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Theorem mrsubfval 31405
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
mrsubfval |- ( ( 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 ) ) ) )
Distinct variable groups:   v, e, A   C, e, v   e, F, v   R, e, v   e, G   T, e, v   e, V, v
Allowed substitution hints:   S( v, e)   G( v)
Proof of Theorem mrsubfval
Dummy variable f is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.c . . . . . 6 |- C = (mCN ` T )
2 mrsubffval.v . . . . . 6 |- V = (mVR ` T )
3 mrsubffval.r . . . . . 6 |- R = (mREx ` T )
4 mrsubffval.s . . . . . 6 |- S = (mRSubst ` T )
5 mrsubffval.g . . . . . 6 |- G = (freeMnd ` ( C u. V ) )
61, 2, 3, 4, 5mrsubffval 31404 . . . . 5 |- ( T e. _V -> 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 ) ) ) ) )
76adantr 481 . . . 4 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> 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 ) ) ) ) )
8 dmeq 5324 . . . . . . . . . . 11 |- ( f = F -> dom f = dom F )
9 fdm 6051 . . . . . . . . . . . 12 |- ( F : A --> R -> dom F = A )
109ad2antrl 764 . . . . . . . . . . 11 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> dom F = A )
118, 10sylan9eqr 2678 . . . . . . . . . 10 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> dom f = A )
1211eleq2d 2687 . . . . . . . . 9 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> ( v e. dom f <-> v e. A ) )
13 simpr 477 . . . . . . . . . 10 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> f = F )
1413fveq1d 6193 . . . . . . . . 9 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> ( f ` v ) = ( F ` v ) )
1512, 14ifbieq1d 4109 . . . . . . . 8 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> if ( v e. dom f , ( f ` v ) , <" v "> ) = if ( v e. A , ( F ` v ) , <" v "> ) )
1615mpteq2dv 4745 . . . . . . 7 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> ( v e. ( C u. V ) |-> if ( v e. dom f , ( f ` v ) , <" v "> ) ) = ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) )
1716coeq1d 5283 . . . . . 6 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> ( ( v e. ( C u. V ) |-> if ( v e. dom f , ( f ` v ) , <" v "> ) ) o. e ) = ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) )
1817oveq2d 6666 . . . . 5 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. dom f , ( f ` v ) , <" v "> ) ) o. e ) ) = ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) )
1918mpteq2dv 4745 . . . 4 |- ( ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) /\ f = F ) -> ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. dom f , ( f ` v ) , <" v "> ) ) o. e ) ) ) = ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) )
20 fvex 6201 . . . . . . 7 |- (mREx ` T ) e. _V
213, 20eqeltri 2697 . . . . . 6 |- R e. _V
2221a1i 11 . . . . 5 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> R e. _V )
23 fvex 6201 . . . . . . 7 |- (mVR ` T ) e. _V
242, 23eqeltri 2697 . . . . . 6 |- V e. _V
2524a1i 11 . . . . 5 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> V e. _V )
26 simprl 794 . . . . 5 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> F : A --> R )
27 simprr 796 . . . . 5 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> A C_ V )
28 elpm2r 7875 . . . . 5 |- ( ( ( R e. _V /\ V e. _V ) /\ ( F : A --> R /\ A C_ V ) ) -> F e. ( R ^pm V ) )
2922, 25, 26, 27, 28syl22anc 1327 . . . 4 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> F e. ( R ^pm V ) )
3021mptex 6486 . . . . 5 |- ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) e. _V
3130a1i 11 . . . 4 |- ( ( T e. _V /\ ( F : A --> R /\ A C_ V ) ) -> ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) e. _V )
327, 19, 29, 31fvmptd 6288 . . 3 |- ( ( T e. _V /\ ( 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 ) ) ) )
3332ex 450 . 2 |- ( T e. _V -> ( ( 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 ) ) ) ) )
34 0fv 6227 . . . 4 |- ( (/) ` F ) = (/)
35 fvprc 6185 . . . . . 6 |- ( -. T e. _V -> (mRSubst ` T ) = (/) )
364, 35syl5eq 2668 . . . . 5 |- ( -. T e. _V -> S = (/) )
3736fveq1d 6193 . . . 4 |- ( -. T e. _V -> ( S ` F ) = ( (/) ` F ) )
38 fvprc 6185 . . . . . . 7 |- ( -. T e. _V -> (mREx ` T ) = (/) )
393, 38syl5eq 2668 . . . . . 6 |- ( -. T e. _V -> R = (/) )
4039mpteq1d 4738 . . . . 5 |- ( -. T e. _V -> ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) = ( e e. (/) |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) )
41 mpt0 6021 . . . . 5 |- ( e e. (/) |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) = (/)
4240, 41syl6eq 2672 . . . 4 |- ( -. T e. _V -> ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) = (/) )
4334, 37, 423eqtr4a 2682 . . 3 |- ( -. T e. _V -> ( S ` F ) = ( e e. R |-> ( G gsumg ( ( v e. ( C u. V ) |-> if ( v e. A , ( F ` v ) , <" v "> ) ) o. e ) ) ) )
4443a1d 25 . 2 |- ( -. T e. _V -> ( ( 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 ) ) ) ) )
4533, 44pm2.61i 176 1 |- ( ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   ifcif 4086   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   <"cs1 13294   gsumg 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:  mrsubval  31406  mrsubrn  31410  elmrsubrn  31417
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