Theorem islindf2 20153

Description: Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)

Hypotheses

Ref	Expression
islindf.b	|- B = ( Base ` W )
islindf.v	|- .x. = ( .s ` W )
islindf.k	|- K = ( LSpan ` W )
islindf.s	|- S = (Scalar ` W )
islindf.n	|- N = ( Base ` S )
islindf.z	|- .0. = ( 0g ` S )

Assertion

Ref	Expression
islindf2	|- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( F LIndF W <-> A. x e. I A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( I \ { x } ) ) ) ) )

Distinct variable groups:   k, F, x   k, N   k, W, x   .0. , k   B, k, x   k, I, x   k, X, x   k, Y, x

Allowed substitution hints:   S( x, k)   .x. ( x, k)   K( x, k)   N( x)   .0. ( x)

Proof of Theorem islindf2

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> W e. Y )
2		simp3 1063	. . . 4 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> F : I --> B )
3		simp2 1062	. . . 4 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> I e. X )
4		fex 6490	. . . 4 |- ( ( F : I --> B /\ I e. X ) -> F e. _V )
5	2, 3, 4	syl2anc 693	. . 3 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> F e. _V )
6		islindf.b	. . . 4 |- B = ( Base ` W )
7		islindf.v	. . . 4 |- .x. = ( .s ` W )
8		islindf.k	. . . 4 |- K = ( LSpan ` W )
9		islindf.s	. . . 4 |- S = (Scalar ` W )
10		islindf.n	. . . 4 |- N = ( Base ` S )
11		islindf.z	. . . 4 |- .0. = ( 0g ` S )
12	6, 7, 8, 9, 10, 11	islindf 20151	. . 3 |- ( ( W e. Y /\ F e. _V ) -> ( F LIndF W <-> ( F : dom F --> B /\ A. x e. dom F A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( dom F \ { x } ) ) ) ) ) )
13	1, 5, 12	syl2anc 693	. 2 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( F LIndF W <-> ( F : dom F --> B /\ A. x e. dom F A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( dom F \ { x } ) ) ) ) ) )
14		ffdm 6062	. . . . 5 |- ( F : I --> B -> ( F : dom F --> B /\ dom F C_ I ) )
15	14	simpld 475	. . . 4 |- ( F : I --> B -> F : dom F --> B )
16	15	3ad2ant3 1084	. . 3 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> F : dom F --> B )
17	16	biantrurd 529	. 2 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( A. x e. dom F A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( dom F \ { x } ) ) ) <-> ( F : dom F --> B /\ A. x e. dom F A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( dom F \ { x } ) ) ) ) ) )
18		fdm 6051	. . . 4 |- ( F : I --> B -> dom F = I )
19	18	3ad2ant3 1084	. . 3 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> dom F = I )
20	19	difeq1d 3727	. . . . . . . 8 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( dom F \ { x } ) = ( I \ { x } ) )
21	20	imaeq2d 5466	. . . . . . 7 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( F " ( dom F \ { x } ) ) = ( F " ( I \ { x } ) ) )
22	21	fveq2d 6195	. . . . . 6 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( K ` ( F " ( dom F \ { x } ) ) ) = ( K ` ( F " ( I \ { x } ) ) ) )
23	22	eleq2d 2687	. . . . 5 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( ( k .x. ( F ` x ) ) e. ( K ` ( F " ( dom F \ { x } ) ) ) <-> ( k .x. ( F ` x ) ) e. ( K ` ( F " ( I \ { x } ) ) ) ) )
24	23	notbid 308	. . . 4 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( dom F \ { x } ) ) ) <-> -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( I \ { x } ) ) ) ) )
25	24	ralbidv 2986	. . 3 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( dom F \ { x } ) ) ) <-> A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( I \ { x } ) ) ) ) )
26	19, 25	raleqbidv 3152	. 2 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( A. x e. dom F A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( dom F \ { x } ) ) ) <-> A. x e. I A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( I \ { x } ) ) ) ) )
27	13, 17, 26	3bitr2d 296	1 |- ( ( W e. Y /\ I e. X /\ F : I --> B ) -> ( F LIndF W <-> A. x e. I A. k e. ( N \ { .0. } ) -. ( k .x. ( F ` x ) ) e. ( K ` ( F " ( I \ { x } ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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

This theorem is referenced by:  lindfmm  20166  islindf4  20177