Theorem lincop 42197

Description: A linear combination as operation. (Contributed by AV, 30-Mar-2019.)

Assertion

Ref	Expression
lincop	|- ( M e. X -> ( linC ` M ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) )

Distinct variable groups:   M, s, v, x   v, X

Allowed substitution hints:   X( x, s)

Proof of Theorem lincop

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-linc 42195	. . 3 |- linC = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( x e. v |-> ( ( s ` x ) ( .s ` m ) x ) ) ) ) )
2	1	a1i 11	. 2 |- ( M e. X -> linC = ( m e. _V |-> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( x e. v |-> ( ( s ` x ) ( .s ` m ) x ) ) ) ) ) )
3		fveq2 6191	. . . . . 6 |- ( m = M -> (Scalar ` m ) = (Scalar ` M ) )
4	3	fveq2d 6195	. . . . 5 |- ( m = M -> ( Base ` (Scalar ` m ) ) = ( Base ` (Scalar ` M ) ) )
5	4	oveq1d 6665	. . . 4 |- ( m = M -> ( ( Base ` (Scalar ` m ) ) ^m v ) = ( ( Base ` (Scalar ` M ) ) ^m v ) )
6		fveq2 6191	. . . . 5 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
7	6	pweqd 4163	. . . 4 |- ( m = M -> ~P ( Base ` m ) = ~P ( Base ` M ) )
8		id 22	. . . . 5 |- ( m = M -> m = M )
9		fveq2 6191	. . . . . . 7 |- ( m = M -> ( .s ` m ) = ( .s ` M ) )
10	9	oveqd 6667	. . . . . 6 |- ( m = M -> ( ( s ` x ) ( .s ` m ) x ) = ( ( s ` x ) ( .s ` M ) x ) )
11	10	mpteq2dv 4745	. . . . 5 |- ( m = M -> ( x e. v |-> ( ( s ` x ) ( .s ` m ) x ) ) = ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) )
12	8, 11	oveq12d 6668	. . . 4 |- ( m = M -> ( m gsumg ( x e. v |-> ( ( s ` x ) ( .s ` m ) x ) ) ) = ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) )
13	5, 7, 12	mpt2eq123dv 6717	. . 3 |- ( m = M -> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( x e. v |-> ( ( s ` x ) ( .s ` m ) x ) ) ) ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) )
14	13	adantl 482	. 2 |- ( ( M e. X /\ m = M ) -> ( s e. ( ( Base ` (Scalar ` m ) ) ^m v ) , v e. ~P ( Base ` m ) |-> ( m gsumg ( x e. v |-> ( ( s ` x ) ( .s ` m ) x ) ) ) ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) )
15		elex 3212	. 2 |- ( M e. X -> M e. _V )
16		fvex 6201	. . . 4 |- ( Base ` M ) e. _V
17	16	pwex 4848	. . 3 |- ~P ( Base ` M ) e. _V
18		ovexd 6680	. . . 4 |- ( M e. X -> ( ( Base ` (Scalar ` M ) ) ^m v ) e. _V )
19	18	ralrimivw 2967	. . 3 |- ( M e. X -> A. v e. ~P ( Base ` M ) ( ( Base ` (Scalar ` M ) ) ^m v ) e. _V )
20		eqid 2622	. . . 4 |- ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) )
21	20	mpt2exxg2 42116	. . 3 |- ( ( ~P ( Base ` M ) e. _V /\ A. v e. ~P ( Base ` M ) ( ( Base ` (Scalar ` M ) ) ^m v ) e. _V ) -> ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) e. _V )
22	17, 19, 21	sylancr 695	. 2 |- ( M e. X -> ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) e. _V )
23	2, 14, 15, 22	fvmptd 6288	1 |- ( M e. X -> ( linC ` M ) = ( s e. ( ( Base ` (Scalar ` M ) ) ^m v ) , v e. ~P ( Base ` M ) |-> ( M gsumg ( x e. v |-> ( ( s ` x ) ( .s ` M ) x ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^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-1st 7168  df-2nd 7169  df-linc 42195

This theorem is referenced by:  lincval  42198