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Theorem cntzi 15130
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 2438 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
2 cntzi.z . . . . . . 7  |-  Z  =  (Cntz `  M )
31, 2cntzrcl 15128 . . . . . 6  |-  ( X  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  ( Base `  M
) ) )
43simprd 451 . . . . 5  |-  ( X  e.  ( Z `  S )  ->  S  C_  ( Base `  M
) )
5 cntzi.p . . . . . 6  |-  .+  =  ( +g  `  M )
61, 5, 2elcntz 15123 . . . . 5  |-  ( S 
C_  ( Base `  M
)  ->  ( X  e.  ( Z `  S
)  <->  ( X  e.  ( Base `  M
)  /\  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) ) )
74, 6syl 16 . . . 4  |-  ( X  e.  ( Z `  S )  ->  ( X  e.  ( Z `  S )  <->  ( X  e.  ( Base `  M
)  /\  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) ) )
87simplbda 609 . . 3  |-  ( ( X  e.  ( Z `
 S )  /\  X  e.  ( Z `  S ) )  ->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) )
98anidms 628 . 2  |-  ( X  e.  ( Z `  S )  ->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) )
10 oveq2 6091 . . . 4  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
11 oveq1 6090 . . . 4  |-  ( y  =  Y  ->  (
y  .+  X )  =  ( Y  .+  X ) )
1210, 11eqeq12d 2452 . . 3  |-  ( y  =  Y  ->  (
( X  .+  y
)  =  ( y 
.+  X )  <->  ( X  .+  Y )  =  ( Y  .+  X ) ) )
1312rspccva 3053 . 2  |-  ( ( A. y  e.  S  ( X  .+  y )  =  ( y  .+  X )  /\  Y  e.  S )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
149, 13sylan 459 1  |-  ( ( X  e.  ( Z `
 S )  /\  Y  e.  S )  ->  ( X  .+  Y
)  =  ( Y 
.+  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531  Cntzccntz 15116
This theorem is referenced by:  cntri  15131  cntz2ss  15133  cntzsubm  15136  cntzsubg  15137  cntzmhm  15139  cntrsubgnsg  15141  lsmsubm  15289  lsmsubg  15290  lsmcom2  15291  subgdisj1  15325  subgdisj2  15326  pj1id  15333  pj1ghm  15337  gsumval3eu  15515  gsumval3  15516  gsumzaddlem  15528  gsumzoppg  15541  dprdfcntz  15575  cntzsubr  15902  cntzsdrg  27489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-cntz 15118
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