![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > linindscl | Structured version Visualization version GIF version |
Description: A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.) |
Ref | Expression |
---|---|
linindscl | β’ (π linIndS π β π β π« (Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2732 | . . 3 β’ (0gβπ) = (0gβπ) | |
3 | eqid 2732 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
4 | eqid 2732 | . . 3 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
5 | eqid 2732 | . . 3 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
6 | 1, 2, 3, 4, 5 | linindsi 47206 | . 2 β’ (π linIndS π β (π β π« (Baseβπ) β§ βπ β ((Baseβ(Scalarβπ)) βm π)((π finSupp (0gβ(Scalarβπ)) β§ (π( linC βπ)π) = (0gβπ)) β βπ₯ β π (πβπ₯) = (0gβ(Scalarβπ))))) |
7 | 6 | simpld 495 | 1 β’ (π linIndS π β π β π« (Baseβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 π« cpw 4602 class class class wbr 5148 βcfv 6543 (class class class)co 7411 βm cmap 8822 finSupp cfsupp 9363 Basecbs 17146 Scalarcsca 17202 0gc0g 17387 linC clinc 47163 linIndS clininds 47199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-iota 6495 df-fv 6551 df-ov 7414 df-lininds 47201 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |