Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lbssp | Structured version Visualization version GIF version |
Description: The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
lbsss.v | ⊢ 𝑉 = (Base‘𝑊) |
lbsss.j | ⊢ 𝐽 = (LBasis‘𝑊) |
lbssp.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lbssp | ⊢ (𝐵 ∈ 𝐽 → (𝑁‘𝐵) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6806 | . . . . 5 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis) | |
2 | lbsss.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | 1, 2 | eleq2s 2857 | . . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis) |
4 | lbsss.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
5 | eqid 2738 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2738 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
7 | eqid 2738 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | lbssp.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
9 | eqid 2738 | . . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 20338 | . . . 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 266 | . 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 1142 | 1 ⊢ (𝐵 ∈ 𝐽 → (𝑁‘𝐵) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∖ cdif 3884 ⊆ wss 3887 {csn 4561 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Scalarcsca 16965 ·𝑠 cvsca 16966 0gc0g 17150 LSpanclspn 20233 LBasisclbs 20336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-lbs 20337 |
This theorem is referenced by: islbs2 20416 islbs3 20417 frlmup3 21007 frlmup4 21008 lmimlbs 21043 lbslcic 21048 lbslsp 31572 lvecdim0i 31689 dimkerim 31708 lindsdom 35771 matunitlindflem2 35774 aacllem 46505 |
Copyright terms: Public domain | W3C validator |