Theorem lbssp 19850

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 ∈ 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 6701	. . . . 5 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis)
2		lbsss.j	. . . . 5 ⊢ 𝐽 = (LBasis‘𝑊)
3	1, 2	eleq2s 2931	. . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis)
4		lbsss.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
5		eqid 2821	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
6		eqid 2821	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
7		eqid 2821	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
8		lbssp.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑊)
9		eqid 2821	. . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
10	4, 5, 6, 7, 2, 8, 9	islbs 19847	. . . 4 ⊢ (𝑊 ∈ dom LBasis → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠 ‘𝑊)𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
11	3, 10	syl 17	. . 3 ⊢ (𝐵 ∈ 𝐽 → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠 ‘𝑊)𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
12	11	ibi 269	. 2 ⊢ (𝐵 ∈ 𝐽 → (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠 ‘𝑊)𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
13	12	simp2d 1139	1 ⊢ (𝐵 ∈ 𝐽 → (𝑁‘𝐵) = 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ∖ cdif 3932   ⊆ wss 3935  {csn 4566  dom cdm 5554  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  Scalarcsca 16567   ·_𝑠 cvsca 16568  0_gc0g 16712  LSpanclspn 19742  LBasisclbs 19845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7158  df-lbs 19846

This theorem is referenced by:  islbs2  19925  islbs3  19926  frlmup3  20943  frlmup4  20944  lmimlbs  20979  lbslcic  20984  lbslsp  30938  lvecdim0i  31004  dimkerim  31023  lindsdom  34885  matunitlindflem2  34888  aacllem  44901