Theorem lbsind 19080

Description: A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)

Hypotheses

Ref	Expression
lbsss.v	|- V = ( Base ` W )
lbsss.j	|- J = (LBasis ` W )
lbssp.n	|- N = ( LSpan ` W )
lbsind.f	|- F = (Scalar ` W )
lbsind.s	|- .x. = ( .s ` W )
lbsind.k	|- K = ( Base ` F )
lbsind.z	|- .0. = ( 0g ` F )

Assertion

Ref	Expression
lbsind	|- ( ( ( B e. J /\ E e. B ) /\ ( A e. K /\ A =/= .0. ) ) -> -. ( A .x. E ) e. ( N ` ( B \ { E } ) ) )

Proof of Theorem lbsind

Dummy variables y x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 4317	. 2 |- ( A e. ( K \ { .0. } ) <-> ( A e. K /\ A =/= .0. ) )
2		elfvdm 6220	. . . . . . . 8 |- ( B e. (LBasis ` W ) -> W e. dom LBasis )
3		lbsss.j	. . . . . . . 8 |- J = (LBasis ` W )
4	2, 3	eleq2s 2719	. . . . . . 7 |- ( B e. J -> W e. dom LBasis )
5		lbsss.v	. . . . . . . 8 |- V = ( Base ` W )
6		lbsind.f	. . . . . . . 8 |- F = (Scalar ` W )
7		lbsind.s	. . . . . . . 8 |- .x. = ( .s ` W )
8		lbsind.k	. . . . . . . 8 |- K = ( Base ` F )
9		lbssp.n	. . . . . . . 8 |- N = ( LSpan ` W )
10		lbsind.z	. . . . . . . 8 |- .0. = ( 0g ` F )
11	5, 6, 7, 8, 3, 9, 10	islbs 19076	. . . . . . 7 |- ( W e. dom LBasis -> ( B e. J <-> ( B C_ V /\ ( N ` B ) = V /\ A. x e. B A. y e. ( K \ { .0. } ) -. ( y .x. x ) e. ( N ` ( B \ { x } ) ) ) ) )
12	4, 11	syl 17	. . . . . 6 |- ( B e. J -> ( B e. J <-> ( B C_ V /\ ( N ` B ) = V /\ A. x e. B A. y e. ( K \ { .0. } ) -. ( y .x. x ) e. ( N ` ( B \ { x } ) ) ) ) )
13	12	ibi 256	. . . . 5 |- ( B e. J -> ( B C_ V /\ ( N ` B ) = V /\ A. x e. B A. y e. ( K \ { .0. } ) -. ( y .x. x ) e. ( N ` ( B \ { x } ) ) ) )
14	13	simp3d 1075	. . . 4 |- ( B e. J -> A. x e. B A. y e. ( K \ { .0. } ) -. ( y .x. x ) e. ( N ` ( B \ { x } ) ) )
15		oveq2 6658	. . . . . . 7 |- ( x = E -> ( y .x. x ) = ( y .x. E ) )
16		sneq 4187	. . . . . . . . 9 |- ( x = E -> { x } = { E } )
17	16	difeq2d 3728	. . . . . . . 8 |- ( x = E -> ( B \ { x } ) = ( B \ { E } ) )
18	17	fveq2d 6195	. . . . . . 7 |- ( x = E -> ( N ` ( B \ { x } ) ) = ( N ` ( B \ { E } ) ) )
19	15, 18	eleq12d 2695	. . . . . 6 |- ( x = E -> ( ( y .x. x ) e. ( N ` ( B \ { x } ) ) <-> ( y .x. E ) e. ( N ` ( B \ { E } ) ) ) )
20	19	notbid 308	. . . . 5 |- ( x = E -> ( -. ( y .x. x ) e. ( N ` ( B \ { x } ) ) <-> -. ( y .x. E ) e. ( N ` ( B \ { E } ) ) ) )
21		oveq1 6657	. . . . . . 7 |- ( y = A -> ( y .x. E ) = ( A .x. E ) )
22	21	eleq1d 2686	. . . . . 6 |- ( y = A -> ( ( y .x. E ) e. ( N ` ( B \ { E } ) ) <-> ( A .x. E ) e. ( N ` ( B \ { E } ) ) ) )
23	22	notbid 308	. . . . 5 |- ( y = A -> ( -. ( y .x. E ) e. ( N ` ( B \ { E } ) ) <-> -. ( A .x. E ) e. ( N ` ( B \ { E } ) ) ) )
24	20, 23	rspc2v 3322	. . . 4 |- ( ( E e. B /\ A e. ( K \ { .0. } ) ) -> ( A. x e. B A. y e. ( K \ { .0. } ) -. ( y .x. x ) e. ( N ` ( B \ { x } ) ) -> -. ( A .x. E ) e. ( N ` ( B \ { E } ) ) ) )
25	14, 24	syl5com 31	. . 3 |- ( B e. J -> ( ( E e. B /\ A e. ( K \ { .0. } ) ) -> -. ( A .x. E ) e. ( N ` ( B \ { E } ) ) ) )
26	25	impl 650	. 2 |- ( ( ( B e. J /\ E e. B ) /\ A e. ( K \ { .0. } ) ) -> -. ( A .x. E ) e. ( N ` ( B \ { E } ) ) )
27	1, 26	sylan2br 493	1 |- ( ( ( B e. J /\ E e. B ) /\ ( A e. K /\ A =/= .0. ) ) -> -. ( A .x. E ) e. ( N ` ( B \ { E } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = 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  LBasisclbs 19074

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

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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-lbs 19075

This theorem is referenced by:  lbsind2  19081