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Theorem cntzssv 14804
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 3483 . . 3  |-  (/)  C_  B
2 sseq1 3199 . . 3  |-  ( ( Z `  S )  =  (/)  ->  ( ( Z `  S ) 
C_  B  <->  (/)  C_  B
) )
31, 2mpbiri 224 . 2  |-  ( ( Z `  S )  =  (/)  ->  ( Z `
 S )  C_  B )
4 n0 3464 . . 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 14803 . . . . . . 7  |-  ( x  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )
87simprd 449 . . . . . 6  |-  ( x  e.  ( Z `  S )  ->  S  C_  B )
9 eqid 2283 . . . . . . 7  |-  ( +g  `  M )  =  ( +g  `  M )
105, 9, 6cntzval 14797 . . . . . 6  |-  ( S 
C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) } )
118, 10syl 15 . . . . 5  |-  ( x  e.  ( Z `  S )  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) } )
12 ssrab2 3258 . . . . . 6  |-  { x  e.  B  |  A. y  e.  S  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) }  C_  B
1312a1i 10 . . . . 5  |-  ( x  e.  ( Z `  S )  ->  { x  e.  B  |  A. y  e.  S  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) }  C_  B )
1411, 13eqsstrd 3212 . . . 4  |-  ( x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B )
1514exlimiv 1666 . . 3  |-  ( E. x  x  e.  ( Z `  S )  ->  ( Z `  S )  C_  B
)
164, 15sylbi 187 . 2  |-  ( ( Z `  S )  =/=  (/)  ->  ( Z `  S )  C_  B
)
173, 16pm2.61ine 2522 1  |-  ( Z `
 S )  C_  B
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791
This theorem is referenced by:  cntz2ss  14808  cntzsubm  14811  cntzsubg  14812  cntzidss  14813  cntzmhm  14814  cntzmhm2  14815  cntzcmn  15136  cntzspan  15137  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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