Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  msrfval Structured version   Unicode version

Theorem msrfval 29625
Description: Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msrfval.v  |-  V  =  (mVars `  T )
msrfval.p  |-  P  =  (mPreSt `  T )
msrfval.r  |-  R  =  (mStRed `  T )
Assertion
Ref Expression
msrfval  |-  R  =  ( s  e.  P  |-> 
[_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
Distinct variable groups:    h, a,
s, z, P    T, a, h, s    z, V
Allowed substitution hints:    R( z, h, s, a)    T( z)    V( h, s, a)

Proof of Theorem msrfval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 msrfval.r . 2  |-  R  =  (mStRed `  T )
2 fveq2 5803 . . . . . 6  |-  ( t  =  T  ->  (mPreSt `  t )  =  (mPreSt `  T ) )
3 msrfval.p . . . . . 6  |-  P  =  (mPreSt `  T )
42, 3syl6eqr 2459 . . . . 5  |-  ( t  =  T  ->  (mPreSt `  t )  =  P )
5 fveq2 5803 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (mVars `  t )  =  (mVars `  T ) )
6 msrfval.v . . . . . . . . . . . . 13  |-  V  =  (mVars `  T )
75, 6syl6eqr 2459 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (mVars `  t )  =  V )
87imaeq1d 5275 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
(mVars `  t ) " ( h  u. 
{ a } ) )  =  ( V
" ( h  u. 
{ a } ) ) )
98unieqd 4198 . . . . . . . . . 10  |-  ( t  =  T  ->  U. (
(mVars `  t ) " ( h  u. 
{ a } ) )  =  U. ( V " ( h  u. 
{ a } ) ) )
109csbeq1d 3377 . . . . . . . . 9  |-  ( t  =  T  ->  [_ U. ( (mVars `  t ) " ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
)  =  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) )
1110ineq2d 3638 . . . . . . . 8  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) )  =  ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) )
1211oteq1d 4168 . . . . . . 7  |-  ( t  =  T  ->  <. (
( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.  =  <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
1312csbeq2dv 3783 . . . . . 6  |-  ( t  =  T  ->  [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( (mVars `  t ) " ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >.  =  [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
1413csbeq2dv 3783 . . . . 5  |-  ( t  =  T  ->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.  =  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
154, 14mpteq12dv 4470 . . . 4  |-  ( t  =  T  ->  (
s  e.  (mPreSt `  t )  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.
)  =  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
)
16 df-msr 29582 . . . 4  |- mStRed  =  ( t  e.  _V  |->  ( s  e.  (mPreSt `  t )  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( (mVars `  t ) " (
h  u.  { a } ) )  / 
z ]_ ( z  X.  z ) ) ,  h ,  a >.
) )
17 fvex 5813 . . . . . 6  |-  (mPreSt `  T )  e.  _V
183, 17eqeltri 2484 . . . . 5  |-  P  e. 
_V
1918mptex 6078 . . . 4  |-  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )  e.  _V
2015, 16, 19fvmpt 5886 . . 3  |-  ( T  e.  _V  ->  (mStRed `  T )  =  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
)
21 mpt0 5645 . . . . 5  |-  ( s  e.  (/)  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )  =  (/)
2221eqcomi 2413 . . . 4  |-  (/)  =  ( s  e.  (/)  |->  [_ ( 2nd `  ( 1st `  s
) )  /  h ]_ [_ ( 2nd `  s
)  /  a ]_ <. ( ( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
23 fvprc 5797 . . . 4  |-  ( -.  T  e.  _V  ->  (mStRed `  T )  =  (/) )
24 fvprc 5797 . . . . . 6  |-  ( -.  T  e.  _V  ->  (mPreSt `  T )  =  (/) )
253, 24syl5eq 2453 . . . . 5  |-  ( -.  T  e.  _V  ->  P  =  (/) )
2625mpteq1d 4473 . . . 4  |-  ( -.  T  e.  _V  ->  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )  =  ( s  e.  (/)  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
)
2722, 23, 263eqtr4a 2467 . . 3  |-  ( -.  T  e.  _V  ->  (mStRed `  T )  =  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
)
2820, 27pm2.61i 164 . 2  |-  (mStRed `  T )  =  ( s  e.  P  |->  [_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
291, 28eqtri 2429 1  |-  R  =  ( s  e.  P  |-> 
[_ ( 2nd `  ( 1st `  s ) )  /  h ]_ [_ ( 2nd `  s )  / 
a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1403    e. wcel 1840   _Vcvv 3056   [_csb 3370    u. cun 3409    i^i cin 3410   (/)c0 3735   {csn 3969   <.cotp 3977   U.cuni 4188    |-> cmpt 4450    X. cxp 4938   "cima 4943   ` cfv 5523   1stc1st 6734   2ndc2nd 6735  mVarscmvrs 29557  mPreStcmpst 29561  mStRedcmsr 29562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-ot 3978  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-msr 29582
This theorem is referenced by:  msrval  29626  msrf  29630
  Copyright terms: Public domain W3C validator