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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 6877 | . . . . 5 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis) | |
2 | lbsss.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | 1, 2 | eleq2s 2856 | . . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis) |
4 | lbsss.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
5 | eqid 2736 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2736 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
7 | eqid 2736 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | eqid 2736 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
9 | eqid 2736 | . . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 20533 | . . . 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 266 | . 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 1142 | 1 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3063 ∖ cdif 3906 ⊆ wss 3909 {csn 4585 dom cdm 5632 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 Scalarcsca 17133 ·𝑠 cvsca 17134 0gc0g 17318 LSpanclspn 20428 LBasisclbs 20531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6446 df-fun 6496 df-fv 6502 df-ov 7357 df-lbs 20532 |
This theorem is referenced by: lbsel 20535 lbspss 20539 islbs2 20611 islbs3 20612 lmimlbs 21238 lbslsp 32060 lvecdim0 32195 lssdimle 32196 lbsdiflsp0 32212 dimkerim 32213 fedgmullem1 32215 fedgmullem2 32216 fedgmul 32217 extdg1id 32243 |
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