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Theorem cntzssv 15129
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 3658 . . 3  |-  (/)  C_  B
2 sseq1 3371 . . 3  |-  ( ( Z `  S )  =  (/)  ->  ( ( Z `  S ) 
C_  B  <->  (/)  C_  B
) )
31, 2mpbiri 226 . 2  |-  ( ( Z `  S )  =  (/)  ->  ( Z `
 S )  C_  B )
4 n0 3639 . . 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 15128 . . . . . . 7  |-  ( x  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )
87simprd 451 . . . . . 6  |-  ( x  e.  ( Z `  S )  ->  S  C_  B )
9 eqid 2438 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
105, 9, 6cntzval 15122 . . . . . 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 3430 . . . . 5  |-  { x  e.  B  |  A. y  e.  S  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) }  C_  B
1311, 12syl6eqss 3400 . . . 4  |-  ( x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B )
1413exlimiv 1645 . . 3  |-  ( E. x  x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B
)
154, 14sylbi 189 . 2  |-  ( ( Z `  S )  =/=  (/)  ->  ( Z `  S )  C_  B
)
163, 15pm2.61ine 2682 1  |-  ( Z `
 S )  C_  B
Colors of variables: wff set class
Syntax hints:   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531  Cntzccntz 15116
This theorem is referenced by:  cntz2ss  15133  cntzsubm  15136  cntzsubg  15137  cntzidss  15138  cntzmhm  15139  cntzmhm2  15140  cntzcmn  15461  cntzspan  15462  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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