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Theorem elcntz 14798
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 14797 . . 3  |-  ( S 
C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
54eleq2d 2350 . 2  |-  ( S 
C_  B  ->  ( A  e.  ( Z `  S )  <->  A  e.  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } ) )
6 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x  .+  y )  =  ( A  .+  y ) )
7 oveq2 5866 . . . . 5  |-  ( x  =  A  ->  (
y  .+  x )  =  ( y  .+  A ) )
86, 7eqeq12d 2297 . . . 4  |-  ( x  =  A  ->  (
( x  .+  y
)  =  ( y 
.+  x )  <->  ( A  .+  y )  =  ( y  .+  A ) ) )
98ralbidv 2563 . . 3  |-  ( x  =  A  ->  ( A. y  e.  S  ( x  .+  y )  =  ( y  .+  x )  <->  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) )
109elrab 2923 . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791
This theorem is referenced by:  cntzel  14799  cntzi  14805  resscntz  14807  cntzsubm  14811  cntzmhm  14814  oppgcntz  14837  dprdfcntz  15250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-cntz 14793
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