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Theorem mrsubffval 30193
Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubffval.c  |-  C  =  (mCN `  T )
mrsubffval.v  |-  V  =  (mVR `  T )
mrsubffval.r  |-  R  =  (mREx `  T )
mrsubffval.s  |-  S  =  (mRSubst `  T )
mrsubffval.g  |-  G  =  (freeMnd `  ( C  u.  V ) )
Assertion
Ref Expression
mrsubffval  |-  ( T  e.  W  ->  S  =  ( f  e.  ( R  ^pm  V
)  |->  ( e  e.  R  |->  ( G  gsumg  ( ( v  e.  ( C  u.  V )  |->  if ( v  e.  dom  f ,  ( f `  v ) ,  <" v "> )
)  o.  e ) ) ) ) )
Distinct variable groups:    e, f,
v, C    R, e,
f, v    e, G, f    T, e, f, v   
e, V, f, v
Allowed substitution hints:    S( v, e, f)    G( v)    W( v, e, f)

Proof of Theorem mrsubffval
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.s . 2  |-  S  =  (mRSubst `  T )
2 elex 3065 . . 3  |-  ( T  e.  W  ->  T  e.  _V )
3 fveq2 5887 . . . . . . 7  |-  ( t  =  T  ->  (mREx `  t )  =  (mREx `  T ) )
4 mrsubffval.r . . . . . . 7  |-  R  =  (mREx `  T )
53, 4syl6eqr 2513 . . . . . 6  |-  ( t  =  T  ->  (mREx `  t )  =  R )
6 fveq2 5887 . . . . . . 7  |-  ( t  =  T  ->  (mVR `  t )  =  (mVR
`  T ) )
7 mrsubffval.v . . . . . . 7  |-  V  =  (mVR `  T )
86, 7syl6eqr 2513 . . . . . 6  |-  ( t  =  T  ->  (mVR `  t )  =  V )
95, 8oveq12d 6332 . . . . 5  |-  ( t  =  T  ->  (
(mREx `  t )  ^pm  (mVR `  t )
)  =  ( R 
^pm  V ) )
10 fveq2 5887 . . . . . . . . . . 11  |-  ( t  =  T  ->  (mCN `  t )  =  (mCN
`  T ) )
11 mrsubffval.c . . . . . . . . . . 11  |-  C  =  (mCN `  T )
1210, 11syl6eqr 2513 . . . . . . . . . 10  |-  ( t  =  T  ->  (mCN `  t )  =  C )
1312, 8uneq12d 3600 . . . . . . . . 9  |-  ( t  =  T  ->  (
(mCN `  t )  u.  (mVR `  t )
)  =  ( C  u.  V ) )
1413fveq2d 5891 . . . . . . . 8  |-  ( t  =  T  ->  (freeMnd `  ( (mCN `  t
)  u.  (mVR `  t ) ) )  =  (freeMnd `  ( C  u.  V )
) )
15 mrsubffval.g . . . . . . . 8  |-  G  =  (freeMnd `  ( C  u.  V ) )
1614, 15syl6eqr 2513 . . . . . . 7  |-  ( t  =  T  ->  (freeMnd `  ( (mCN `  t
)  u.  (mVR `  t ) ) )  =  G )
1713mpteq1d 4497 . . . . . . . 8  |-  ( t  =  T  ->  (
v  e.  ( (mCN
`  t )  u.  (mVR `  t )
)  |->  if ( v  e.  dom  f ,  ( f `  v
) ,  <" v "> ) )  =  ( v  e.  ( C  u.  V ) 
|->  if ( v  e. 
dom  f ,  ( f `  v ) ,  <" v "> ) ) )
1817coeq1d 5014 . . . . . . 7  |-  ( t  =  T  ->  (
( v  e.  ( (mCN `  t )  u.  (mVR `  t )
)  |->  if ( v  e.  dom  f ,  ( f `  v
) ,  <" v "> ) )  o.  e )  =  ( ( v  e.  ( C  u.  V ) 
|->  if ( v  e. 
