Theorem dflinc2 42199

Description: Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)

Assertion

Ref	Expression
dflinc2	|- linC = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( s oF ( .s ` m ) ( _I |` v ) ) ) ) )

Distinct variable group:   m, s, v

Proof of Theorem dflinc2

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-linc 42195	. 2 |- linC = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( i e. v |-> ( ( s ` i ) ( .s ` m ) i ) ) ) ) )
2		elmapfn 7880	. . . . . . . 8 |- ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) -> s Fn v )
3	2	adantr 481	. . . . . . 7 |- ( ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) /\ v e. ~P ( Base ` m ) ) -> s Fn v )
4		fnresi 6008	. . . . . . . 8 |- ( _I |` v ) Fn v
5	4	a1i 11	. . . . . . 7 |- ( ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) /\ v e. ~P ( Base ` m ) ) -> ( _I |` v ) Fn v )
6		vex 3203	. . . . . . . 8 |- v e. _V
7	6	a1i 11	. . . . . . 7 |- ( ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) /\ v e. ~P ( Base ` m ) ) -> v e. _V )
8		inidm 3822	. . . . . . 7 |- ( v i^i v ) = v
9		eqidd 2623	. . . . . . 7 |- ( ( ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) /\ v e. ~P ( Base ` m ) ) /\ i e. v ) -> ( s ` i ) = ( s ` i ) )
10		fvresi 6439	. . . . . . . 8 |- ( i e. v -> ( ( _I |` v ) ` i ) = i )
11	10	adantl 482	. . . . . . 7 |- ( ( ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) /\ v e. ~P ( Base ` m ) ) /\ i e. v ) -> ( ( _I |` v ) ` i ) = i )
12	3, 5, 7, 7, 8, 9, 11	offval 6904	. . . . . 6 |- ( ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) /\ v e. ~P ( Base ` m ) ) -> ( s oF ( .s ` m ) ( _I |` v ) ) = ( i e. v |-> ( ( s ` i ) ( .s ` m ) i ) ) )
13	12	eqcomd 2628	. . . . 5 |- ( ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) /\ v e. ~P ( Base ` m ) ) -> ( i e. v |-> ( ( s ` i ) ( .s ` m ) i ) ) = ( s oF ( .s ` m ) ( _I |` v ) ) )
14	13	oveq2d 6666	. . . 4 |- ( ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) /\ v e. ~P ( Base ` m ) ) -> ( m gsumg ( i e. v |-> ( ( s ` i ) ( .s ` m ) i ) ) ) = ( m gsumg ( s oF ( .s ` m ) ( _I |` v ) ) ) )
15	14	mpt2eq3ia 6720	. . 3 |- ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( i e. v |-> ( ( s ` i ) ( .s ` m ) i ) ) ) ) = ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( s oF ( .s ` m ) ( _I |` v ) ) ) )
16	15	mpteq2i 4741	. 2 |- ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( i e. v |-> ( ( s ` i ) ( .s ` m ) i ) ) ) ) ) = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( s oF ( .s ` m ) ( _I |` v ) ) ) ) )
17	1, 16	eqtri 2644	1 |- linC = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( s oF ( .s ` m ) ( _I |` v ) ) ) ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   |-> cmpt 4729   _I cid 5023   |` cres 5116   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   ^m cmap 7857   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   gsum_g cgsu 16101   linC clinc 42193

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-of 6897  df-1st 7168  df-2nd 7169  df-map 7859  df-linc 42195

This theorem is referenced by: (None)