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Theorem cntzrcl 14803
Description: Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypotheses
Ref Expression
cntzrcl.b  |-  B  =  ( Base `  M
)
cntzrcl.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
cntzrcl  |-  ( X  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )

Proof of Theorem cntzrcl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3459 . . . 4  |-  -.  X  e.  (/)
2 cntzrcl.z . . . . . . . 8  |-  Z  =  (Cntz `  M )
3 fvprc 5519 . . . . . . . 8  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
42, 3syl5eq 2327 . . . . . . 7  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
54fveq1d 5527 . . . . . 6  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  ( (/) `  S
) )
6 fv01 5559 . . . . . 6  |-  ( (/) `  S )  =  (/)
75, 6syl6eq 2331 . . . . 5  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  (/) )
87eleq2d 2350 . . . 4  |-  ( -.  M  e.  _V  ->  ( X  e.  ( Z `
 S )  <->  X  e.  (/) ) )
91, 8mtbiri 294 . . 3  |-  ( -.  M  e.  _V  ->  -.  X  e.  ( Z `
 S ) )
109con4i 122 . 2  |-  ( X  e.  ( Z `  S )  ->  M  e.  _V )
11 cntzrcl.b . . . . . . . 8  |-  B  =  ( Base `  M
)
12 eqid 2283 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1311, 12, 2cntzfval 14796 . . . . . . 7  |-  ( M  e.  _V  ->  Z  =  ( x  e. 
~P B  |->  { y  e.  B  |  A. z  e.  x  (
y ( +g  `  M
) z )  =  ( z ( +g  `  M ) y ) } ) )
1410, 13syl 15 . . . . . 6  |-  ( X  e.  ( Z `  S )  ->  Z  =  ( x  e. 
~P B  |->  { y  e.  B  |  A. z  e.  x  (
y ( +g  `  M
) z )  =  ( z ( +g  `  M ) y ) } ) )
1514dmeqd 4881 . . . . 5  |-  ( X  e.  ( Z `  S )  ->  dom  Z  =  dom  ( x  e.  ~P B  |->  { y  e.  B  |  A. z  e.  x  ( y ( +g  `  M ) z )  =  ( z ( +g  `  M ) y ) } ) )
16 eqid 2283 . . . . . . 7  |-  ( x  e.  ~P B  |->  { y  e.  B  |  A. z  e.  x  ( y ( +g  `  M ) z )  =  ( z ( +g  `  M ) y ) } )  =  ( x  e. 
~P B  |->  { y  e.  B  |  A. z  e.  x  (
y ( +g  `  M
) z )  =  ( z ( +g  `  M ) y ) } )
1716dmmptss 5169 . . . . . 6  |-  dom  (
x  e.  ~P B  |->  { y  e.  B  |  A. z  e.  x  ( y ( +g  `  M ) z )  =  ( z ( +g  `  M ) y ) } ) 
C_  ~P B
1817a1i 10 . . . . 5  |-  ( X  e.  ( Z `  S )  ->  dom  ( x  e.  ~P B  |->  { y  e.  B  |  A. z  e.  x  ( y
( +g  `  M ) z )  =  ( z ( +g  `  M
) y ) } )  C_  ~P B
)
1915, 18eqsstrd 3212 . . . 4  |-  ( X  e.  ( Z `  S )  ->  dom  Z 
C_  ~P B )
20 elfvdm 5554 . . . 4  |-  ( X  e.  ( Z `  S )  ->  S  e.  dom  Z )
2119, 20sseldd 3181 . . 3  |-  ( X  e.  ( Z `  S )  ->  S  e.  ~P B )
22 elpwi 3633 . . 3  |-  ( S  e.  ~P B  ->  S  C_  B )
2321, 22syl 15 . 2  |-  ( X  e.  ( Z `  S )  ->  S  C_  B )
2410, 23jca 518 1  |-  ( X  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791
This theorem is referenced by:  cntzssv  14804  cntzi  14805  resscntz  14807  cntzmhm  14814  oppgcntz  14837
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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