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Theorem cntzi 14805
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 2283 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
2 cntzi.z . . . . . . 7  |-  Z  =  (Cntz `  M )
31, 2cntzrcl 14803 . . . . . 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 14798 . . . . 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 5866 . . . 4  |-  ( y  =  Y  ->  ( X  .+  y )  =  ( X  .+  Y
) )
11 oveq1 5865 . . . 4  |-  ( y  =  Y  ->  (
y  .+  X )  =  ( Y  .+  X ) )
1210, 11eqeq12d 2297 . . 3  |-  ( y  =  Y  ->  (
( X  .+  y
)  =  ( y 
.+  X )  <->  ( X  .+  Y )  =  ( Y  .+  X ) ) )
1312rspccva 2883 . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791
This theorem is referenced by:  cntri  14806  cntz2ss  14808  cntzsubm  14811  cntzsubg  14812  cntzmhm  14814  cntrsubgnsg  14816  lsmsubm  14964  lsmsubg  14965  lsmcom2  14966  subgdisj1  15000  subgdisj2  15001  pj1id  15008  pj1ghm  15012  gsumval3eu  15190  gsumval3  15191  gsumzaddlem  15203  gsumzoppg  15216  dprdfcntz  15250  cntzsubr  15577  cntzsdrg  27510
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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