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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > linindsi | Structured version Visualization version GIF version |
Description: The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
islininds.b | β’ π΅ = (Baseβπ) |
islininds.z | β’ π = (0gβπ) |
islininds.r | β’ π = (Scalarβπ) |
islininds.e | β’ πΈ = (Baseβπ ) |
islininds.0 | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
linindsi | β’ (π linIndS π β (π β π« π΅ β§ βπ β (πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | linindsv 46762 | . . 3 β’ (π linIndS π β (π β V β§ π β V)) | |
2 | islininds.b | . . . 4 β’ π΅ = (Baseβπ) | |
3 | islininds.z | . . . 4 β’ π = (0gβπ) | |
4 | islininds.r | . . . 4 β’ π = (Scalarβπ) | |
5 | islininds.e | . . . 4 β’ πΈ = (Baseβπ ) | |
6 | islininds.0 | . . . 4 β’ 0 = (0gβπ ) | |
7 | 2, 3, 4, 5, 6 | islininds 46763 | . . 3 β’ ((π β V β§ π β V) β (π linIndS π β (π β π« π΅ β§ βπ β (πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 )))) |
8 | 1, 7 | syl 17 | . 2 β’ (π linIndS π β (π linIndS π β (π β π« π΅ β§ βπ β (πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 )))) |
9 | 8 | ibi 266 | 1 β’ (π linIndS π β (π β π« π΅ β§ βπ β (πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3060 Vcvv 3472 π« cpw 4595 class class class wbr 5140 βcfv 6531 (class class class)co 7392 βm cmap 8802 finSupp cfsupp 9343 Basecbs 17125 Scalarcsca 17181 0gc0g 17366 linC clinc 46721 linIndS clininds 46757 |
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 2702 ax-sep 5291 ax-nul 5298 ax-pr 5419 |
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 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3474 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5141 df-opab 5203 df-xp 5674 df-rel 5675 df-iota 6483 df-fv 6539 df-ov 7395 df-lininds 46759 |
This theorem is referenced by: linindslinci 46765 linindscl 46768 |
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