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Theorem cntzval 14813
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 14812 . . . 4  |-  ( M  e.  _V  ->  Z  =  ( s  e. 
~P B  |->  { x  e.  B  |  A. y  e.  s  (
x  .+  y )  =  ( y  .+  x ) } ) )
54fveq1d 5543 . . 3  |-  ( M  e.  _V  ->  ( Z `  S )  =  ( ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } ) `  S ) )
6 fvex 5555 . . . . . 6  |-  ( Base `  M )  e.  _V
71, 6eqeltri 2366 . . . . 5  |-  B  e. 
_V
87elpw2 4191 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
9 raleq 2749 . . . . . 6  |-  ( s  =  S  ->  ( A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x )  <->  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) ) )
109rabbidv 2793 . . . . 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 2296 . . . . 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 4181 . . . . 5  |-  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) }  e.  _V
1310, 11, 12fvmpt 5618 . . . 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 204 . . 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 2348 . 2  |-  ( ( M  e.  _V  /\  S  C_  B )  -> 
( Z `  S
)  =  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) } )
16 fv01 5575 . . . 4  |-  ( (/) `  S )  =  (/)
17 fvprc 5535 . . . . . 6  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
183, 17syl5eq 2340 . . . . 5  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
1918fveq1d 5543 . . . 4  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  ( (/) `  S
) )
20 ssrab2 3271 . . . . . 6  |-  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) }  C_  B
21 fvprc 5535 . . . . . . 7  |-  ( -.  M  e.  _V  ->  (
Base `  M )  =  (/) )
221, 21syl5eq 2340 . . . . . 6  |-  ( -.  M  e.  _V  ->  B  =  (/) )
2320, 22syl5sseq 3239 . . . . 5  |-  ( -.  M  e.  _V  ->  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  C_  (/) )
24 ss0 3498 . . . . 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 15 . . . 4  |-  ( -.  M  e.  _V  ->  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  =  (/) )
2616, 19, 253eqtr4a 2354 . . 3  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
2726adantr 451 . 2  |-  ( ( -.  M  e.  _V  /\  S  C_  B )  ->  ( Z `  S
)  =  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) } )
2815, 27pm2.61ian 765 1  |-  ( S 
C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Cntzccntz 14807
This theorem is referenced by:  elcntz  14814  cntzsnval  14816  sscntz  14818  cntzssv  14820  cntziinsn  14826
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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