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Theorem cntzssv 15846
Description: The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypotheses
Ref Expression
cntzrcl.b  |-  B  =  ( Base `  M
)
cntzrcl.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
cntzssv  |-  ( Z `
 S )  C_  B

Proof of Theorem cntzssv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3666 . . 3  |-  (/)  C_  B
2 sseq1 3377 . . 3  |-  ( ( Z `  S )  =  (/)  ->  ( ( Z `  S ) 
C_  B  <->  (/)  C_  B
) )
31, 2mpbiri 233 . 2  |-  ( ( Z `  S )  =  (/)  ->  ( Z `
 S )  C_  B )
4 n0 3646 . . 3  |-  ( ( Z `  S )  =/=  (/)  <->  E. x  x  e.  ( Z `  S
) )
5 cntzrcl.b . . . . . . . 8  |-  B  =  ( Base `  M
)
6 cntzrcl.z . . . . . . . 8  |-  Z  =  (Cntz `  M )
75, 6cntzrcl 15845 . . . . . . 7  |-  ( x  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )
87simprd 463 . . . . . 6  |-  ( x  e.  ( Z `  S )  ->  S  C_  B )
9 eqid 2443 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
105, 9, 6cntzval 15839 . . . . . 6  |-  ( S 
C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) } )
118, 10syl 16 . . . . 5  |-  ( x  e.  ( Z `  S )  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) } )
12 ssrab2 3437 . . . . 5  |-  { x  e.  B  |  A. y  e.  S  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) }  C_  B
1311, 12syl6eqss 3406 . . . 4  |-  ( x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B )
1413exlimiv 1688 . . 3  |-  ( E. x  x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B
)
154, 14sylbi 195 . 2  |-  ( ( Z `  S )  =/=  (/)  ->  ( Z `  S )  C_  B
)
163, 15pm2.61ine 2687 1  |-  ( Z `
 S )  C_  B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719   _Vcvv 2972    C_ wss 3328   (/)c0 3637   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238  Cntzccntz 15833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-cntz 15835
This theorem is referenced by:  cntz2ss  15850  cntzsubm  15853  cntzsubg  15854  cntzidss  15855  cntzmhm  15856  cntzmhm2  15857  cntzcmn  16324  cntzspan  16326  cntzsubr  16897  cntzsdrg  29559
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