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Theorem cntzssv 16158
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 3814 . . 3  |-  (/)  C_  B
2 sseq1 3525 . . 3  |-  ( ( Z `  S )  =  (/)  ->  ( ( Z `  S ) 
C_  B  <->  (/)  C_  B
) )
31, 2mpbiri 233 . 2  |-  ( ( Z `  S )  =  (/)  ->  ( Z `
 S )  C_  B )
4 n0 3794 . . 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 16157 . . . . . . 7  |-  ( x  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )
87simprd 463 . . . . . 6  |-  ( x  e.  ( Z `  S )  ->  S  C_  B )
9 eqid 2467 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
105, 9, 6cntzval 16151 . . . . . 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 3585 . . . . 5  |-  { x  e.  B  |  A. y  e.  S  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) }  C_  B
1311, 12syl6eqss 3554 . . . 4  |-  ( x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B )
1413exlimiv 1698 . . 3  |-  ( E. x  x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B
)
154, 14sylbi 195 . 2  |-  ( ( Z `  S )  =/=  (/)  ->  ( Z `  S )  C_  B
)
163, 15pm2.61ine 2780 1  |-  ( Z `
 S )  C_  B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548  Cntzccntz 16145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-cntz 16147
This theorem is referenced by:  cntz2ss  16162  cntzsubm  16165  cntzsubg  16166  cntzidss  16167  cntzmhm  16168  cntzmhm2  16169  cntzcmn  16638  cntzspan  16640  cntzsubr  17241  cntzsdrg  30756
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