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 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 20794 | . . 3 ⊢ (𝑎 ∈ 𝐽 ↔ (𝑎 ∈ (LIndS‘𝑊) ∧ ((LSpan‘𝑊)‘𝑎) = (Base‘𝑊))) |
5 | 4 | simplbi 501 | . 2 ⊢ (𝑎 ∈ 𝐽 → 𝑎 ∈ (LIndS‘𝑊)) |
6 | 5 | ssriv 3905 | 1 ⊢ 𝐽 ⊆ (LIndS‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ‘cfv 6380 Basecbs 16760 LSpanclspn 20008 LBasisclbs 20111 LIndSclinds 20767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-lbs 20112 df-lindf 20768 df-linds 20769 |
This theorem is referenced by: islinds4 20797 lmimlbs 20798 lbslcic 20803 lvecdim0 31404 lssdimle 31405 lbsdiflsp0 31421 dimkerim 31422 fedgmullem2 31425 fedgmul 31426 extdg1id 31452 |
Copyright terms: Public domain | W3C validator |