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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 6570 | . . . 4 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS) | |
2 | eqid 2795 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | 2 | islinds 20635 | . . . 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 268 | . 2 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
6 | 5 | simprd 496 | 1 ⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2081 ⊆ wss 3859 class class class wbr 4962 I cid 5347 dom cdm 5443 ↾ cres 5445 ‘cfv 6225 Basecbs 16312 LIndF clindf 20630 LIndSclinds 20631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-res 5455 df-iota 6189 df-fun 6227 df-fv 6233 df-linds 20633 |
This theorem is referenced by: lindsind2 20645 lindsss 20650 f1linds 20651 |
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