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Theorem msrval 30184
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 )
msrval.z  |-  Z  = 
U. ( V "
( H  u.  { A } ) )
Assertion
Ref Expression
msrval  |-  ( <. D ,  H ,  A >.  e.  P  -> 
( R `  <. D ,  H ,  A >. )  =  <. ( D  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )

Proof of Theorem msrval
Dummy variables  h  a  s  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 msrfval.v . . . 4  |-  V  =  (mVars `  T )
2 msrfval.p . . . 4  |-  P  =  (mPreSt `  T )
3 msrfval.r . . . 4  |-  R  =  (mStRed `  T )
41, 2, 3msrfval 30183 . . 3  |-  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 >. )
54a1i 11 . 2  |-  ( <. D ,  H ,  A >.  e.  P  ->  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 >. )
)
6 fvex 5891 . . . 4  |-  ( 2nd `  ( 1st `  s
) )  e.  _V
76a1i 11 . . 3  |-  ( (
<. D ,  H ,  A >.  e.  P  /\  s  =  <. D ,  H ,  A >. )  ->  ( 2nd `  ( 1st `  s ) )  e.  _V )
8 fvex 5891 . . . . 5  |-  ( 2nd `  s )  e.  _V
98a1i 11 . . . 4  |-  ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  =  <. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  ->  ( 2nd `  s )  e.  _V )
10 simpllr 767 . . . . . . . . 9  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  s  =  <. D ,  H ,  A >. )
1110fveq2d 5885 . . . . . . . 8  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( 1st `  s )  =  ( 1st `  <. D ,  H ,  A >. ) )
1211fveq2d 5885 . . . . . . 7  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( 1st `  ( 1st `  s
) )  =  ( 1st `  ( 1st `  <. D ,  H ,  A >. ) ) )
13 eqid 2422 . . . . . . . . . . . . 13  |-  (mDV `  T )  =  (mDV
`  T )
14 eqid 2422 . . . . . . . . . . . . 13  |-  (mEx `  T )  =  (mEx
`  T )
1513, 14, 2elmpst 30182 . . . . . . . . . . . 12  |-  ( <. D ,  H ,  A >.  e.  P  <->  ( ( D  C_  (mDV `  T
)  /\  `' D  =  D )  /\  ( H  C_  (mEx `  T
)  /\  H  e.  Fin )  /\  A  e.  (mEx `  T )
) )
1615simp1bi 1020 . . . . . . . . . . 11  |-  ( <. D ,  H ,  A >.  e.  P  -> 
( D  C_  (mDV `  T )  /\  `' D  =  D )
)
1716simpld 460 . . . . . . . . . 10  |-  ( <. D ,  H ,  A >.  e.  P  ->  D  C_  (mDV `  T
) )
1817ad3antrrr 734 . . . . . . . . 9  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  D  C_  (mDV `  T )
)
19 fvex 5891 . . . . . . . . . 10  |-  (mDV `  T )  e.  _V
2019ssex 4568 . . . . . . . . 9  |-  ( D 
C_  (mDV `  T
)  ->  D  e.  _V )
2118, 20syl 17 . . . . . . . 8  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  D  e.  _V )
2215simp2bi 1021 . . . . . . . . . 10  |-  ( <. D ,  H ,  A >.  e.  P  -> 
( H  C_  (mEx `  T )  /\  H  e.  Fin ) )
2322simprd 464 . . . . . . . . 9  |-  ( <. D ,  H ,  A >.  e.  P  ->  H  e.  Fin )
2423ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  H  e.  Fin )
2515simp3bi 1022 . . . . . . . . 9  |-  ( <. D ,  H ,  A >.  e.  P  ->  A  e.  (mEx `  T
) )
2625ad3antrrr 734 . . . . . . . 8  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  A  e.  (mEx `  T )
)
27 ot1stg 6821 . . . . . . . 8  |-  ( ( D  e.  _V  /\  H  e.  Fin  /\  A  e.  (mEx `  T )
)  ->  ( 1st `  ( 1st `  <. D ,  H ,  A >. ) )  =  D )
2821, 24, 26, 27syl3anc 1264 . . . . . . 7  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( 1st `  ( 1st `  <. D ,  H ,  A >. ) )  =  D )
2912, 28eqtrd 2463 . . . . . 6  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( 1st `  ( 1st `  s
) )  =  D )
30 fvex 5891 . . . . . . . . . . 11  |-  (mVars `  T )  e.  _V
311, 30eqeltri 2503 . . . . . . . . . 10  |-  V  e. 
_V
32 imaexg 6744 . . . . . . . . . 10  |-  ( V  e.  _V  ->  ( V " ( h  u. 
{ a } ) )  e.  _V )
3331, 32ax-mp 5 . . . . . . . . 9  |-  ( V
" ( h  u. 
{ a } ) )  e.  _V
3433uniex 6601 . . . . . . . 8  |-  U. ( V " ( h  u. 
{ a } ) )  e.  _V
3534a1i 11 . . . . . . 7  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  U. ( V " ( h  u. 
{ a } ) )  e.  _V )
36 id 22 . . . . . . . . 9  |-  ( z  =  U. ( V
" ( h  u. 
