Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6695 | . . . . 5 ⊢ (𝐵 ∈ (LBasis‘𝑊) → 𝑊 ∈ dom LBasis) | |
2 | lbsss.j | . . . . 5 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | 1, 2 | eleq2s 2930 | . . . 4 ⊢ (𝐵 ∈ 𝐽 → 𝑊 ∈ dom LBasis) |
4 | lbsss.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
5 | eqid 2820 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2820 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
7 | eqid 2820 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | eqid 2820 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
9 | eqid 2820 | . . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 19841 | . . . 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 269 | . 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 1137 | 1 ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∖ cdif 3926 ⊆ wss 3929 {csn 4560 dom cdm 5548 ‘cfv 6348 (class class class)co 7149 Basecbs 16476 Scalarcsca 16561 ·𝑠 cvsca 16562 0gc0g 16706 LSpanclspn 19736 LBasisclbs 19839 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-lbs 19840 |
This theorem is referenced by: lbsel 19843 lbspss 19847 islbs2 19919 islbs3 19920 lmimlbs 20973 lbslsp 30958 lvecdim0 31027 lssdimle 31028 lbsdiflsp0 31044 dimkerim 31045 fedgmullem1 31047 fedgmullem2 31048 fedgmul 31049 extdg1id 31075 |
Copyright terms: Public domain | W3C validator |