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Theorem cntzval 14797
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 14796 . . . 4  |-  ( M  e.  _V  ->  Z  =  ( s  e. 
~P B  |->  { x  e.  B  |  A. y  e.  s  (
x  .+  y )  =  ( y  .+  x ) } ) )
54fveq1d 5527 . . 3  |-  ( M  e.  _V  ->  ( Z `  S )  =  ( ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } ) `  S ) )
6 fvex 5539 . . . . . 6  |-  ( Base `  M )  e.  _V
71, 6eqeltri 2353 . . . . 5  |-  B  e. 
_V
87elpw2 4175 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
9 raleq 2736 . . . . . 6  |-  ( s  =  S  ->  ( A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x )  <->  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) ) )
109rabbidv 2780 . . . . 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 2283 . . . . 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 4165 . . . . 5  |-  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) }  e.  _V
1310, 11, 12fvmpt 5602 . . . 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 2335 . 2  |-  ( ( M  e.  _V  /\  S  C_  B )  -> 
( Z `  S
)  =  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) } )
16 fv01 5559 . . . 4  |-  ( (/) `  S )  =  (/)
17 fvprc 5519 . . . . . 6  |-  ( -.  M  e.  _V  ->  (Cntz `  M )  =  (/) )
183, 17syl5eq 2327 . . . . 5  |-  ( -.  M  e.  _V  ->  Z  =  (/) )
1918fveq1d 5527 . . . 4  |-  ( -.  M  e.  _V  ->  ( Z `  S )  =  ( (/) `  S
) )
20 ssrab2 3258 . . . . . 6  |-  { x  e.  B  |  A. y  e.  S  (
x  .+  y )  =  ( y  .+  x ) }  C_  B
21 fvprc 5519 . . . . . . 7  |-  ( -.  M  e.  _V  ->  (
Base `  M )  =  (/) )
221, 21syl5eq 2327 . . . . . 6  |-  ( -.  M  e.  _V  ->  B  =  (/) )
2320, 22syl5sseq 3226 . . . . 5  |-  ( -.  M  e.  _V  ->  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  C_  (/) )
24 ss0 3485 . . . . 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 2341 . . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791
This theorem is referenced by:  elcntz  14798  cntzsnval  14800  sscntz  14802  cntzssv  14804  cntziinsn  14810
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-cntz 14793
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