Theorem linindslinci 42237

Description: The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-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
linindslinci	|- ( ( S linIndS M /\ ( F e. ( E ^m S ) /\ F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) ) -> A. x e. S ( F ` x ) = .0. )

Distinct variable groups:   x, M   x, S   x, F

Allowed substitution hints:   B( x)   R( x)   E( x)   .0. ( x)   Z( x)

Proof of Theorem linindslinci

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		islininds.b	. . . 4 |- B = ( Base ` M )
2		islininds.z	. . . 4 |- Z = ( 0g ` M )
3		islininds.r	. . . 4 |- R = (Scalar ` M )
4		islininds.e	. . . 4 |- E = ( Base ` R )
5		islininds.0	. . . 4 |- .0. = ( 0g ` R )
6	1, 2, 3, 4, 5	linindsi 42236	. . 3 |- ( 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. ) ) )
7		breq1 4656	. . . . . . . . . 10 |- ( f = F -> ( f finSupp .0. <-> F finSupp .0. ) )
8		oveq1 6657	. . . . . . . . . . 11 |- ( f = F -> ( f ( linC ` M ) S ) = ( F ( linC ` M ) S ) )
9	8	eqeq1d 2624	. . . . . . . . . 10 |- ( f = F -> ( ( f ( linC ` M ) S ) = Z <-> ( F ( linC ` M ) S ) = Z ) )
10	7, 9	anbi12d 747	. . . . . . . . 9 |- ( f = F -> ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) <-> ( F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) ) )
11		fveq1 6190	. . . . . . . . . . 11 |- ( f = F -> ( f ` x ) = ( F ` x ) )
12	11	eqeq1d 2624	. . . . . . . . . 10 |- ( f = F -> ( ( f ` x ) = .0. <-> ( F ` x ) = .0. ) )
13	12	ralbidv 2986	. . . . . . . . 9 |- ( f = F -> ( A. x e. S ( f ` x ) = .0. <-> A. x e. S ( F ` x ) = .0. ) )
14	10, 13	imbi12d 334	. . . . . . . 8 |- ( f = F -> ( ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) <-> ( ( F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) -> A. x e. S ( F ` x ) = .0. ) ) )
15	14	rspcv 3305	. . . . . . 7 |- ( F e. ( E ^m S ) -> ( A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) -> ( ( F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) -> A. x e. S ( F ` x ) = .0. ) ) )
16	15	com23 86	. . . . . 6 |- ( F e. ( E ^m S ) -> ( ( F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) -> ( A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) -> A. x e. S ( F ` x ) = .0. ) ) )
17	16	3impib 1262	. . . . 5 |- ( ( F e. ( E ^m S ) /\ F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) -> ( A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) -> A. x e. S ( F ` x ) = .0. ) )
18	17	com12 32	. . . 4 |- ( A. f e. ( E ^m S ) ( ( f finSupp .0. /\ ( f ( linC ` M ) S ) = Z ) -> A. x e. S ( f ` x ) = .0. ) -> ( ( F e. ( E ^m S ) /\ F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) -> A. x e. S ( F ` x ) = .0. ) )
19	18	adantl 482	. . 3 |- ( ( 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. ) ) -> ( ( F e. ( E ^m S ) /\ F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) -> A. x e. S ( F ` x ) = .0. ) )
20	6, 19	syl 17	. 2 |- ( S linIndS M -> ( ( F e. ( E ^m S ) /\ F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) -> A. x e. S ( F ` x ) = .0. ) )
21	20	imp 445	1 |- ( ( S linIndS M /\ ( F e. ( E ^m S ) /\ F finSupp .0. /\ ( F ( linC ` M ) S ) = Z ) ) -> A. x e. S ( F ` x ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = 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-xp 5120  df-rel 5121  df-iota 5851  df-fv 5896  df-ov 6653  df-lininds 42231

This theorem is referenced by: (None)