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Mirrors > Home > MPE Home > Th. List > linds1 | Structured version Visualization version GIF version |
Description: An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
islinds.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
linds1 | ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6879 | . . . 4 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS) | |
2 | islinds.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | islinds 21213 | . . . 4 ⊢ (𝑊 ∈ dom LIndS → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
5 | 4 | ibi 266 | . 2 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
6 | 5 | simpld 495 | 1 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 class class class wbr 5105 I cid 5530 dom cdm 5633 ↾ cres 5635 ‘cfv 6496 Basecbs 17082 LIndF clindf 21208 LIndSclinds 21209 |
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 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 |
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 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-res 5645 df-iota 6448 df-fun 6498 df-fv 6504 df-linds 21211 |
This theorem is referenced by: lindsss 21228 lindsmm2 21233 islinds3 21238 islinds4 21239 0nellinds 32103 linds2eq 32112 lindsunlem 32259 lindsun 32260 dimkerim 32262 lindsadd 36061 lindsdom 36062 lindsenlbs 36063 |
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