dom  f ,  ( f `  v ) ,  <" v "> ) )  o.  e ) )
1916, 18oveq12d 6332 . . . . . 6  |-  ( t  =  T  ->  (
(freeMnd `  ( (mCN `  t )  u.  (mVR `  t ) ) ) 
gsumg  ( ( v  e.  ( (mCN `  t
)  u.  (mVR `  t ) )  |->  if ( v  e.  dom  f ,  ( f `  v ) ,  <" v "> )
)  o.  e ) )  =  ( G 
gsumg  ( ( v  e.  ( C  u.  V
)  |->  if ( v  e.  dom  f ,  ( f `  v
) ,  <" v "> ) )  o.  e ) ) )
205, 19mpteq12dv 4494 . . . . 5  |-  ( t  =  T  ->  (
e  e.  (mREx `  t )  |->  ( (freeMnd `  ( (mCN `  t
)  u.  (mVR `  t ) ) ) 
gsumg  ( ( v  e.  ( (mCN `  t
)  u.  (mVR `  t ) )  |->  if ( v  e.  dom  f ,  ( f `  v ) ,  <" v "> )
)  o.  e ) ) )  =  ( e  e.  R  |->  ( G  gsumg  ( ( v  e.  ( C  u.  V
)  |->  if ( v  e.  dom  f ,  ( f `  v
) ,  <" v "> ) )  o.  e ) ) ) )
219, 20mpteq12dv 4494 . . . 4  |-  ( t  =  T  ->  (
f  e.  ( (mREx `  t )  ^pm  (mVR `  t ) )  |->  ( e  e.  (mREx `  t )  |->  ( (freeMnd `  ( (mCN `  t
)  u.  (mVR `  t ) ) ) 
gsumg  ( ( v  e.  ( (mCN `  t
)  u.  (mVR `  t ) )  |->  if ( v  e.  dom  f ,  ( f `  v ) ,  <" v "> )
)  o.  e ) ) ) )  =  ( f  e.  ( R  ^pm  V )  |->  ( e  e.  R  |->  ( G  gsumg  ( ( v  e.  ( C  u.  V
)  |->  if ( v  e.  dom  f ,  ( f `  v
) ,  <" v "> ) )  o.  e ) ) ) ) )
22 df-mrsub 30176 . . . 4  |- mRSubst  =  ( t  e.  _V  |->  ( f  e.  ( (mREx `  t )  ^pm  (mVR `  t ) )  |->  ( e  e.  (mREx `  t )  |->  ( (freeMnd `  ( (mCN `  t
)  u.  (mVR `  t ) ) ) 
gsumg  ( ( v  e.  ( (mCN `  t
)  u.  (mVR `  t ) )  |->  if ( v  e.  dom  f ,  ( f `  v ) ,  <" v "> )
)  o.  e ) ) ) ) )
23 ovex 6342 . . . . 5  |-  ( R 
^pm  V )  e. 
_V
2423mptex 6160 . . . 4  |-  ( f  e.  ( R  ^pm  V )  |->  ( e  e.  R  |->  ( G  gsumg  ( ( v  e.  ( C  u.  V )  |->  if ( v  e.  dom  f ,  ( f `  v ) ,  <" v "> )
)  o.  e ) ) ) )  e. 
_V
2521, 22, 24fvmpt 5970 . . 3  |-  ( T  e.  _V  ->  (mRSubst `  T )  =  ( f  e.  ( R 
^pm  V )  |->  ( e  e.  R  |->  ( G  gsumg  ( ( v  e.  ( C  u.  V
)  |->  if ( v  e.  dom  f ,  ( f `  v
) ,  <" v "> ) )  o.  e ) ) ) ) )
262, 25syl 17 . 2  |-  ( T  e.  W  ->  (mRSubst `  T )  =  ( f  e.  ( R 
^pm  V )  |->  ( e  e.  R  |->  ( G  gsumg  ( ( v  e.  ( C  u.  V
)  |->  if ( v  e.  dom  f ,  ( f `  v
) ,  <" v "> ) )  o.  e ) ) ) ) )
271, 26syl5eq 2507 1  |-  ( T  e.  W  ->  S  =  ( f  e.  ( R  ^pm  V
)  |->  ( e  e.  R  |->  ( G  gsumg  ( ( v  e.  ( C  u.  V )  |->  if ( v  e.  dom  f ,  ( f `  v ) ,  <" v "> )
)  o.  e ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897   _Vcvv 3056    u. cun 3413   ifcif 3892    |-> cmpt 4474   dom cdm 4852    o. ccom 4856   ` cfv 5600  (class class class)co 6314    ^pm cpm 7498   <"cs1 12691    gsumg cgsu 15387  freeMndcfrmd 16679  mCNcmcn 30146  mVRcmvar 30147  mRExcmrex 30152  mRSubstcmrsub 30156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-mrsub 30176
This theorem is referenced by:  mrsubfval  30194  mrsubff  30198
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