Step | Hyp | Ref
| Expression |
1 | | imasbas.u |
. . 3
β’ (π β π = (πΉ βs π
)) |
2 | | imasbas.v |
. . 3
β’ (π β π = (Baseβπ
)) |
3 | | eqid 2737 |
. . 3
β’
(+gβπ
) = (+gβπ
) |
4 | | eqid 2737 |
. . 3
β’
(.rβπ
) = (.rβπ
) |
5 | | eqid 2737 |
. . 3
β’
(Scalarβπ
) =
(Scalarβπ
) |
6 | | imasvsca.k |
. . . 4
β’ πΎ = (BaseβπΊ) |
7 | | imassca.g |
. . . . 5
β’ πΊ = (Scalarβπ
) |
8 | 7 | fveq2i 6846 |
. . . 4
β’
(BaseβπΊ) =
(Baseβ(Scalarβπ
)) |
9 | 6, 8 | eqtri 2765 |
. . 3
β’ πΎ =
(Baseβ(Scalarβπ
)) |
10 | | imasvsca.q |
. . 3
β’ Β· = (
Β·π βπ
) |
11 | | eqid 2737 |
. . 3
β’
(Β·πβπ
) =
(Β·πβπ
) |
12 | | eqid 2737 |
. . 3
β’
(TopOpenβπ
) =
(TopOpenβπ
) |
13 | | eqid 2737 |
. . 3
β’
(distβπ
) =
(distβπ
) |
14 | | eqid 2737 |
. . 3
β’
(leβπ
) =
(leβπ
) |
15 | | imasbas.f |
. . . 4
β’ (π β πΉ:πβontoβπ΅) |
16 | | imasbas.r |
. . . 4
β’ (π β π
β π) |
17 | | eqid 2737 |
. . . 4
β’
(+gβπ) = (+gβπ) |
18 | 1, 2, 15, 16, 3, 17 | imasplusg 17400 |
. . 3
β’ (π β (+gβπ) = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}) |
19 | | eqid 2737 |
. . . 4
β’
(.rβπ) = (.rβπ) |
20 | 1, 2, 15, 16, 4, 19 | imasmulr 17401 |
. . 3
β’ (π β (.rβπ) = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}) |
21 | | eqidd 2738 |
. . 3
β’ (π β βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π))) = βͺ
π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))) |
22 | | eqidd 2738 |
. . 3
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©}) |
23 | | eqidd 2738 |
. . 3
β’ (π β ((TopOpenβπ
) qTop πΉ) = ((TopOpenβπ
) qTop πΉ)) |
24 | | eqid 2737 |
. . . 4
β’
(distβπ) =
(distβπ) |
25 | 1, 2, 15, 16, 13, 24 | imasds 17396 |
. . 3
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*,
< ))) |
26 | | eqidd 2738 |
. . 3
β’ (π β ((πΉ β (leβπ
)) β β‘πΉ) = ((πΉ β (leβπ
)) β β‘πΉ)) |
27 | 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 18, 20, 21, 22, 23, 25, 26, 15, 16 | imasval 17394 |
. 2
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), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))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), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©})) |
28 | | eqid 2737 |
. . 3
β’
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))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), ((TopOpenβπ
) 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), βͺ π
β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))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), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
29 | 28 | imasvalstr 17334 |
. 2
β’
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))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), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) Struct
β¨1, ;12β© |
30 | | vscaid 17202 |
. 2
β’
Β·π = Slot (
Β·π βndx) |
31 | | snsstp2 4778 |
. . 3
β’ {β¨(
Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))β©} β {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©} |
32 | | ssun2 4134 |
. . . 4
β’
{β¨(Scalarβndx), (Scalarβπ
)β©, β¨(
Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©} β
({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©,
β¨(.rβndx), (.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), βͺ π
β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))β©,
β¨(Β·πβndx), βͺ π
β π βͺ π
β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) |
33 | | ssun1 4133 |
. . . 4
β’
({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) β
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©,
β¨(.rβndx), (.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), βͺ π
β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))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), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
34 | 32, 33 | sstri 3954 |
. . 3
β’
{β¨(Scalarβndx), (Scalarβπ
)β©, β¨(
Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©} β
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©,
β¨(.rβndx), (.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), βͺ π
β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))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), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
35 | 31, 34 | sstri 3954 |
. 2
β’ {β¨(
Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))β©} β (({β¨(Baseβndx),
π΅β©,
β¨(+gβndx), (+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
(Scalarβπ
)β©,
β¨( Β·π βndx), βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))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), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
36 | | fvex 6856 |
. . . 4
β’
(Baseβπ
)
β V |
37 | 2, 36 | eqeltrdi 2846 |
. . 3
β’ (π β π β V) |
38 | 6 | fvexi 6857 |
. . . . 5
β’ πΎ β V |
39 | | snex 5389 |
. . . . 5
β’ {(πΉβπ)} β V |
40 | 38, 39 | mpoex 8013 |
. . . 4
β’ (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π))) β V |
41 | 40 | rgenw 3069 |
. . 3
β’
βπ β
π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π))) β V |
42 | | iunexg 7897 |
. . 3
β’ ((π β V β§ βπ β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π))) β V) β βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π))) β V) |
43 | 37, 41, 42 | sylancl 587 |
. 2
β’ (π β βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π))) β V) |
44 | | imasvsca.s |
. 2
β’ β = (
Β·π βπ) |
45 | 27, 29, 30, 35, 43, 44 | strfv3 17078 |
1
β’ (π β β = βͺ π β π (π β πΎ, π₯ β {(πΉβπ)} β¦ (πΉβ(π Β· π)))) |