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Theorem cntzssv 16493
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 3823 . . 3  |-  (/)  C_  B
2 sseq1 3520 . . 3  |-  ( ( Z `  S )  =  (/)  ->  ( ( Z `  S ) 
C_  B  <->  (/)  C_  B
) )
31, 2mpbiri 233 . 2  |-  ( ( Z `  S )  =  (/)  ->  ( Z `
 S )  C_  B )
4 n0 3803 . . 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 16492 . . . . . . 7  |-  ( x  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )
87simprd 463 . . . . . 6  |-  ( x  e.  ( Z `  S )  ->  S  C_  B )
9 eqid 2457 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
105, 9, 6cntzval 16486 . . . . . 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 3581 . . . . 5  |-  { x  e.  B  |  A. y  e.  S  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) }  C_  B
1311, 12syl6eqss 3549 . . . 4  |-  ( x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B )
1413exlimiv 1723 . . 3  |-  ( E. x  x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B
)
154, 14sylbi 195 . 2  |-  ( ( Z `  S )  =/=  (/)  ->  ( Z `  S )  C_  B
)
163, 15pm2.61ine 2770 1  |-  ( Z `
 S )  C_  B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712  Cntzccntz 16480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-cntz 16482
This theorem is referenced by:  cntz2ss  16497  cntzsubm  16500  cntzsubg  16501  cntzidss  16502  cntzmhm  16503  cntzmhm2  16504  cntzcmn  16975  cntzspan  16977  cntzsubr  17588  cntzsdrg  31355
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