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Theorem cntzrcl 14819
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 3472 . . . 4  |-  -.  X  e.  (/)
2 cntzrcl.z . . . . . . . 8  |-  Z  =  (Cntz `  M )
3 fvprc 5535 . . . . . . . 8  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
42, 3syl5eq 2340 . . . . . . 7  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
54fveq1d 5543 . . . . . 6  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  ( (/) `  S
) )
6 fv01 5575 . . . . . 6  |-  ( (/) `  S )  =  (/)
75, 6syl6eq 2344 . . . . 5  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  (/) )
87eleq2d 2363 . . . 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 2296 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1311, 12, 2cntzfval 14812 . . . . . . 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 4897 . . . . 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 2296 . . . . . . 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 5185 . . . . . 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 3225 . . . 4  |-  ( X  e.  ( Z `  S )  ->  dom  Z 
C_  ~P B )
20 elfvdm 5570 . . . 4  |-  ( X  e.  ( Z `  S )  ->  S  e.  dom  Z )
2119, 20sseldd 3194 . . 3  |-  ( X  e.  ( Z `  S )  ->  S  e.  ~P B )
22 elpwi 3646 . . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Cntzccntz 14807
This theorem is referenced by:  cntzssv  14820  cntzi  14821  resscntz  14823  cntzmhm  14830  oppgcntz  14853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-cntz 14809
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