Step | Hyp | Ref
| Expression |
1 | | imassca.g |
. . . 4
β’ πΊ = (Scalarβπ
) |
2 | 1 | fvexi 6857 |
. . 3
β’ πΊ β V |
3 | | eqid 2737 |
. . . . 5
β’
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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), πΊβ©, β¨(
Β·π βndx), βͺ π
β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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©}) |
4 | 3 | imasvalstr 17334 |
. . . 4
β’
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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© |
5 | | scaid 17197 |
. . . 4
β’ Scalar =
Slot (Scalarβndx) |
6 | | snsstp1 4777 |
. . . . . 6
β’
{β¨(Scalarβndx), πΊβ©} β {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)π)))β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©} |
7 | | ssun2 4134 |
. . . . . 6
β’
{β¨(Scalarβndx), πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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), πΊβ©, β¨(
Β·π βndx), βͺ π
β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)π)))β©,
β¨(Β·πβndx), βͺ π
β π βͺ π
β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) |
8 | 6, 7 | sstri 3954 |
. . . . 5
β’
{β¨(Scalarβndx), πΊβ©} β ({β¨(Baseβndx),
π΅β©,
β¨(+gβndx), (+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)π)))β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π(Β·πβπ
)π)β©}β©}) |
9 | | ssun1 4133 |
. . . . 5
β’
({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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), πΊβ©, β¨(
Β·π βndx), βͺ π
β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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©}) |
10 | 8, 9 | sstri 3954 |
. . . 4
β’
{β¨(Scalarβndx), πΊβ©} β (({β¨(Baseβndx),
π΅β©,
β¨(+gβndx), (+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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©}) |
11 | 4, 5, 10 | strfv 17077 |
. . 3
β’ (πΊ β V β πΊ =
(Scalarβ(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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©}))) |
12 | 2, 11 | ax-mp 5 |
. 2
β’ πΊ =
(Scalarβ(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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©})) |
13 | | imasbas.u |
. . . 4
β’ (π β π = (πΉ βs π
)) |
14 | | imasbas.v |
. . . 4
β’ (π β π = (Baseβπ
)) |
15 | | eqid 2737 |
. . . 4
β’
(+gβπ
) = (+gβπ
) |
16 | | eqid 2737 |
. . . 4
β’
(.rβπ
) = (.rβπ
) |
17 | | eqid 2737 |
. . . 4
β’
(BaseβπΊ) =
(BaseβπΊ) |
18 | | eqid 2737 |
. . . 4
β’ (
Β·π βπ
) = ( Β·π
βπ
) |
19 | | eqid 2737 |
. . . 4
β’
(Β·πβπ
) =
(Β·πβπ
) |
20 | | eqid 2737 |
. . . 4
β’
(TopOpenβπ
) =
(TopOpenβπ
) |
21 | | eqid 2737 |
. . . 4
β’
(distβπ
) =
(distβπ
) |
22 | | eqid 2737 |
. . . 4
β’
(leβπ
) =
(leβπ
) |
23 | | imasbas.f |
. . . . 5
β’ (π β πΉ:πβontoβπ΅) |
24 | | imasbas.r |
. . . . 5
β’ (π β π
β π) |
25 | | eqid 2737 |
. . . . 5
β’
(+gβπ) = (+gβπ) |
26 | 13, 14, 23, 24, 15, 25 | 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©}) |
27 | | eqid 2737 |
. . . . 5
β’
(.rβπ) = (.rβπ) |
28 | 13, 14, 23, 24, 16, 27 | 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©}) |
29 | | eqidd 2738 |
. . . 4
β’ (π β βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)π))) = βͺ
π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)π)))) |
30 | | 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©}) |
31 | | eqidd 2738 |
. . . 4
β’ (π β ((TopOpenβπ
) qTop πΉ) = ((TopOpenβπ
) qTop πΉ)) |
32 | | eqid 2737 |
. . . . 5
β’
(distβπ) =
(distβπ) |
33 | 13, 14, 23, 24, 21, 32 | imasds 17396 |
. . . 4
β’ (π β (distβπ) = (π₯ β π΅, π¦ β π΅ β¦ inf(βͺ π β β ran (π β {β β ((π Γ π) βm (1...π)) β£ ((πΉβ(1st β(ββ1))) = π₯ β§ (πΉβ(2nd β(ββπ))) = π¦ β§ βπ β (1...(π β 1))(πΉβ(2nd β(ββπ))) = (πΉβ(1st β(ββ(π + 1)))))} β¦
(β*π Ξ£g
((distβπ
) β
π))), β*,
< ))) |
34 | | eqidd 2738 |
. . . 4
β’ (π β ((πΉ β (leβπ
)) β β‘πΉ) = ((πΉ β (leβπ
)) β β‘πΉ)) |
35 | 13, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 26, 28, 29, 30, 31, 33, 34, 23, 24 | imasval 17394 |
. . 3
β’ (π β π = (({β¨(Baseβndx), π΅β©,
β¨(+gβndx), (+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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 | 35 | fveq2d 6847 |
. 2
β’ (π β (Scalarβπ) =
(Scalarβ(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
(+gβπ)β©, β¨(.rβndx),
(.rβπ)β©} βͺ {β¨(Scalarβndx),
πΊβ©, β¨(
Β·π βndx), βͺ π β π (π β (BaseβπΊ), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)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©}))) |
37 | 12, 36 | eqtr4id 2796 |
1
β’ (π β πΊ = (Scalarβπ)) |