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Theorem cntzrcl 16155
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 3784 . . . 4  |-  -.  X  e.  (/)
2 cntzrcl.z . . . . . . . 8  |-  Z  =  (Cntz `  M )
3 fvprc 5853 . . . . . . . 8  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
42, 3syl5eq 2515 . . . . . . 7  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
54fveq1d 5861 . . . . . 6  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  ( (/) `  S
) )
6 0fv 5892 . . . . . 6  |-  ( (/) `  S )  =  (/)
75, 6syl6eq 2519 . . . . 5  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  (/) )
87eleq2d 2532 . . . 4  |-  ( -.  M  e.  _V  ->  ( X  e.  ( Z `
 S )  <->  X  e.  (/) ) )
91, 8mtbiri 303 . . 3  |-  ( -.  M  e.  _V  ->  -.  X  e.  ( Z `
 S ) )
109con4i 130 . 2  |-  ( X  e.  ( Z `  S )  ->  M  e.  _V )
11 cntzrcl.b . . . . . . . 8  |-  B  =  ( Base `  M
)
12 eqid 2462 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1311, 12, 2cntzfval 16148 . . . . . . 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 5198 . . . . 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 2462 . . . . . 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 5496 . . . . 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 3549 . . . 4  |-  ( X  e.  ( Z `  S )  ->  dom  Z 
C_  ~P B )
19 elfvdm 5885 . . . 4  |-  ( X  e.  ( Z `  S )  ->  S  e.  dom  Z )
2018, 19sseldd 3500 . . 3  |-  ( X  e.  ( Z `  S )  ->  S  e.  ~P B )
2120elpwid 4015 . 2  |-  ( X  e.  ( Z `  S )  ->  S  C_  B )
2210, 21jca 532 1  |-  ( X  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   {crab 2813   _Vcvv 3108    C_ wss 3471   (/)c0 3780   ~Pcpw 4005    |-> cmpt 4500   dom cdm 4994   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546  Cntzccntz 16143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-cntz 16145
This theorem is referenced by:  cntzssv  16156  cntzi  16157  resscntz  16159  cntzmhm  16166  oppgcntz  16189
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