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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 6943 | . . . 4 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS) | |
2 | islinds.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | islinds 21846 | . . . 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 267 | . 2 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
6 | 5 | simpld 494 | 1 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 class class class wbr 5147 I cid 5581 dom cdm 5688 ↾ cres 5690 ‘cfv 6562 Basecbs 17244 LIndF clindf 21841 LIndSclinds 21842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-res 5700 df-iota 6515 df-fun 6564 df-fv 6570 df-linds 21844 |
This theorem is referenced by: lindsss 21861 lindsmm2 21866 islinds3 21871 islinds4 21872 0nellinds 33377 linds2eq 33388 lindsunlem 33651 lindsun 33652 dimkerim 33654 lindsadd 37599 lindsdom 37600 lindsenlbs 37601 |
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