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| 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 6943 | . . . . 5 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis) | |
| 2 | lbsss.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
| 3 | 1, 2 | eleq2s 2859 | . . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis) |
| 4 | lbsss.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
| 5 | eqid 2737 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 6 | eqid 2737 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 7 | eqid 2737 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 8 | lbssp.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
| 10 | 4, 5, 6, 7, 2, 8, 9 | islbs 21075 | . . . 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 267 | . 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 1144 | 1 ⊢ (𝐵 ∈ 𝐽 → (𝑁‘𝐵) = 𝑉) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LSpanclspn 20969 LBasisclbs 21073 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-lbs 21074 |
| This theorem is referenced by: islbs2 21156 islbs3 21157 frlmup3 21820 frlmup4 21821 lmimlbs 21856 lbslcic 21861 lbslsp 33405 lvecdim0i 33656 dimkerim 33678 dimlssid 33683 fldextrspunlsplem 33723 lindsdom 37621 matunitlindflem2 37624 aacllem 49320 |
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