{ a } ) )  ->  z  =  U. ( V " (
h  u.  { a } ) ) )
37 simplr 760 . . . . . . . . . . . . . 14  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  h  =  ( 2nd `  ( 1st `  s ) ) )
3811fveq2d 5885 . . . . . . . . . . . . . 14  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  <. D ,  H ,  A >. ) ) )
39 ot2ndg 6822 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  _V  /\  H  e.  Fin  /\  A  e.  (mEx `  T )
)  ->  ( 2nd `  ( 1st `  <. D ,  H ,  A >. ) )  =  H )
4021, 24, 26, 39syl3anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( 2nd `  ( 1st `  <. D ,  H ,  A >. ) )  =  H )
4137, 38, 403eqtrd 2467 . . . . . . . . . . . . 13  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  h  =  H )
42 simpr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  a  =  ( 2nd `  s
) )
4310fveq2d 5885 . . . . . . . . . . . . . . 15  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( 2nd `  s )  =  ( 2nd `  <. D ,  H ,  A >. ) )
44 ot3rdg 6823 . . . . . . . . . . . . . . . 16  |-  ( A  e.  (mEx `  T
)  ->  ( 2nd ` 
<. D ,  H ,  A >. )  =  A )
4526, 44syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( 2nd `  <. D ,  H ,  A >. )  =  A )
4642, 43, 453eqtrd 2467 . . . . . . . . . . . . . 14  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  a  =  A )
4746sneqd 4010 . . . . . . . . . . . . 13  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  { a }  =  { A } )
4841, 47uneq12d 3621 . . . . . . . . . . . 12  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  (
h  u.  { a } )  =  ( H  u.  { A } ) )
4948imaeq2d 5187 . . . . . . . . . . 11  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  ( V " ( h  u. 
{ a } ) )  =  ( V
" ( H  u.  { A } ) ) )
5049unieqd 4229 . . . . . . . . . 10  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  U. ( V " ( h  u. 
{ a } ) )  =  U. ( V " ( H  u.  { A } ) ) )
51 msrval.z . . . . . . . . . 10  |-  Z  = 
U. ( V "
( H  u.  { A } ) )
5250, 51syl6eqr 2481 . . . . . . . . 9  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  U. ( V " ( h  u. 
{ a } ) )  =  Z )
5336, 52sylan9eqr 2485 . . . . . . . 8  |-  ( ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  =  <. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s ) )  /\  z  = 
U. ( V "
( h  u.  {
a } ) ) )  ->  z  =  Z )
5453sqxpeqd 4879 . . . . . . 7  |-  ( ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  =  <. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s ) )  /\  z  = 
U. ( V "
( h  u.  {
a } ) ) )  ->  ( z  X.  z )  =  ( Z  X.  Z ) )
5535, 54csbied 3422 . . . . . 6  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
)  =  ( Z  X.  Z ) )
5629, 55ineq12d 3665 . . . . 5  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  (
( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) )  =  ( D  i^i  ( Z  X.  Z ) ) )
5756, 41, 46oteq123d 4202 . . . 4  |-  ( ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  = 
<. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  /\  a  =  ( 2nd `  s
) )  ->  <. (
( 1st `  ( 1st `  s ) )  i^i  [_ U. ( V
" ( h  u. 
{ a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >.  =  <. ( D  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
589, 57csbied 3422 . . 3  |-  ( ( ( <. D ,  H ,  A >.  e.  P  /\  s  =  <. D ,  H ,  A >. )  /\  h  =  ( 2nd `  ( 1st `  s ) ) )  ->  [_ ( 2nd `  s )  /  a ]_ <. ( ( 1st `  ( 1st `  s
) )  i^i  [_ U. ( V " ( h  u.  { a } ) )  /  z ]_ ( z  X.  z
) ) ,  h ,  a >.  =  <. ( D  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
597, 58csbied 3422 . 2  |-  ( (
<. D ,  H ,  A >.  e.  P  /\  s  =  <. D ,  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 >.  =  <. ( D  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
60 id 22 . 2  |-  ( <. D ,  H ,  A >.  e.  P  ->  <. D ,  H ,  A >.  e.  P )
61 otex 4686 . . 3  |-  <. ( D  i^i  ( Z  X.  Z ) ) ,  H ,  A >.  e. 
_V
6261a1i 11 . 2  |-  ( <. D ,  H ,  A >.  e.  P  ->  <. ( D  i^i  ( Z  X.  Z ) ) ,  H ,  A >.  e.  _V )
635, 59, 60, 62fvmptd 5970 1  |-  ( <. D ,  H ,  A >.  e.  P  -> 
( R `  <. D ,  H ,  A >. )  =  <. ( D  i^i  ( Z  X.  Z ) ) ,  H ,  A >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080   [_csb 3395    u. cun 3434    i^i cin 3435    C_ wss 3436   {csn 3998   <.cotp 4006   U.cuni 4219    |-> cmpt 4482    X. cxp 4851   `'ccnv 4852   "cima 4856   ` cfv 5601   1stc1st 6805   2ndc2nd 6806   Fincfn 7580  mExcmex 30113  mDVcmdv 30114  mVarscmvrs 30115  mPreStcmpst 30119  mStRedcmsr 30120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-ot 4007  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1st 6807  df-2nd 6808  df-mpst 30139  df-msr 30140
This theorem is referenced by:  msrf  30188  msrid  30191  elmsta  30194  mthmpps  30228
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