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Theorem cntzrcl 15116
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 3624 . . . 4  |-  -.  X  e.  (/)
2 cntzrcl.z . . . . . . . 8  |-  Z  =  (Cntz `  M )
3 fvprc 5714 . . . . . . . 8  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
42, 3syl5eq 2479 . . . . . . 7  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
54fveq1d 5722 . . . . . 6  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  ( (/) `  S
) )
6 fv01 5755 . . . . . 6  |-  ( (/) `  S )  =  (/)
75, 6syl6eq 2483 . . . . 5  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  (/) )
87eleq2d 2502 . . . 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 2435 . . . . . . . 8  |-  ( +g  `  M )  =  ( +g  `  M )
1311, 12, 2cntzfval 15109 . . . . . . 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 5064 . . . . 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 2435 . . . . . 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 5358 . . . . 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 3390 . . . 4  |-  ( X  e.  ( Z `  S )  ->  dom  Z 
C_  ~P B )
19 elfvdm 5749 . . . 4  |-  ( X  e.  ( Z `  S )  ->  S  e.  dom  Z )
2018, 19sseldd 3341 . . 3  |-  ( X  e.  ( Z `  S )  ->  S  e.  ~P B )
2120elpwid 3800 . 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 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791    e. cmpt 4258   dom cdm 4870   ` cfv 5446  (class class class)co 6073   Basecbs 13459   +g cplusg 13519  Cntzccntz 15104
This theorem is referenced by:  cntzssv  15117  cntzi  15118  resscntz  15120  cntzmhm  15127  oppgcntz  15150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-cntz 15106
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