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Mirrors > Home > MPE Home > Th. List > lbslinds | Structured version Visualization version GIF version |
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 2824 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lbslinds.j | . . . 4 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | eqid 2824 | . . . 4 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
4 | 1, 2, 3 | islbs4 20979 | . . 3 ⊢ (𝑎 ∈ 𝐽 ↔ (𝑎 ∈ (LIndS‘𝑊) ∧ ((LSpan‘𝑊)‘𝑎) = (Base‘𝑊))) |
5 | 4 | simplbi 500 | . 2 ⊢ (𝑎 ∈ 𝐽 → 𝑎 ∈ (LIndS‘𝑊)) |
6 | 5 | ssriv 3974 | 1 ⊢ 𝐽 ⊆ (LIndS‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ‘cfv 6358 Basecbs 16486 LSpanclspn 19746 LBasisclbs 19849 LIndSclinds 20952 |
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 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-lbs 19850 df-lindf 20953 df-linds 20954 |
This theorem is referenced by: islinds4 20982 lmimlbs 20983 lbslcic 20988 lvecdim0 31009 lssdimle 31010 lbsdiflsp0 31026 dimkerim 31027 fedgmullem2 31030 fedgmul 31031 extdg1id 31057 |
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