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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 6220 | . . . . 5 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis) | |
2 | lbsss.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | 1, 2 | eleq2s 2719 | . . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis) |
4 | lbsss.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
5 | eqid 2622 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2622 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
7 | eqid 2622 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | eqid 2622 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
9 | eqid 2622 | . . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 19076 | . . . 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 256 | . 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 1073 | 1 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 LSpanclspn 18971 LBasisclbs 19074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-lbs 19075 |
This theorem is referenced by: lbsel 19078 lbspss 19082 islbs2 19154 islbs3 19155 lmimlbs 20175 |
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