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Theorem cntzrcl 15053
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 3575 . . . 4  |-  -.  X  e.  (/)
2 cntzrcl.z . . . . . . . 8  |-  Z  =  (Cntz `  M )
3 fvprc 5662 . . . . . . . 8  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
42, 3syl5eq 2431 . . . . . . 7  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
54fveq1d 5670 . . . . . 6  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  ( (/) `  S
) )
6 fv01 5702 . . . . . 6  |-  ( (/) `  S )  =  (/)
75, 6syl6eq 2435 . . . . 5  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  (/) )
87eleq2d 2454 . . . 4  |-  ( -.  M  e.  _V  ->  ( X  e.  ( Z `
 S )  <->  X  e.  (/) ) )
91, 8mtbiri 295 . . 3  |-  ( -.  M  e.  _V  ->  -.  X  e.  ( Z `
 S ) )
109con4i 124 . 2  |-  ( X  e.  ( Z `  S )  ->  M  e.  _V )
11 cntzrcl.b . . . . . . . 8  |-  B  =  ( Base `  M
)
12 eqid 2387 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1311, 12, 2cntzfval 15046 . . . . . . 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 16 . . . . . 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 5012 . . . . 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 2387 . . . . . 6  |-  ( 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 5306 . . . . 5  |-  dom  (
x  e.  ~P B  |->  { y  e.  B  |  A. z  e.  x  ( y ( +g  `  M ) z )  =  ( z ( +g  `  M ) y ) } ) 
C_  ~P B
1815, 17syl6eqss 3341 . . . 4  |-  ( X  e.  ( Z `  S )  ->  dom  Z 
C_  ~P B )
19 elfvdm 5697 . . . 4  |-  ( X  e.  ( Z `  S )  ->  S  e.  dom  Z )
2018, 19sseldd 3292 . . 3  |-  ( X  e.  ( Z `  S )  ->  S  e.  ~P B )
2120elpwid 3751 . 2  |-  ( X  e.  ( Z `  S )  ->  S  C_  B )
2210, 21jca 519 1  |-  ( X  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653   _Vcvv 2899    C_ wss 3263   (/)c0 3571   ~Pcpw 3742    e. cmpt 4207   dom cdm 4818   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456  Cntzccntz 15041
This theorem is referenced by:  cntzssv  15054  cntzi  15055  resscntz  15057  cntzmhm  15064  oppgcntz  15087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-cntz 15043
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