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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 6699 | . . . 4 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS) | |
2 | islinds.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2 | islinds 20949 | . . . 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 269 | . 2 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
6 | 5 | simpld 497 | 1 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3933 class class class wbr 5063 I cid 5456 dom cdm 5552 ↾ cres 5554 ‘cfv 6352 Basecbs 16479 LIndF clindf 20944 LIndSclinds 20945 |
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 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 |
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 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-res 5564 df-iota 6311 df-fun 6354 df-fv 6360 df-linds 20947 |
This theorem is referenced by: lindsss 20964 lindsmm2 20969 islinds3 20974 islinds4 20975 0nellinds 30956 linds2eq 30963 lindsunlem 31044 lindsun 31045 dimkerim 31047 lindsadd 34923 lindsdom 34924 lindsenlbs 34925 |
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