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Theorem cntzval 17053
Description: Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b  |-  B  =  ( Base `  M
)
cntzfval.p  |-  .+  =  ( +g  `  M )
cntzfval.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
cntzval  |-  ( S 
C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
Distinct variable groups:    x, y,  .+    x, B    x, M, y    x, S, y
Allowed substitution hints:    B( y)    Z( x, y)

Proof of Theorem cntzval
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 cntzfval.b . . . . 5  |-  B  =  ( Base `  M
)
2 cntzfval.p . . . . 5  |-  .+  =  ( +g  `  M )
3 cntzfval.z . . . . 5  |-  Z  =  (Cntz `  M )
41, 2, 3cntzfval 17052 . . . 4  |-  ( M  e.  _V  ->  Z  =  ( s  e. 
~P B  |->  { x  e.  B  |  A. y  e.  s  (
x  .+  y )  =  ( y  .+  x ) } ) )
54fveq1d 5881 . . 3  |-  ( M  e.  _V  ->  ( Z `  S )  =  ( ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } ) `  S ) )
6 fvex 5889 . . . . . 6  |-  ( Base `  M )  e.  _V
71, 6eqeltri 2545 . . . . 5  |-  B  e. 
_V
87elpw2 4565 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
9 raleq 2973 . . . . . 6  |-  ( s  =  S  ->  ( A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x )  <->  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) ) )
109rabbidv 3022 . . . . 5  |-  ( s  =  S  ->  { x  e.  B  |  A. y  e.  s  (
x  .+  y )  =  ( y  .+  x ) }  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
11 eqid 2471 . . . . 5  |-  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } )  =  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } )
127rabex 4550 . . . . 5  |-  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) }  e.  _V
1310, 11, 12fvmpt 5963 . . . 4  |-  ( S  e.  ~P B  -> 
( ( s  e. 
~P B  |->  { x  e.  B  |  A. y  e.  s  (
x  .+  y )  =  ( y  .+  x ) } ) `
 S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
148, 13sylbir 218 . . 3  |-  ( S 
C_  B  ->  (
( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s  ( x  .+  y )  =  ( y  .+  x ) } ) `  S
)  =  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) } )
155, 14sylan9eq 2525 . 2  |-  ( ( M  e.  _V  /\  S  C_  B )  -> 
( Z `  S
)  =  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) } )
16 0fv 5912 . . . 4  |-  ( (/) `  S )  =  (/)
17 fvprc 5873 . . . . . 6  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
183, 17syl5eq 2517 . . . . 5  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
1918fveq1d 5881 . . . 4  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  ( (/) `  S
) )
20 ssrab2 3500 . . . . . 6  |-  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) }  C_  B
21 fvprc 5873 . . . . . . 7  |-  ( -.  M  e.  _V  ->  (
Base `  M )  =  (/) )
221, 21syl5eq 2517 . . . . . 6  |-  ( -.  M  e.  _V  ->  B  =  (/) )
2320, 22syl5sseq 3466 . . . . 5  |-  ( -.  M  e.  _V  ->  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  C_  (/) )
24 ss0 3768 . . . . 5  |-  ( { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  C_  (/) 
->  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  =  (/) )
2523, 24syl 17 . . . 4  |-  ( -.  M  e.  _V  ->  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  =  (/) )
2616, 19, 253eqtr4a 2531 . . 3  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
2726adantr 472 . 2  |-  ( ( -.  M  e.  _V  /\  S  C_  B )  ->  ( Z `  S
)  =  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) } )
2815, 27pm2.61ian 807 1  |-  ( S 
C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268  Cntzccntz 17047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-cntz 17049
This theorem is referenced by:  elcntz  17054  cntzsnval  17056  sscntz  17058  cntzssv  17060  cntziinsn  17066
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