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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 6706 | . . . . 5 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis) | |
2 | lbsss.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | 1, 2 | eleq2s 2851 | . . . 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 19967 | . . . 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 270 | . 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 209 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∖ cdif 3840 ⊆ wss 3843 {csn 4516 dom cdm 5525 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 Scalarcsca 16671 ·𝑠 cvsca 16672 0gc0g 16816 LSpanclspn 19862 LBasisclbs 19965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7173 df-lbs 19966 |
This theorem is referenced by: islbs2 20045 islbs3 20046 frlmup3 20616 frlmup4 20617 lmimlbs 20652 lbslcic 20657 lbslsp 31144 lvecdim0i 31261 dimkerim 31280 lindsdom 35394 matunitlindflem2 35397 aacllem 45958 |
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