Theorem lbssp 19079

Description: The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
lbsss.v	|- V = ( Base ` W )
lbsss.j	|- J = (LBasis ` W )
lbssp.n	|- N = ( LSpan ` W )

Assertion

Ref	Expression
lbssp	|- ( B e. J -> ( N ` B ) = V )

Proof of Theorem lbssp

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

Step	Hyp	Ref	Expression
1		elfvdm 6220	. . . . 5 |- ( B e. (LBasis ` W ) -> W e. dom LBasis )
2		lbsss.j	. . . . 5 |- J = (LBasis ` W )
3	1, 2	eleq2s 2719	. . . 4 |- ( B e. J -> W e. dom LBasis )
4		lbsss.v	. . . . 5 |- V = ( Base ` W )
5		eqid 2622	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
6		eqid 2622	. . . . 5 |- ( .s ` W ) = ( .s ` W )
7		eqid 2622	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
8		lbssp.n	. . . . 5 |- N = ( LSpan ` W )
9		eqid 2622	. . . . 5 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
10	4, 5, 6, 7, 2, 8, 9	islbs 19076	. . . 4 |- ( W e. dom LBasis -> ( B e. J <-> ( B C_ V /\ ( N ` B ) = V /\ A. x e. B A. y e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( y ( .s ` W ) x ) e. ( N ` ( B \ { x } ) ) ) ) )
11	3, 10	syl 17	. . 3 |- ( B e. J -> ( B e. J <-> ( B C_ V /\ ( N ` B ) = V /\ A. x e. B A. y e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( y ( .s ` W ) x ) e. ( N ` ( B \ { x } ) ) ) ) )
12	11	ibi 256	. 2 |- ( B e. J -> ( B C_ V /\ ( N ` B ) = V /\ A. x e. B A. y e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) -. ( y ( .s ` W ) x ) e. ( N ` ( B \ { x } ) ) ) )
13	12	simp2d 1074	1 |- ( B e. J -> ( N ` B ) = V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   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:  islbs2  19154  islbs3  19155  frlmup3  20139  frlmup4  20140  lmimlbs  20175  lbslcic  20180  lindsdom  33403  matunitlindflem2  33406  aacllem  42547