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Theorem cntzfval 14796
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 2796 . . 3  |-  ( M  e.  V  ->  M  e.  _V )
3 fveq2 5525 . . . . . . 7  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
4 cntzfval.b . . . . . . 7  |-  B  =  ( Base `  M
)
53, 4syl6eqr 2333 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  B )
65pweqd 3630 . . . . 5  |-  ( m  =  M  ->  ~P ( Base `  m )  =  ~P B )
7 fveq2 5525 . . . . . . . . . 10  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
8 cntzfval.p . . . . . . . . . 10  |-  .+  =  ( +g  `  M )
97, 8syl6eqr 2333 . . . . . . . . 9  |-  ( m  =  M  ->  ( +g  `  m )  = 
.+  )
109oveqd 5875 . . . . . . . 8  |-  ( m  =  M  ->  (
x ( +g  `  m
) y )  =  ( x  .+  y
) )
119oveqd 5875 . . . . . . . 8  |-  ( m  =  M  ->  (
y ( +g  `  m
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2297 . . . . . . 7  |-  ( m  =  M  ->  (
( x ( +g  `  m ) y )  =  ( y ( +g  `  m ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
1312ralbidv 2563 . . . . . 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 2783 . . . . 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 4098 . . . 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 14793 . . . 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 5539 . . . . . . 7  |-  ( Base `  M )  e.  _V
184, 17eqeltri 2353 . . . . . 6  |-  B  e. 
_V
1918pwex 4193 . . . . 5  |-  ~P B  e.  _V
2019mptex 5746 . . . 4  |-  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } )  e.  _V
2115, 16, 20fvmpt 5602 . . 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 2327 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791
This theorem is referenced by:  cntzval  14797  cntzrcl  14803
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-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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