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Theorem cntzfval 15109
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 2956 . . 3  |-  ( M  e.  V  ->  M  e.  _V )
3 fveq2 5720 . . . . . . 7  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
4 cntzfval.b . . . . . . 7  |-  B  =  ( Base `  M
)
53, 4syl6eqr 2485 . . . . . 6  |-  ( m  =  M  ->  ( Base `  m )  =  B )
65pweqd 3796 . . . . 5  |-  ( m  =  M  ->  ~P ( Base `  m )  =  ~P B )
7 fveq2 5720 . . . . . . . . . 10  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
8 cntzfval.p . . . . . . . . . 10  |-  .+  =  ( +g  `  M )
97, 8syl6eqr 2485 . . . . . . . . 9  |-  ( m  =  M  ->  ( +g  `  m )  = 
.+  )
109oveqd 6090 . . . . . . . 8  |-  ( m  =  M  ->  (
x ( +g  `  m
) y )  =  ( x  .+  y
) )
119oveqd 6090 . . . . . . . 8  |-  ( m  =  M  ->  (
y ( +g  `  m
) x )  =  ( y  .+  x
) )
1210, 11eqeq12d 2449 . . . . . . 7  |-  ( m  =  M  ->  (
( x ( +g  `  m ) y )  =  ( y ( +g  `  m ) x )  <->  ( x  .+  y )  =  ( y  .+  x ) ) )
1312ralbidv 2717 . . . . . 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 2943 . . . . 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 4279 . . . 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 15106 . . . 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 5734 . . . . . . 7  |-  ( Base `  M )  e.  _V
184, 17eqeltri 2505 . . . . . 6  |-  B  e. 
_V
1918pwex 4374 . . . . 5  |-  ~P B  e.  _V
2019mptex 5958 . . . 4  |-  ( s  e.  ~P B  |->  { x  e.  B  |  A. y  e.  s 
( x  .+  y
)  =  ( y 
.+  x ) } )  e.  _V
2115, 16, 20fvmpt 5798 . . 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 2479 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 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948   ~Pcpw 3791    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Basecbs 13459   +g cplusg 13519  Cntzccntz 15104
This theorem is referenced by:  cntzval  15110  cntzrcl  15116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-cntz 15106
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