Theorem lindsind 20156

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 792	. 2 |- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> E e. F )
2		eldifsn 4317	. . . 4 |- ( A e. ( K \ { .0. } ) <-> ( A e. K /\ A =/= .0. ) )
3	2	biimpri 218	. . 3 |- ( ( A e. K /\ A =/= .0. ) -> A e. ( K \ { .0. } ) )
4	3	adantl 482	. 2 |- ( ( ( F e. (LIndS ` W ) /\ E e. F ) /\ ( A e. K /\ A =/= .0. ) ) -> A e. ( K \ { .0. } ) )
5		elfvdm 6220	. . . . . 6 |- ( F e. (LIndS ` W ) -> W e. dom LIndS )
6		eqid 2622	. . . . . . 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 20152	. . . . . 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 17	. . . . 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 256	. . . 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 479	. . 3 |- ( F e. (LIndS ` W ) -> A. e e. F A. a e. ( K \ { .0. } ) -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) )
16	15	ad2antrr 762	. 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 6658	. . . . 5 |- ( e = E -> ( a .x. e ) = ( a .x. E ) )
18		sneq 4187	. . . . . . 7 |- ( e = E -> { e } = { E } )
19	18	difeq2d 3728	. . . . . 6 |- ( e = E -> ( F \ { e } ) = ( F \ { E } ) )
20	19	fveq2d 6195	. . . . 5 |- ( e = E -> ( N ` ( F \ { e } ) ) = ( N ` ( F \ { E } ) ) )
21	17, 20	eleq12d 2695	. . . 4 |- ( e = E -> ( ( a .x. e ) e. ( N ` ( F \ { e } ) ) <-> ( a .x. E ) e. ( N ` ( F \ { E } ) ) ) )
22	21	notbid 308	. . 3 |- ( e = E -> ( -. ( a .x. e ) e. ( N ` ( F \ { e } ) ) <-> -. ( a .x. E ) e. ( N ` ( F \ { E } ) ) ) )
23		oveq1 6657	. . . . 5 |- ( a = A -> ( a .x. E ) = ( A .x. E ) )
24	23	eleq1d 2686	. . . 4 |- ( a = A -> ( ( a .x. E ) e. ( N ` ( F \ { E } ) ) <-> ( A .x. E ) e. ( N ` ( F \ { E } ) ) ) )
25	24	notbid 308	. . 3 |- ( a = A -> ( -. ( a .x. E ) e. ( N ` ( F \ { E } ) ) <-> -. ( A .x. E ) e. ( N ` ( F \ { E } ) ) ) )
26	22, 25	rspc2va 3323	. 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 1325	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   C_ wss 3574   {csn 4177   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LSpanclspn 18971  LIndSclinds 20144

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-lindf 20145  df-linds 20146

This theorem is referenced by: (None)