Theorem mrsubffval 31404

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.

Step	Hyp	Ref	Expression
1		mrsubffval.s	. 2 |- S = (mRSubst ` T )
2		elex 3212	. . 3 |- ( T e. W -> T e. _V )
3		fveq2 6191	. . . . . . 7 |- ( t = T -> (mREx ` t ) = (mREx ` T ) )
4		mrsubffval.r	. . . . . . 7 |- R = (mREx ` T )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( t = T -> (mREx ` t ) = R )
6		fveq2 6191	. . . . . . 7 |- ( t = T -> (mVR ` t ) = (mVR ` T ) )
7		mrsubffval.v	. . . . . . 7 |- V = (mVR ` T )
8	6, 7	syl6eqr 2674	. . . . . 6 |- ( t = T -> (mVR ` t ) = V )
9	5, 8	oveq12d 6668	. . . . 5 |- ( t = T -> ( (mREx ` t ) ^pm (mVR ` t ) ) = ( R ^pm V ) )
10		fveq2 6191	. . . . . . . . . . 11 |- ( t = T -> (mCN ` t ) = (mCN ` T ) )
11		mrsubffval.c	. . . . . . . . . . 11 |- C = (mCN ` T )
12	10, 11	syl6eqr 2674	. . . . . . . . . 10 |- ( t = T -> (mCN ` t ) = C )
13	12, 8	uneq12d 3768	. . . . . . . . 9 |- ( t = T -> ( (mCN ` t ) u. (mVR ` t ) ) = ( C u. V ) )
14	13	fveq2d 6195	. . . . . . . 8 |- ( t = T -> (freeMnd ` ( (mCN ` t ) u. (mVR ` t ) ) ) = (freeMnd ` ( C u. V ) ) )
15		mrsubffval.g	. . . . . . . 8 |- G = (freeMnd ` ( C u. V ) )
16	14, 15	syl6eqr 2674	. . . . . . 7 |- ( t = T -> (freeMnd ` ( (mCN ` t ) u. (mVR ` t ) ) ) = G )
17	13	mpteq1d 4738	. . . . . . . 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 "> ) ) )
18	17	coeq1d 5283	. . . . . . 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 ) )
19	16, 18	oveq12d 6668	. . . . . 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 ) ) )
20	5, 19	mpteq12dv 4733	. . . . 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 ) ) ) )
21	9, 20	mpteq12dv 4733	. . . 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 31387	. . . 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 6678	. . . . 5 |- ( R ^pm V ) e. _V
24	23	mptex 6486	. . . 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
25	21, 22, 24	fvmpt 6282	. . 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 ) ) ) ) )
26	2, 25	syl 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 ) ) ) ) )
27	1, 26	syl5eq 2668	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 ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   ifcif 4086   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   <"cs1 13294   gsum_g cgsu 16101  freeMndcfrmd 17384  mCNcmcn 31357  mVRcmvar 31358  mRExcmrex 31363  mRSubstcmrsub 31367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-mrsub 31387

This theorem is referenced by:  mrsubfval  31405  mrsubff  31409