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Theorem cntzi 14821
Description: Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
cntzi.p  |-  .+  =  ( +g  `  M )
cntzi.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
cntzi  |-  ( ( X  e.  ( Z `
 S )  /\  Y  e.  S )  ->  ( X  .+  Y
)  =  ( Y 
.+  X ) )

Proof of Theorem cntzi
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
2 cntzi.z . . . . . . 7  |-  Z  =  (Cntz `  M )
31, 2cntzrcl 14819 . . . . . 6  |-  ( X  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  ( Base `  M
) ) )
43simprd 449 . . . . 5  |-  ( X  e.  ( Z `  S )  ->  S  C_  ( Base `  M
) )
5 cntzi.p . . . . . 6  |-  .+  =  ( +g  `  M )
61, 5, 2elcntz 14814 . . . . 5  |-  ( S 
C_  ( Base `  M
)  ->  ( X  e.  ( Z `  S
)  <->  ( X  e.  ( Base `  M
)  /\  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) ) )
74, 6syl 15 . . . 4  |-  ( X  e.  ( Z `  S )  ->  ( X  e.  ( Z `  S )  <->  ( X  e.  ( Base `  M
)  /\  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) ) )
87simplbda 607 . . 3  |-  ( ( X  e.  ( Z `
 S )  /\  X  e.  ( Z `  S ) )  ->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) )
98anidms 626 . 2  |-  ( X  e.  ( Z `  S )  ->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) )
10 oveq2 5882 . . . 4  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
11 oveq1 5881 . . . 4  |-  ( y  =  Y  ->  (
y  .+  X )  =  ( Y  .+  X ) )
1210, 11eqeq12d 2310 . . 3  |-  ( y  =  Y  ->  (
( X  .+  y
)  =  ( y 
.+  X )  <->  ( X  .+  Y )  =  ( Y  .+  X ) ) )
1312rspccva 2896 . 2  |-  ( ( A. y  e.  S  ( X  .+  y )  =  ( y  .+  X )  /\  Y  e.  S )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
149, 13sylan 457 1  |-  ( ( X  e.  ( Z `
 S )  /\  Y  e.  S )  ->  ( X  .+  Y
)  =  ( Y 
.+  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Cntzccntz 14807
This theorem is referenced by:  cntri  14822  cntz2ss  14824  cntzsubm  14827  cntzsubg  14828  cntzmhm  14830  cntrsubgnsg  14832  lsmsubm  14980  lsmsubg  14981  lsmcom2  14982  subgdisj1  15016  subgdisj2  15017  pj1id  15024  pj1ghm  15028  gsumval3eu  15206  gsumval3  15207  gsumzaddlem  15219  gsumzoppg  15232  dprdfcntz  15266  cntzsubr  15593  cntzsdrg  27613
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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