Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lbslinds.j | . . . 4 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
4 | 1, 2, 3 | islbs4 20975 | . . 3 ⊢ (𝑎 ∈ 𝐽 ↔ (𝑎 ∈ (LIndS‘𝑊) ∧ ((LSpan‘𝑊)‘𝑎) = (Base‘𝑊))) |
5 | 4 | simplbi 500 | . 2 ⊢ (𝑎 ∈ 𝐽 → 𝑎 ∈ (LIndS‘𝑊)) |
6 | 5 | ssriv 3970 | 1 ⊢ 𝐽 ⊆ (LIndS‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6354 Basecbs 16482 LSpanclspn 19742 LBasisclbs 19845 LIndSclinds 20948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-lbs 19846 df-lindf 20949 df-linds 20950 |
This theorem is referenced by: islinds4 20978 lmimlbs 20979 lbslcic 20984 lvecdim0 31005 lssdimle 31006 lbsdiflsp0 31022 dimkerim 31023 fedgmullem2 31026 fedgmul 31027 extdg1id 31053 |
Copyright terms: Public domain | W3C validator |