Step | Hyp | Ref
| Expression |
1 | | hvmapval.h |
. . . 4
β’ π» = (LHypβπΎ) |
2 | | hvmapval.u |
. . . 4
β’ π = ((DVecHβπΎ)βπ) |
3 | | hvmapval.o |
. . . 4
β’ π = ((ocHβπΎ)βπ) |
4 | | hvmapval.v |
. . . 4
β’ π = (Baseβπ) |
5 | | hvmapval.p |
. . . 4
β’ + =
(+gβπ) |
6 | | hvmapval.t |
. . . 4
β’ Β· = (
Β·π βπ) |
7 | | hvmapval.z |
. . . 4
β’ 0 =
(0gβπ) |
8 | | hvmapval.s |
. . . 4
β’ π = (Scalarβπ) |
9 | | hvmapval.r |
. . . 4
β’ π
= (Baseβπ) |
10 | | hvmapval.m |
. . . 4
β’ π = ((HVMapβπΎ)βπ) |
11 | | hvmapval.k |
. . . 4
β’ (π β (πΎ β π΄ β§ π β π»)) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 | hvmapfval 40272 |
. . 3
β’ (π β π = (π₯ β (π β { 0 }) β¦ (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯)))))) |
13 | 12 | fveq1d 6848 |
. 2
β’ (π β (πβπ) = ((π₯ β (π β { 0 }) β¦ (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯)))))βπ)) |
14 | | hvmapval.x |
. . 3
β’ (π β π β (π β { 0 })) |
15 | 4 | fvexi 6860 |
. . . 4
β’ π β V |
16 | 15 | mptex 7177 |
. . 3
β’ (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π})π£ = (π‘ + (π Β· π)))) β V |
17 | | sneq 4600 |
. . . . . . . 8
β’ (π₯ = π β {π₯} = {π}) |
18 | 17 | fveq2d 6850 |
. . . . . . 7
β’ (π₯ = π β (πβ{π₯}) = (πβ{π})) |
19 | | oveq2 7369 |
. . . . . . . . 9
β’ (π₯ = π β (π Β· π₯) = (π Β· π)) |
20 | 19 | oveq2d 7377 |
. . . . . . . 8
β’ (π₯ = π β (π‘ + (π Β· π₯)) = (π‘ + (π Β· π))) |
21 | 20 | eqeq2d 2744 |
. . . . . . 7
β’ (π₯ = π β (π£ = (π‘ + (π Β· π₯)) β π£ = (π‘ + (π Β· π)))) |
22 | 18, 21 | rexeqbidv 3319 |
. . . . . 6
β’ (π₯ = π β (βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯)) β βπ‘ β (πβ{π})π£ = (π‘ + (π Β· π)))) |
23 | 22 | riotabidv 7319 |
. . . . 5
β’ (π₯ = π β (β©π β π
βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯))) = (β©π β π
βπ‘ β (πβ{π})π£ = (π‘ + (π Β· π)))) |
24 | 23 | mpteq2dv 5211 |
. . . 4
β’ (π₯ = π β (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯)))) = (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π})π£ = (π‘ + (π Β· π))))) |
25 | | eqid 2733 |
. . . 4
β’ (π₯ β (π β { 0 }) β¦ (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯))))) = (π₯ β (π β { 0 }) β¦ (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯))))) |
26 | 24, 25 | fvmptg 6950 |
. . 3
β’ ((π β (π β { 0 }) β§ (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π})π£ = (π‘ + (π Β· π)))) β V) β ((π₯ β (π β { 0 }) β¦ (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯)))))βπ) = (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π})π£ = (π‘ + (π Β· π))))) |
27 | 14, 16, 26 | sylancl 587 |
. 2
β’ (π β ((π₯ β (π β { 0 }) β¦ (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π₯})π£ = (π‘ + (π Β· π₯)))))βπ) = (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π})π£ = (π‘ + (π Β· π))))) |
28 | 13, 27 | eqtrd 2773 |
1
β’ (π β (πβπ) = (π£ β π β¦ (β©π β π
βπ‘ β (πβ{π})π£ = (π‘ + (π Β· π))))) |