Theorem lindsind 27278

Description: A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
lindfind.s	|- .x. = ( .s ` W )
lindfind.n	|- N = ( LSpan ` W )
lindfind.l	|- L = (Scalar ` W )
lindfind.z	|- .0. = ( 0g ` L )
lindfind.k	|- K = ( Base ` L )

Assertion

Ref	Expression
lindsind	|- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> -. ( A .x. E ) e. ( N ` ( F \ { E } ) ) )

Proof of Theorem lindsind

Dummy variables a e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 733	. 2 |- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> E e. F )
2		eldifsn 3929	. . . 4 |- ( A e. ( K \ { .0. } ) <-> ( A e. K /\ A =/= .0. ) )
3	2	biimpri 199	. . 3 |- ( ( A e. K /\ A =/= .0. ) -> A e. ( K \ { .0. } ) )
4	3	adantl 454	. 2 |- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> A e. ( K \ { .0. } ) )
5		elfvdm 5760	. . . . . 6 |- ( F e. (LIndS ` W ) -> W e. dom LIndS )
6		eqid 2438	. . . . . . 7 |- ( Base ` W ) = ( Base ` W )
7		lindfind.s	. . . . . . 7 |- .x. = ( .s ` W )
8		lindfind.n	. . . . . . 7 |- N = ( LSpan ` W )
9		lindfind.l	. . . . . . 7 |- L = (Scalar ` W )
10		lindfind.k	. . . . . . 7 |- K = ( Base ` L )
11		lindfind.z	. . . . . . 7 |- .0. = ( 0g ` L )
12	6, 7, 8, 9, 10, 11	islinds2 27274	. . . . . 6 |- ( W e. dom LIndS -> ( F e. (LIndS ` W ) <-> ( F C_ ( Base ` W ) /\ A. e e. F A. a e. ( K \ { .0. } ) -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) ) ) )
13	5, 12	syl 16	. . . . 5 |- ( F e. (LIndS ` W ) -> ( F e. (LIndS ` W ) <-> ( F C_ ( Base ` W ) /\ A. e e. F A. a e. ( K \ { .0. } ) -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) ) ) )
14	13	ibi 234	. . . 4 |- ( F e. (LIndS ` W ) -> ( F C_ ( Base ` W ) /\ A. e e. F A. a e. ( K \ { .0. } ) -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) ) )
15	14	simprd 451	. . 3 |- ( F e. (LIndS ` W ) -> A. e e. F A. a e. ( K \ { .0. } ) -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) )
16	15	ad2antrr 708	. 2 |- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> A. e e. F A. a e. ( K \ { .0. } ) -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) )
17		oveq2 6092	. . . . 5 |- ( e = E -> ( a .x. e ) = ( a .x. E ) )
18		sneq 3827	. . . . . . 7 |- ( e = E -> { e } = { E } )
19	18	difeq2d 3467	. . . . . 6 |- ( e = E -> ( F \ { e } ) = ( F \ { E } ) )
20	19	fveq2d 5735	. . . . 5 |- ( e = E -> ( N ` ( F \ { e } ) ) = ( N ` ( F \ { E } ) ) )
21	17, 20	eleq12d 2506	. . . 4 |- ( e = E -> ( ( a .x. e ) e. ( N ` ( F \ { e } ) ) <-> ( a .x. E ) e. ( N ` ( F \ { E } ) ) ) )
22	21	notbid 287	. . 3 |- ( e = E -> ( -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) <-> -. ( a .x. E ) e. ( N ` ( F \ { E } ) ) ) )
23		oveq1 6091	. . . . 5 |- ( a = A -> ( a .x. E ) = ( A .x. E ) )
24	23	eleq1d 2504	. . . 4 |- ( a = A -> ( ( a .x. E ) e. ( N ` ( F \ { E } ) ) <-> ( A .x. E ) e. ( N ` ( F \ { E } ) ) ) )
25	24	notbid 287	. . 3 |- ( a = A -> ( -. ( a .x. E ) e. ( N ` ( F \ { E } ) ) <-> -. ( A .x. E ) e. ( N ` ( F \ { E } ) ) ) )
26	22, 25	rspc2va 3061	. 2 |- ( ( ( E e. F /\ A e. ( K \ { .0. } ) ) /\ A. e e. F A. a e. ( K \ { .0. } ) -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) ) -> -. ( A .x. E ) e. ( N ` ( F \ { E } ) ) )
27	1, 4, 16, 26	syl21anc 1184	1 |- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> -. ( A .x. E ) e. ( N ` ( F \ { E } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 178   /\ wa 360   = wceq 1653   e. wcel 1726   =/= wne 2601   A.wral 2707   \ cdif 3319   C_ wss 3322   {csn 3816   dom cdm 4881   ` cfv 5457  (class class class)co 6084   Basecbs 13474  Scalarcsca 13537   .scvsca 13538   0gc0g 13728   LSpanclspn 16052  LIndSclinds 27266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-lindf 27267  df-linds 27268