Step | Hyp | Ref
| Expression |
1 | | imasbas.u |
. . . 4
β’ (π β π = (πΉ βs π
)) |
2 | | imasbas.v |
. . . 4
β’ (π β π = (Baseβπ
)) |
3 | | eqid 2737 |
. . . 4
β’
(+gβπ
) = (+gβπ
) |
4 | | eqid 2737 |
. . . 4
β’
(.rβπ
) = (.rβπ
) |
5 | | eqid 2737 |
. . . 4
β’
(Scalarβπ
) =
(Scalarβπ
) |
6 | | eqid 2737 |
. . . 4
β’
(Baseβ(Scalarβπ
)) = (Baseβ(Scalarβπ
)) |
7 | | eqid 2737 |
. . . 4
β’ (
Β·π βπ
) = ( Β·π
βπ
) |
8 | | eqid 2737 |
. . . 4
β’
(Β·πβπ
) =
(Β·πβπ
) |
9 | | imastset.j |
. . . 4
β’ π½ = (TopOpenβπ
) |
10 | | eqid 2737 |
. . . 4
β’
(distβπ
) =
(distβπ
) |
11 | | eqid 2737 |
. . . 4
β’
(leβπ
) =
(leβπ
) |
12 | | imasbas.f |
. . . . 5
β’ (π β πΉ:πβontoβπ΅) |
13 | | imasbas.r |
. . . . 5
β’ (π β π
β π) |
14 | | eqid 2737 |
. . . . 5
β’
(+gβπ) = (+gβπ) |
15 | 1, 2, 12, 13, 3, 14 | imasplusg 17400 |
. . . 4
β’ (π β (+gβπ) = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}) |
16 | | eqid 2737 |
. . . . 5
β’
(.rβπ) = (.rβπ) |
17 | 1, 2, 12, 13, 4, 16 | imasmulr 17401 |
. . . 4
β’ (π β (.rβπ) = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}) |
18 | | eqid 2737 |
. . . . 5
β’ (
Β·π βπ) = ( Β·π
βπ) |
19 | 1, 2, 12, 13, 5, 6, 7, 18 | imasvsca 17403 |
. . . 4
β’ (π β (
Β·π βπ) = βͺ
π β π (π β (Baseβ(Scalarβπ
)), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)π)))) |
20 | | eqidd 2738 |
. . . 4
β’ (π β βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©} = βͺ
π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}) |
21 | | eqidd 2738 |
. . . 4
β’ (π β (π½ qTop πΉ) = (π½ qTop πΉ)) |
22 | | eqid 2737 |
. . . . 5
β’
(distβπ) =
(distβπ) |
23 | 1, 2, 12, 13, 10, 22 | imasds 17396 |
. . . 4
β’ (π β (distβπ) = (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π’ β β ran (π§ β {π€ β ((π Γ π) βm (1...π’)) β£ ((πΉβ(1st β(π€β1))) = π₯ β§ (πΉβ(2nd β(π€βπ’))) = π¦ β§ βπ£ β (1...(π’ β 1))(πΉβ(2nd β(π€βπ£))) = (πΉβ(1st β(π€β(π£ + 1)))))} β¦
(β*π Ξ£g
((distβπ
) β
π§))), β*,
< ))) |
24 | | eqidd 2738 |
. . . 4
β’ (π β ((πΉ β (leβπ
)) β β‘πΉ) = ((πΉ β (leβπ
)) β β‘πΉ)) |
25 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 15, 17, 19, 20, 21, 23, 24, 12, 13 | imasval 17394 |
. . 3
β’ (π β π = (({β¨(Baseβndx), π΅β©,
β¨(+gβndx), (+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©})) |
26 | 25 | fveq2d 6847 |
. 2
β’ (π β (TopSetβπ) =
(TopSetβ(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}))) |
27 | | imastset.o |
. 2
β’ π = (TopSetβπ) |
28 | | ovex 7391 |
. . 3
β’ (π½ qTop πΉ) β V |
29 | | eqid 2737 |
. . . . 5
β’
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) =
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©,
β¨(.rβndx), (.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π
β π βͺ π
β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
30 | 29 | imasvalstr 17334 |
. . . 4
β’
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) Struct
β¨1, ;12β© |
31 | | tsetid 17235 |
. . . 4
β’ TopSet =
Slot (TopSetβndx) |
32 | | snsstp1 4777 |
. . . . 5
β’
{β¨(TopSetβndx), (π½ qTop πΉ)β©} β {β¨(TopSetβndx),
(π½ qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©} |
33 | | ssun2 4134 |
. . . . 5
β’
{β¨(TopSetβndx), (π½ qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}
β (({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
34 | 32, 33 | sstri 3954 |
. . . 4
β’
{β¨(TopSetβndx), (π½ qTop πΉ)β©} β (({β¨(Baseβndx),
π΅β©,
β¨(+gβndx), (+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
35 | 30, 31, 34 | strfv 17077 |
. . 3
β’ ((π½ qTop πΉ) β V β (π½ qTop πΉ) = (TopSetβ(({β¨(Baseβndx),
π΅β©,
β¨(+gβndx), (+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}))) |
36 | 28, 35 | ax-mp 5 |
. 2
β’ (π½ qTop πΉ) = (TopSetβ(({β¨(Baseβndx),
π΅β©,
β¨(+gβndx), (+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) βͺ
{β¨(TopSetβndx), (π½
qTop πΉ)β©,
β¨(leβndx), ((πΉ
β (leβπ
)) β
β‘πΉ)β©, β¨(distβndx),
(distβπ)β©})) |
37 | 26, 27, 36 | 3eqtr4g 2802 |
1
β’ (π β π = (π½ qTop πΉ)) |