Theorem lindfind 20155

Description: A linearly independent family 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
lindfind	|- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> -. ( A .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) )

Proof of Theorem lindfind

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

Step	Hyp	Ref	Expression
1		simplr 792	. 2 |- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> E e. dom 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 LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> A e. ( K \ { .0. } ) )
5		simpll 790	. . . 4 |- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> F LIndF W )
6		lindfind.l	. . . . . . 7 |- L = (Scalar ` W )
7		lindfind.k	. . . . . . 7 |- K = ( Base ` L )
8	6, 7	elbasfv 15920	. . . . . 6 |- ( A e. K -> W e. _V )
9	8	ad2antrl 764	. . . . 5 |- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> W e. _V )
10		rellindf 20147	. . . . . . 7 |- Rel LIndF
11	10	brrelexi 5158	. . . . . 6 |- ( F LIndF W -> F e. _V )
12	11	ad2antrr 762	. . . . 5 |- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> F e. _V )
13		eqid 2622	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
14		lindfind.s	. . . . . 6 |- .x. = ( .s ` W )
15		lindfind.n	. . . . . 6 |- N = ( LSpan ` W )
16		lindfind.z	. . . . . 6 |- .0. = ( 0g ` L )
17	13, 14, 15, 6, 7, 16	islindf 20151	. . . . 5 |- ( ( W e. _V /\ F e. _V ) -> ( F LIndF W <-> ( F : dom F --> ( Base ` W ) /\ A. e e. dom F A. a e. ( K \ { .0. } ) -. ( a .x. ( F ` e ) ) e. ( N ` ( F " ( dom F \ { e } ) ) ) ) ) )
18	9, 12, 17	syl2anc 693	. . . 4 |- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> ( F LIndF W <-> ( F : dom F --> ( Base ` W ) /\ A. e e. dom F A. a e. ( K \ { .0. } ) -. ( a .x. ( F ` e ) ) e. ( N ` ( F " ( dom F \ { e } ) ) ) ) ) )
19	5, 18	mpbid 222	. . 3 |- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> ( F : dom F --> ( Base ` W ) /\ A. e e. dom F A. a e. ( K \ { .0. } ) -. ( a .x. ( F ` e ) ) e. ( N ` ( F " ( dom F \ { e } ) ) ) ) )
20	19	simprd 479	. 2 |- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> A. e e. dom F A. a e. ( K \ { .0. } ) -. ( a .x. ( F ` e ) ) e. ( N ` ( F " ( dom F \ { e } ) ) ) )
21		fveq2 6191	. . . . . 6 |- ( e = E -> ( F ` e ) = ( F ` E ) )
22	21	oveq2d 6666	. . . . 5 |- ( e = E -> ( a .x. ( F ` e ) ) = ( a .x. ( F ` E ) ) )
23		sneq 4187	. . . . . . . 8 |- ( e = E -> { e } = { E } )
24	23	difeq2d 3728	. . . . . . 7 |- ( e = E -> ( dom F \ { e } ) = ( dom F \ { E } ) )
25	24	imaeq2d 5466	. . . . . 6 |- ( e = E -> ( F " ( dom F \ { e } ) ) = ( F " ( dom F \ { E } ) ) )
26	25	fveq2d 6195	. . . . 5 |- ( e = E -> ( N ` ( F " ( dom F \ { e } ) ) ) = ( N ` ( F " ( dom F \ { E } ) ) ) )
27	22, 26	eleq12d 2695	. . . 4 |- ( e = E -> ( ( a .x. ( F ` e ) ) e. ( N ` ( F " ( dom F \ { e } ) ) ) <-> ( a .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) ) )
28	27	notbid 308	. . 3 |- ( e = E -> ( -. ( a .x. ( F ` e ) ) e. ( N ` ( F " ( dom F \ { e } ) ) ) <-> -. ( a .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) ) )
29		oveq1 6657	. . . . 5 |- ( a = A -> ( a .x. ( F ` E ) ) = ( A .x. ( F ` E ) ) )
30	29	eleq1d 2686	. . . 4 |- ( a = A -> ( ( a .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) <-> ( A .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) ) )
31	30	notbid 308	. . 3 |- ( a = A -> ( -. ( a .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) <-> -. ( A .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) ) )
32	28, 31	rspc2va 3323	. 2 |- ( ( ( E e. dom F /\ A e. ( K \ { .0. } ) ) /\ A. e e. dom F A. a e. ( K \ { .0. } ) -. ( a .x. ( F ` e ) ) e. ( N ` ( F " ( dom F \ { e } ) ) ) ) -> -. ( A .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) )
33	1, 4, 20, 32	syl21anc 1325	1 |- ( ( ( F LIndF W /\ E e. dom F ) /\ ( A e. K /\ A =/= .0. ) ) -> -. ( A .x. ( F ` E ) ) e. ( N ` ( F " ( dom F \ { E } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   {csn 4177   class class class wbr 4653   dom cdm 5114   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LSpanclspn 18971   LIndF clindf 20143

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-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-fv 5896  df-ov 6653  df-slot 15861  df-base 15863  df-lindf 20145

This theorem is referenced by:  lindfind2  20157  lindfrn  20160  f1lindf  20161