Theorem lindsind 27159

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 732	. 2 |- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> E e. F )
2		eldifsn 3891	. . . 4 |- ( A e. ( K \ { .0. } ) <-> ( A e. K /\ A =/= .0. ) )
3	2	biimpri 198	. . 3 |- ( ( A e. K /\ A =/= .0. ) -> A e. ( K \ { .0. } ) )
4	3	adantl 453	. 2 |- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> A e. ( K \ { .0. } ) )
5		elfvdm 5720	. . . . . 6 |- ( F e. (LIndS ` W ) -> W e. dom LIndS )
6		eqid 2408	. . . . . . 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 27155	. . . . . 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 233	. . . 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 450	. . 3 |- ( F e. (LIndS ` W ) -> A. e e. F A. a e. ( K \ { .0. } ) -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) )
16	15	ad2antrr 707	. 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 6052	. . . . 5 |- ( e = E -> ( a .x. e ) = ( a .x. E ) )
18		sneq 3789	. . . . . . 7 |- ( e = E -> { e } = { E } )
19	18	difeq2d 3429	. . . . . 6 |- ( e = E -> ( F \ { e } ) = ( F \ { E } ) )
20	19	fveq2d 5695	. . . . 5 |- ( e = E -> ( N ` ( F \ { e } ) ) = ( N ` ( F \ { E } ) ) )
21	17, 20	eleq12d 2476	. . . 4 |- ( e = E -> ( ( a .x. e ) e. ( N ` ( F \ { e } ) ) <-> ( a .x. E ) e. ( N ` ( F \ { E } ) ) ) )
22	21	notbid 286	. . 3 |- ( e = E -> ( -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) <-> -. ( a .x. E ) e. ( N ` ( F \ { E } ) ) ) )
23		oveq1 6051	. . . . 5 |- ( a = A -> ( a .x. E ) = ( A .x. E ) )
24	23	eleq1d 2474	. . . 4 |- ( a = A -> ( ( a .x. E ) e. ( N ` ( F \ { E } ) ) <-> ( A .x. E ) e. ( N ` ( F \ { E } ) ) ) )
25	24	notbid 286	. . 3 |- ( a = A -> ( -. ( a .x. E ) e. ( N ` ( F \ { E } ) ) <-> -. ( A .x. E ) e. ( N ` ( F \ { E } ) ) ) )
26	22, 25	rspc2va 3023	. 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 1183	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 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2571   A.wral 2670   \ cdif 3281   C_ wss 3284   {csn 3778   dom cdm 4841   ` cfv 5417  (class class class)co 6044   Basecbs 13428  Scalarcsca 13491   .scvsca 13492   0gc0g 13682   LSpanclspn 16006  LIndSclinds 27147

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-lindf 27148  df-linds 27149