Theorem mrsubval 31406

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
mrsubval	|- ( ( F : A --> R /\ A C_ V /\ X e. R ) -> ( ( S ` F ) ` X ) = ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. X ) ) )

Distinct variable groups:   v, A   v, C   v, F   v, R   v, X   v, T   v, V

Allowed substitution hints:   S( v)   G( v)

Proof of Theorem mrsubval

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrsubffval.c	. . . 4 |- C = (mCN ` T )
2		mrsubffval.v	. . . 4 |- V = (mVR ` T )
3		mrsubffval.r	. . . 4 |- R = (mREx ` T )
4		mrsubffval.s	. . . 4 |- S = (mRSubst ` T )
5		mrsubffval.g	. . . 4 |- G = (freeMnd ` ( C u. V ) )
6	1, 2, 3, 4, 5	mrsubfval 31405	. . 3 |- ( ( F : A --> R /\ A C_ V ) -> ( S ` F ) = ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) )
7	6	3adant3 1081	. 2 |- ( ( F : A --> R /\ A C_ V /\ X e. R ) -> ( S ` F ) = ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) )
8		simpr 477	. . . 4 |- ( ( ( F : A --> R /\ A C_ V /\ X e. R ) /\ e = X ) -> e = X )
9	8	coeq2d 5284	. . 3 |- ( ( ( F : A --> R /\ A C_ V /\ X e. R ) /\ e = X ) -> ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) = ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. X ) )
10	9	oveq2d 6666	. 2 |- ( ( ( F : A --> R /\ A C_ V /\ X e. R ) /\ e = X ) -> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) = ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. X ) ) )
11		simp3 1063	. 2 |- ( ( F : A --> R /\ A C_ V /\ X e. R ) -> X e. R )
12		ovexd 6680	. 2 |- ( ( F : A --> R /\ A C_ V /\ X e. R ) -> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. X ) ) e. _V )
13	7, 10, 11, 12	fvmptd 6288	1 |- ( ( F : A --> R /\ A C_ V /\ X e. R ) -> ( ( S ` F ) ` X ) = ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   ifcif 4086   |-> cmpt 4729   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   <"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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  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-oprab 6654  df-mpt2 6655  df-pm 7860  df-mrsub 31387

This theorem is referenced by:  mrsubcv  31407  mrsub0  31413  mrsubccat  31415