Theorem lcoop 42200

Description: A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)

Hypotheses

Ref	Expression
lcoop.b	|- B = ( Base ` M )
lcoop.s	|- S = (Scalar ` M )
lcoop.r	|- R = ( Base ` S )

Assertion

Ref	Expression
lcoop	|- ( ( M e. X /\ V e. ~P B ) -> ( M LinCo V ) = { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } )

Distinct variable groups:   B, c   M, c, s   R, c, s   V, c, s

Allowed substitution hints:   B( s)   S( s, c)   X( s, c)

Proof of Theorem lcoop

Dummy variables m v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( M e. X -> M e. _V )
2	1	adantr 481	. 2 |- ( ( M e. X /\ V e. ~P B ) -> M e. _V )
3		lcoop.b	. . . . . 6 |- B = ( Base ` M )
4	3	pweqi 4162	. . . . 5 |- ~P B = ~P ( Base ` M )
5	4	eleq2i 2693	. . . 4 |- ( V e. ~P B <-> V e. ~P ( Base ` M ) )
6	5	biimpi 206	. . 3 |- ( V e. ~P B -> V e. ~P ( Base ` M ) )
7	6	adantl 482	. 2 |- ( ( M e. X /\ V e. ~P B ) -> V e. ~P ( Base ` M ) )
8		fvex 6201	. . . 4 |- ( Base ` M ) e. _V
9	3, 8	eqeltri 2697	. . 3 |- B e. _V
10		rabexg 4812	. . 3 |- ( B e. _V -> { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } e. _V )
11	9, 10	mp1i 13	. 2 |- ( ( M e. X /\ V e. ~P B ) -> { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } e. _V )
12		fveq2 6191	. . . . . 6 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
13	12, 3	syl6eqr 2674	. . . . 5 |- ( m = M -> ( Base ` m ) = B )
14	13	adantr 481	. . . 4 |- ( ( m = M /\ v = V ) -> ( Base ` m ) = B )
15		fveq2 6191	. . . . . . . . 9 |- ( m = M -> (Scalar ` m ) = (Scalar ` M ) )
16	15	fveq2d 6195	. . . . . . . 8 |- ( m = M -> ( Base ` (Scalar ` m ) ) = ( Base ` (Scalar ` M ) ) )
17	16	adantr 481	. . . . . . 7 |- ( ( m = M /\ v = V ) -> ( Base ` (Scalar ` m ) ) = ( Base ` (Scalar ` M ) ) )
18		lcoop.r	. . . . . . . 8 |- R = ( Base ` S )
19		lcoop.s	. . . . . . . . 9 |- S = (Scalar ` M )
20	19	fveq2i 6194	. . . . . . . 8 |- ( Base ` S ) = ( Base ` (Scalar ` M ) )
21	18, 20	eqtri 2644	. . . . . . 7 |- R = ( Base ` (Scalar ` M ) )
22	17, 21	syl6eqr 2674	. . . . . 6 |- ( ( m = M /\ v = V ) -> ( Base ` (Scalar ` m ) ) = R )
23		simpr 477	. . . . . 6 |- ( ( m = M /\ v = V ) -> v = V )
24	22, 23	oveq12d 6668	. . . . 5 |- ( ( m = M /\ v = V ) -> ( ( Base ` (Scalar ` m ) ) ^m v ) = ( R ^m V ) )
25	15	fveq2d 6195	. . . . . . . . 9 |- ( m = M -> ( 0g ` (Scalar ` m ) ) = ( 0g ` (Scalar ` M ) ) )
26	19	a1i 11	. . . . . . . . . . 11 |- ( m = M -> S = (Scalar ` M ) )
27	26	eqcomd 2628	. . . . . . . . . 10 |- ( m = M -> (Scalar ` M ) = S )
28	27	fveq2d 6195	. . . . . . . . 9 |- ( m = M -> ( 0g ` (Scalar ` M ) ) = ( 0g ` S ) )
29	25, 28	eqtrd 2656	. . . . . . . 8 |- ( m = M -> ( 0g ` (Scalar ` m ) ) = ( 0g ` S ) )
30	29	adantr 481	. . . . . . 7 |- ( ( m = M /\ v = V ) -> ( 0g ` (Scalar ` m ) ) = ( 0g ` S ) )
31	30	breq2d 4665	. . . . . 6 |- ( ( m = M /\ v = V ) -> ( s finSupp ( 0g ` (Scalar ` m ) ) <-> s finSupp ( 0g ` S ) ) )
32		fveq2 6191	. . . . . . . . 9 |- ( m = M -> ( linC ` m ) = ( linC ` M ) )
33	32	adantr 481	. . . . . . . 8 |- ( ( m = M /\ v = V ) -> ( linC ` m ) = ( linC ` M ) )
34		eqidd 2623	. . . . . . . 8 |- ( ( m = M /\ v = V ) -> s = s )
35	33, 34, 23	oveq123d 6671	. . . . . . 7 |- ( ( m = M /\ v = V ) -> ( s ( linC ` m ) v ) = ( s ( linC ` M ) V ) )
36	35	eqeq2d 2632	. . . . . 6 |- ( ( m = M /\ v = V ) -> ( c = ( s ( linC ` m ) v ) <-> c = ( s ( linC ` M ) V ) ) )
37	31, 36	anbi12d 747	. . . . 5 |- ( ( m = M /\ v = V ) -> ( ( s finSupp ( 0g ` (Scalar ` m ) ) /\ c = ( s ( linC ` m ) v ) ) <-> ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) ) )
38	24, 37	rexeqbidv 3153	. . . 4 |- ( ( m = M /\ v = V ) -> ( E. s e. ( ( Base ` (Scalar ` m ) ) ^m v ) ( s finSupp ( 0g ` (Scalar ` m ) ) /\ c = ( s ( linC ` m ) v ) ) <-> E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) ) )
39	14, 38	rabeqbidv 3195	. . 3 |- ( ( m = M /\ v = V ) -> { c e. ( Base ` m ) | E. s e. ( ( Base ` (Scalar ` m ) ) ^m v ) ( s finSupp ( 0g ` (Scalar ` m ) ) /\ c = ( s ( linC ` m ) v ) ) } = { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } )
40	12	pweqd 4163	. . 3 |- ( m = M -> ~P ( Base ` m ) = ~P ( Base ` M ) )
41		df-lco 42196	. . 3 |- LinCo = ( m e. _V , v e. ~P ( Base ` m ) |-> { c e. ( Base ` m ) | E. s e. ( ( Base ` (Scalar ` m ) ) ^m v ) ( s finSupp ( 0g ` (Scalar ` m ) ) /\ c = ( s ( linC ` m ) v ) ) } )
42	39, 40, 41	ovmpt2x 6789	. 2 |- ( ( M e. _V /\ V e. ~P ( Base ` M ) /\ { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } e. _V ) -> ( M LinCo V ) = { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } )
43	2, 7, 11, 42	syl3anc 1326	1 |- ( ( M e. X /\ V e. ~P B ) -> ( M LinCo V ) = { c e. B | E. s e. ( R ^m V ) ( s finSupp ( 0g ` S ) /\ c = ( s ( linC ` M ) V ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   finSupp cfsupp 8275   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   linC clinc 42193   LinCo clinco 42194

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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-lco 42196

This theorem is referenced by:  lcoval  42201  lco0  42216