Theorem lindff 20154

Description: Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypothesis

Ref	Expression
lindff.b	|- B = ( Base ` W )

Assertion

Ref	Expression
lindff	|- ( ( F LIndF W /\ W e. Y ) -> F : dom F --> B )

Proof of Theorem lindff

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

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( F LIndF W /\ W e. Y ) -> F LIndF W )
2		rellindf 20147	. . . . . 6 |- Rel LIndF
3	2	brrelexi 5158	. . . . 5 |- ( F LIndF W -> F e. _V )
4		lindff.b	. . . . . 6 |- B = ( Base ` W )
5		eqid 2622	. . . . . 6 |- ( .s ` W ) = ( .s ` W )
6		eqid 2622	. . . . . 6 |- ( LSpan ` W ) = ( LSpan ` W )
7		eqid 2622	. . . . . 6 |- (Scalar ` W ) = (Scalar ` W )
8		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
9		eqid 2622	. . . . . 6 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
10	4, 5, 6, 7, 8, 9	islindf 20151	. . . . 5 |- ( ( W e. Y /\ F e. _V ) -> ( F LIndF W <-> ( F : dom F --> B /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) ) )
11	3, 10	sylan2 491	. . . 4 |- ( ( W e. Y /\ F LIndF W ) -> ( F LIndF W <-> ( F : dom F --> B /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) ) )
12	11	ancoms 469	. . 3 |- ( ( F LIndF W /\ W e. Y ) -> ( F LIndF W <-> ( F : dom F --> B /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) ) )
13	1, 12	mpbid 222	. 2 |- ( ( F LIndF W /\ W e. Y ) -> ( F : dom F --> B /\ A. x e. dom F A. k e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( k ( .s ` W ) ( F ` x ) ) e. ( ( LSpan ` W ) ` ( F " ( dom F \ { x } ) ) ) ) )
14	13	simpld 475	1 |- ( ( F LIndF W /\ W e. Y ) -> F : dom F --> B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-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-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-lindf 20145

This theorem is referenced by:  lindfind2  20157  lindff1  20159  lindfrn  20160  f1lindf  20161  indlcim  20179