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Mirrors > Home > MPE Home > Th. List > linds2 | Structured version Visualization version GIF version |
Description: An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
linds2 | ⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6695 | . . . 4 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS) | |
2 | eqid 2819 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | 2 | islinds 20945 | . . . 4 ⊢ (𝑊 ∈ dom LIndS → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
5 | 4 | ibi 269 | . 2 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
6 | 5 | simprd 498 | 1 ⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2108 ⊆ wss 3934 class class class wbr 5057 I cid 5452 dom cdm 5548 ↾ cres 5550 ‘cfv 6348 Basecbs 16475 LIndF clindf 20940 LIndSclinds 20941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-linds 20943 |
This theorem is referenced by: lindsind2 20955 lindsss 20960 f1linds 20961 |
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