Step | Hyp | Ref
| Expression |
1 | | islininds.b |
. . 3
β’ π΅ = (Baseβπ) |
2 | | islininds.z |
. . 3
β’ π = (0gβπ) |
3 | | islininds.r |
. . 3
β’ π
= (Scalarβπ) |
4 | | islininds.e |
. . 3
β’ πΈ = (Baseβπ
) |
5 | | islininds.0 |
. . 3
β’ 0 =
(0gβπ
) |
6 | 1, 2, 3, 4, 5 | islininds 46617 |
. 2
β’ ((π β Fin β§ π β 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 )))) |
7 | | pm4.79 1003 |
. . . . . . 7
β’ (((π finSupp 0 β βπ₯ β π (πβπ₯) = 0 ) β¨ ((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 )) β ((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 )) |
8 | | elmapi 8793 |
. . . . . . . . . . . . 13
β’ (π β (πΈ βm π) β π:πβΆπΈ) |
9 | 8 | adantl 483 |
. . . . . . . . . . . 12
β’ (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β π:πβΆπΈ) |
10 | | simpll 766 |
. . . . . . . . . . . 12
β’ (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β π β Fin) |
11 | 5 | fvexi 6860 |
. . . . . . . . . . . . 13
β’ 0 β
V |
12 | 11 | a1i 11 |
. . . . . . . . . . . 12
β’ (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β 0 β V) |
13 | 9, 10, 12 | fdmfifsupp 9323 |
. . . . . . . . . . 11
β’ (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β π finSupp 0 ) |
14 | 13 | adantr 482 |
. . . . . . . . . 10
β’ ((((π β Fin β§ π β π) β§ π β (πΈ βm π)) β§ (π( linC βπ)π) = π) β π finSupp 0 ) |
15 | 14 | imim1i 63 |
. . . . . . . . 9
β’ ((π finSupp 0 β βπ₯ β π (πβπ₯) = 0 ) β ((((π β Fin β§ π β π) β§ π β (πΈ βm π)) β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 )) |
16 | 15 | expd 417 |
. . . . . . . 8
β’ ((π finSupp 0 β βπ₯ β π (πβπ₯) = 0 ) β (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β ((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ))) |
17 | | ax-1 6 |
. . . . . . . 8
β’ (((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ) β (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β ((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ))) |
18 | 16, 17 | jaoi 856 |
. . . . . . 7
β’ (((π finSupp 0 β βπ₯ β π (πβπ₯) = 0 ) β¨ ((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 )) β (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β ((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ))) |
19 | 7, 18 | sylbir 234 |
. . . . . 6
β’ (((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ) β (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β ((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ))) |
20 | 19 | com12 32 |
. . . . 5
β’ (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β (((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ) β ((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ))) |
21 | | pm3.42 495 |
. . . . 5
β’ (((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ) β ((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 )) |
22 | 20, 21 | impbid1 224 |
. . . 4
β’ (((π β Fin β§ π β π) β§ π β (πΈ βm π)) β (((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ) β ((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ))) |
23 | 22 | ralbidva 3169 |
. . 3
β’ ((π β Fin β§ π β π) β (βπ β (πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ) β βπ β (πΈ βm π)((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 ))) |
24 | 23 | anbi2d 630 |
. 2
β’ ((π β Fin β§ π β 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 )) β (π β π« π΅ β§ βπ β (πΈ βm π)((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 )))) |
25 | 6, 24 | bitrd 279 |
1
β’ ((π β Fin β§ π β π) β (π linIndS π β (π β π« π΅ β§ βπ β (πΈ βm π)((π( linC βπ)π) = π β βπ₯ β π (πβπ₯) = 0 )))) |