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| Mirrors > Home > MPE Home > Th. List > lbsss | Structured version Visualization version GIF version | ||
| Description: A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.) |
| Ref | Expression |
|---|---|
| lbsss.v | ⊢ 𝑉 = (Base‘𝑊) |
| lbsss.j | ⊢ 𝐽 = (LBasis‘𝑊) |
| Ref | Expression |
|---|---|
| lbsss | ⊢ (𝐵 ∈ 𝐽 → 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 | eqid 2737 | . . . . 5 ⊢ (LSpan‘𝑊) = (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 ∧ ((LSpan‘𝑊)‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(𝐵 ∖ {𝑥}))))) |
| 11 | 3, 10 | syl 17 | . . 3 ⊢ (𝐵 ∈ 𝐽 → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ ((LSpan‘𝑊)‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(𝐵 ∖ {𝑥}))))) |
| 12 | 11 | ibi 267 | . 2 ⊢ (𝐵 ∈ 𝐽 → (𝐵 ⊆ 𝑉 ∧ ((LSpan‘𝑊)‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑦( ·𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(𝐵 ∖ {𝑥})))) |
| 13 | 12 | simp1d 1143 | 1 ⊢ (𝐵 ∈ 𝐽 → 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: lbsel 21077 lbspss 21081 islbs2 21156 islbs3 21157 lmimlbs 21856 lbslsp 33405 lmimdim 33654 lvecdim0 33657 lssdimle 33658 lbsdiflsp0 33677 dimkerim 33678 fedgmullem1 33680 fedgmullem2 33681 fedgmul 33682 dimlssid 33683 extdg1id 33716 fldextrspunlsplem 33723 fldextrspunlsp 33724 fldextrspunlem1 33725 |
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