Theorem islininds 42235

Description: The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)

Hypotheses

Ref	Expression
islininds.b	|- B = ( Base ` M )
islininds.z	|- Z = ( 0g ` M )
islininds.r	|- R = (Scalar ` M )
islininds.e	|- E = ( Base ` R )
islininds.0	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
islininds	|- ( ( S e. V /\ M e. W ) -> ( S linIndS M <-> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) ) )

Distinct variable groups:   f, E   f, M, x   S, f, x

Allowed substitution hints:   B( x, f)   R( x, f)   E( x)   V( x, f)   W( x, f)   .0. ( x, f)   Z( x, f)

Proof of Theorem islininds

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( s = S /\ m = M ) -> s = S )
2		fveq2 6191	. . . . . . 7 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
3		islininds.b	. . . . . . 7 |- B = ( Base ` M )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( m = M -> ( Base ` m ) = B )
5	4	adantl 482	. . . . 5 |- ( ( s = S /\ m = M ) -> ( Base ` m ) = B )
6	5	pweqd 4163	. . . 4 |- ( ( s = S /\ m = M ) -> ~P ( Base ` m ) = ~P B )
7	1, 6	eleq12d 2695	. . 3 |- ( ( s = S /\ m = M ) -> ( s e. ~P ( Base ` m ) <-> S e. ~P B ) )
8		fveq2 6191	. . . . . . . . 9 |- ( m = M -> (Scalar ` m ) = (Scalar ` M ) )
9		islininds.r	. . . . . . . . 9 |- R = (Scalar ` M )
10	8, 9	syl6eqr 2674	. . . . . . . 8 |- ( m = M -> (Scalar ` m ) = R )
11	10	fveq2d 6195	. . . . . . 7 |- ( m = M -> ( Base ` (Scalar ` m ) ) = ( Base ` R ) )
12		islininds.e	. . . . . . 7 |- E = ( Base ` R )
13	11, 12	syl6eqr 2674	. . . . . 6 |- ( m = M -> ( Base ` (Scalar ` m ) ) = E )
14	13	adantl 482	. . . . 5 |- ( ( s = S /\ m = M ) -> ( Base ` (Scalar ` m ) ) = E )
15	14, 1	oveq12d 6668	. . . 4 |- ( ( s = S /\ m = M ) -> ( ( Base ` (Scalar ` m ) ) ^m s ) = ( E ^m S ) )
16	8	adantl 482	. . . . . . . . . 10 |- ( ( s = S /\ m = M ) -> (Scalar ` m ) = (Scalar ` M ) )
17	16, 9	syl6eqr 2674	. . . . . . . . 9 |- ( ( s = S /\ m = M ) -> (Scalar ` m ) = R )
18	17	fveq2d 6195	. . . . . . . 8 |- ( ( s = S /\ m = M ) -> ( 0g ` (Scalar ` m ) ) = ( 0g ` R ) )
19		islininds.0	. . . . . . . 8 |- .0. = ( 0g ` R )
20	18, 19	syl6eqr 2674	. . . . . . 7 |- ( ( s = S /\ m = M ) -> ( 0g ` (Scalar ` m ) ) = .0. )
21	20	breq2d 4665	. . . . . 6 |- ( ( s = S /\ m = M ) -> ( f finSupp ( 0g ` (Scalar ` m ) ) <-> f finSupp .0. ) )
22		fveq2 6191	. . . . . . . . 9 |- ( m = M -> ( linC ` m ) = ( linC ` M ) )
23	22	adantl 482	. . . . . . . 8 |- ( ( s = S /\ m = M ) -> ( linC ` m ) = ( linC ` M ) )
24		eqidd 2623	. . . . . . . 8 |- ( ( s = S /\ m = M ) -> f = f )
25	23, 24, 1	oveq123d 6671	. . . . . . 7 |- ( ( s = S /\ m = M ) -> ( f ( linC ` m ) s ) = ( f ( linC ` M ) S ) )
26		fveq2 6191	. . . . . . . . 9 |- ( m = M -> ( 0g ` m ) = ( 0g ` M ) )
