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Theorem cntzfval 14812
Description: First level 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
cntzfval  |-  ( M  e.  V  ->  Z  =  ( s  e. 
~P B  |->  { x  e.  B  |  A. y  e.  s  (
x  .+  y )  =  ( y  .+  x ) } ) )
Distinct variable groups:    x, s,
y,  .+    B, s, x    M, s, x, y
Allowed substitution hints:    B( y)    V( x, y, s)    Z( x, y, s)

Proof of Theorem cntzfval
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 cntzfval.z . 2  |-  Z  =  (Cntz `  M )
2 elex 2809 . . 3  |-  ( M  e.  V  ->  M  e.  _V )
3 fveq2 5541 . . . . . . 7  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
4 cntzfval.b . . . . . . 7  |-  B  =  ( Base `  M
)
53, 4syl6eqr 2346 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  B )
65pweqd 3643 . . . . 5  |-  ( m  =  M  ->  ~P ( Base `  m )  =  ~P B )
7 fveq2 5541 . . . . . . . . . 10  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
8 cntzfval.p . . . . . . . . . 10  |-  .+  =  ( +g  `  M )
97, 8syl6eqr 2346 . . . . . . . . 9  |-  ( m  =  M  ->  ( +g  `  m )  = 
.+  )
109oveqd 5891 . . . . . . . 8  |-  ( m  =  M  ->  (
x ( +g  `  m
) y )  =  ( x  .+  y
) )
119oveqd 5891 . . . . . . . 8  |-  ( m  =  M  ->  (
y ( +g  `  m
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2310 . . . . . . 7  |-  ( m  =  M  ->  (
( x ( +g  `  m ) y )  =  ( y ( +g  `  m ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
1312ralbidv 2576 . . . . . 6  |-  ( m  =  M  ->  ( A. y  e.  s 
( x ( +g  `  m ) y )  =  ( y ( +g  `  m ) x )  <->  A. y  e.  s  ( x  .+  y )  =  ( y  .+  x ) ) )
145, 13rabeqbidv 2796 . . . . 5  |-  ( m  =  M  ->  { x  e.  ( Base `  m
)  |  A. y  e.  s  ( x
( +g  `  m ) y )  =  ( y ( +g  `  m
) x ) }  =  { x  e.  B  |  A. y  e.  s  ( x  .+  y )  =  ( y  .+  x ) } )
156, 14mpteq12dv 4114 . . . 4  |-  ( m  =  M  ->  (
s  e.  ~P ( Base `  m )  |->  { x  e.  ( Base `  m )  |  A. y  e.  s  (
x ( +g  `  m
) y )  =  ( y ( +g  `  m ) x ) } )  =  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s  ( x  .+  y
)  =  ( y 
.+  x ) } ) )
16 df-cntz 14809 . . . 4  |- Cntz  =  ( m  e.  _V  |->  ( s  e.  ~P ( Base `  m )  |->  { x  e.  ( Base `  m )  |  A. y  e.  s  (
x ( +g  `  m
) y )  =  ( y ( +g  `  m ) x ) } ) )
17 fvex 5555 . . . . . . 7  |-  ( Base `  M )  e.  _V
184, 17eqeltri 2366 . . . . . 6  |-  B  e. 
_V
1918pwex 4209 . . . . 5  |-  ~P B  e.  _V
2019mptex 5762 . . . 4  |-  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } )  e.  _V
2115, 16, 20fvmpt 5618 . . 3  |-  ( M  e.  _V  ->  (Cntz `  M )  =  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s  ( x  .+  y
)  =  ( y 
.+  x ) } ) )
222, 21syl 15 . 2  |-  ( M  e.  V  ->  (Cntz `  M )  =  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s  ( x  .+  y
)  =  ( y 
.+  x ) } ) )
231, 22syl5eq 2340 1  |-  ( M  e.  V  ->  Z  =  ( s  e. 
~P B  |->  { x  e.  B  |  A. y  e.  s  (
x  .+  y )  =  ( y  .+  x ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   ~Pcpw 3638    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Cntzccntz 14807
This theorem is referenced by:  cntzval  14813  cntzrcl  14819
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-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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