Step | Hyp | Ref
| Expression |
1 | | islininds.b |
. . . 4
β’ π΅ = (Baseβπ) |
2 | | islininds.z |
. . . 4
β’ π = (0gβπ) |
3 | | islininds.r |
. . . 4
β’ π
= (Scalarβπ) |
4 | | islininds.e |
. . . 4
β’ πΈ = (Baseβπ
) |
5 | | islininds.0 |
. . . 4
β’ 0 =
(0gβπ
) |
6 | 1, 2, 3, 4, 5 | linindsi 47128 |
. . 3
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 | | breq1 5152 |
. . . . . . . . 9
β’ (π = πΉ β (π finSupp 0 β πΉ finSupp 0 )) |
8 | | oveq1 7416 |
. . . . . . . . . 10
β’ (π = πΉ β (π( linC βπ)π) = (πΉ( linC βπ)π)) |
9 | 8 | eqeq1d 2735 |
. . . . . . . . 9
β’ (π = πΉ β ((π( linC βπ)π) = π β (πΉ( linC βπ)π) = π)) |
10 | 7, 9 | anbi12d 632 |
. . . . . . . 8
β’ (π = πΉ β ((π finSupp 0 β§ (π( linC βπ)π) = π) β (πΉ finSupp 0 β§ (πΉ( linC βπ)π) = π))) |
11 | | fveq1 6891 |
. . . . . . . . . 10
β’ (π = πΉ β (πβπ₯) = (πΉβπ₯)) |
12 | 11 | eqeq1d 2735 |
. . . . . . . . 9
β’ (π = πΉ β ((πβπ₯) = 0 β (πΉβπ₯) = 0 )) |
13 | 12 | ralbidv 3178 |
. . . . . . . 8
β’ (π = πΉ β (βπ₯ β π (πβπ₯) = 0 β βπ₯ β π (πΉβπ₯) = 0 )) |
14 | 10, 13 | imbi12d 345 |
. . . . . . 7
β’ (π = πΉ β (((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ) β ((πΉ finSupp 0 β§ (πΉ( linC βπ)π) = π) β βπ₯ β π (πΉβπ₯) = 0 ))) |
15 | 14 | rspcv 3609 |
. . . . . 6
β’ (πΉ β (πΈ βm π) β (βπ β (πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ) β ((πΉ finSupp 0 β§ (πΉ( linC βπ)π) = π) β βπ₯ β π (πΉβπ₯) = 0 ))) |
16 | 15 | com23 86 |
. . . . 5
β’ (πΉ β (πΈ βm π) β ((πΉ finSupp 0 β§ (πΉ( linC βπ)π) = 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 π (πΉβπ₯) = 0 ))) |
17 | 16 | 3impib 1117 |
. . . 4
β’ ((πΉ β (πΈ βm π) β§ πΉ finSupp 0 β§ (πΉ( linC βπ)π) = 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 π (πΉβπ₯) = 0 )) |
18 | 17 | com12 32 |
. . 3
β’
(βπ β
(πΈ βm π)((π finSupp 0 β§ (π( linC βπ)π) = π) β βπ₯ β π (πβπ₯) = 0 ) β ((πΉ β (πΈ βm π) β§ πΉ finSupp 0 β§ (πΉ( linC βπ)π) = π) β βπ₯ β π (πΉβπ₯) = 0 )) |
19 | 6, 18 | simpl2im 505 |
. 2
β’ (π linIndS π β ((πΉ β (πΈ βm π) β§ πΉ finSupp 0 β§ (πΉ( linC βπ)π) = π) β βπ₯ β π (πΉβπ₯) = 0 )) |
20 | 19 | imp 408 |
1
β’ ((π linIndS π β§ (πΉ β (πΈ βm π) β§ πΉ finSupp 0 β§ (πΉ( linC βπ)π) = π)) β βπ₯ β π (πΉβπ₯) = 0 ) |