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| Description: A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| lbslinds.j | ⊢ 𝐽 = (LBasis‘𝑊) | 
| Ref | Expression | 
|---|---|
| lbslinds | ⊢ 𝐽 ⊆ (LIndS‘𝑊) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | lbslinds.j | . . . 4 ⊢ 𝐽 = (LBasis‘𝑊) | |
| 3 | eqid 2737 | . . . 4 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
| 4 | 1, 2, 3 | islbs4 21852 | . . 3 ⊢ (𝑎 ∈ 𝐽 ↔ (𝑎 ∈ (LIndS‘𝑊) ∧ ((LSpan‘𝑊)‘𝑎) = (Base‘𝑊))) | 
| 5 | 4 | simplbi 497 | . 2 ⊢ (𝑎 ∈ 𝐽 → 𝑎 ∈ (LIndS‘𝑊)) | 
| 6 | 5 | ssriv 3987 | 1 ⊢ 𝐽 ⊆ (LIndS‘𝑊) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 Basecbs 17247 LSpanclspn 20969 LBasisclbs 21073 LIndSclinds 21825 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-lbs 21074 df-lindf 21826 df-linds 21827 | 
| This theorem is referenced by: islinds4 21855 lmimlbs 21856 lbslcic 21861 lvecdim0 33657 lssdimle 33658 lbsdiflsp0 33677 dimkerim 33678 fedgmullem2 33681 fedgmul 33682 extdg1id 33716 | 
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