27	26	adantl 482	. . . . . . . 8 |- ( ( s = S /\ m = M ) -> ( 0g ` m ) = ( 0g ` M ) )
28		islininds.z	. . . . . . . 8 |- Z = ( 0g ` M )
29	27, 28	syl6eqr 2674	. . . . . . 7 |- ( ( s = S /\ m = M ) -> ( 0g ` m ) = Z )
30	25, 29	eqeq12d 2637	. . . . . 6 |- ( ( s = S /\ m = M ) -> ( ( f ( linC ` m ) s ) = ( 0g ` m ) <-> ( f ( linC ` M ) S ) = Z ) )
31	21, 30	anbi12d 747	. . . . 5 |- ( ( s = S /\ m = M ) -> ( ( f finSupp ( 0g ` (Scalar ` m ) ) /\ ( f ( linC ` m ) s ) = ( 0g ` m ) ) <-> ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) ) )
32	10	fveq2d 6195	. . . . . . . . 9 |- ( m = M -> ( 0g ` (Scalar ` m ) ) = ( 0g ` R ) )
33	32, 19	syl6eqr 2674	. . . . . . . 8 |- ( m = M -> ( 0g ` (Scalar ` m ) ) = .0. )
34	33	adantl 482	. . . . . . 7 |- ( ( s = S /\ m = M ) -> ( 0g ` (Scalar ` m ) ) = .0. )
35	34	eqeq2d 2632	. . . . . 6 |- ( ( s = S /\ m = M ) -> ( ( f ` x ) = ( 0g ` (Scalar ` m ) ) <-> ( f ` x ) = .0. ) )
36	1, 35	raleqbidv 3152	. . . . 5 |- ( ( s = S /\ m = M ) -> ( A. x e. s ( f ` x ) = ( 0g ` (Scalar ` m ) ) <-> A. x e. S ( f ` x ) = .0. ) )
37	31, 36	imbi12d 334	. . . 4 |- ( ( s = S /\ m = M ) -> ( ( ( f finSupp ( 0g ` (Scalar ` m ) ) /\ ( f ( linC ` m ) s ) = ( 0g ` m ) ) -> A. x e. s ( f ` x ) = ( 0g ` (Scalar ` m ) ) ) <-> ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) )
38	15, 37	raleqbidv 3152	. . 3 |- ( ( s = S /\ m = M ) -> ( A. f e. ( ( Base ` (Scalar ` m ) ) ^m s ) ( ( f finSupp ( 0g ` (Scalar ` m ) ) /\ ( f ( linC ` m ) s ) = ( 0g ` m ) ) -> A. x e. s ( f ` x ) = ( 0g ` (Scalar ` m ) ) ) <-> A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) )
39	7, 38	anbi12d 747	. 2 |- ( ( s = S /\ m = M ) -> ( ( s e. ~P ( Base ` m ) /\ A. f e. ( ( Base ` (Scalar ` m ) ) ^m s ) ( ( f finSupp ( 0g ` (Scalar ` m ) ) /\ ( f ( linC ` m ) s ) = ( 0g ` m ) ) -> A. x e. s ( f ` x ) = ( 0g ` (Scalar ` m ) ) ) ) <-> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) ) )
40		df-lininds 42231	. 2 |- linIndS = { <. s , m >. | ( s e. ~P ( Base ` m ) /\ A. f e. ( ( Base ` (Scalar ` m ) ) ^m s ) ( ( f finSupp ( 0g ` (Scalar ` m ) ) /\ ( f ( linC ` m ) s ) = ( 0g ` m ) ) -> A. x e. s ( f ` x ) = ( 0g ` (Scalar ` m ) ) ) ) }
41	39, 40	brabga 4989	1 |- ( ( S e. V /\ M e. W ) -> ( S linIndS M <-> ( S e. ~P B /\ A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ~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   linIndS clininds 42229

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-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-iota 5851  df-fv 5896  df-ov 6653  df-lininds 42231

This theorem is referenced by:  linindsi  42236  islinindfis  42238  islindeps  42242  lindslininds  42253  linds0  42254  lindsrng01  42257  snlindsntor  42260