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Theorem cntzfval 15838
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 2981 . . 3  |-  ( M  e.  V  ->  M  e.  _V )
3 fveq2 5691 . . . . . . 7  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
4 cntzfval.b . . . . . . 7  |-  B  =  ( Base `  M
)
53, 4syl6eqr 2493 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  B )
65pweqd 3865 . . . . 5  |-  ( m  =  M  ->  ~P ( Base `  m )  =  ~P B )
7 fveq2 5691 . . . . . . . . . 10  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
8 cntzfval.p . . . . . . . . . 10  |-  .+  =  ( +g  `  M )
97, 8syl6eqr 2493 . . . . . . . . 9  |-  ( m  =  M  ->  ( +g  `  m )  = 
.+  )
109oveqd 6108 . . . . . . . 8  |-  ( m  =  M  ->  (
x ( +g  `  m
) y )  =  ( x  .+  y
) )
119oveqd 6108 . . . . . . . 8  |-  ( m  =  M  ->  (
y ( +g  `  m
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2457 . . . . . . 7  |-  ( m  =  M  ->  (
( x ( +g  `  m ) y )  =  ( y ( +g  `  m ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
1312ralbidv 2735 . . . . . 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 2967 . . . . 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 4370 . . . 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 15835 . . . 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 5701 . . . . . . 7  |-  ( Base `  M )  e.  _V
184, 17eqeltri 2513 . . . . . 6  |-  B  e. 
_V
1918pwex 4475 . . . . 5  |-  ~P B  e.  _V
2019mptex 5948 . . . 4  |-  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } )  e.  _V
2115, 16, 20fvmpt 5774 . . 3  |-  ( M  e.  _V  ->  (Cntz `  M )  =  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s  ( x  .+  y
)  =  ( y 
.+  x ) } ) )
222, 21syl 16 . 2  |-  ( M  e.  V  ->  (Cntz `  M )  =  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s  ( x  .+  y
)  =  ( y 
.+  x ) } ) )
231, 22syl5eq 2487 1  |-  ( M  e.  V  ->  Z  =  ( s  e. 
~P B  |->  { x  e.  B  |  A. y  e.  s  (
x  .+  y )  =  ( y  .+  x ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   _Vcvv 2972   ~Pcpw 3860    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238  Cntzccntz 15833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-cntz 15835
This theorem is referenced by:  cntzval  15839  cntzrcl  15845
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