |-  .+  =  ( +g  `  M )
cntzfval.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
elcntz  |-  ( S 
C_  B  ->  ( A  e.  ( Z `  S )  <->  ( A  e.  B  /\  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) ) )
Distinct variable groups:    y,  .+    y, A    y, M    y, S
Allowed substitution hints:    B( y)    Z( y)

Proof of Theorem elcntz
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, 3cntzval 14813 . . 3  |-  ( S 
C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
54eleq2d 2363 . 2  |-  ( S 
C_  B  ->  ( A  e.  ( Z `  S )  <->  A  e.  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } ) )
6 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x  .+  y )  =  ( A  .+  y ) )
7 oveq2 5882 . . . . 5  |-  ( x  =  A  ->  (
y  .+  x )  =  ( y  .+  A ) )
86, 7eqeq12d 2310 . . . 4  |-  ( x  =  A  ->  (
( x  .+  y
)  =  ( y 
.+  x )  <->  ( A  .+  y )  =  ( y  .+  A ) ) )
98ralbidv 2576 . . 3  |-  ( x  =  A  ->  ( A. y  e.  S  ( x  .+  y )  =  ( y  .+  x )  <->  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) )
109elrab 2936 . 2  |-  ( A  e.  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  <->  ( A  e.  B  /\  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) )
115, 10syl6bb 252 1  |-  ( S 
C_  B  ->  ( A  e.  ( Z `  S )  <->  ( A  e.  B  /\  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Cntzccntz 14807
This theorem is referenced by:  cntzel  14815  cntzi  14821  resscntz  14823  cntzsubm  14827  cntzmhm  14830  oppgcntz  14853  dprdfcntz  15266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-cntz 14809
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