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 | | eqid 2737 |
. . 3
β’
(Baseβ(Scalarβπ
)) = (Baseβ(Scalarβπ
)) |
7 | | eqid 2737 |
. . 3
β’ (
Β·π βπ
) = ( Β·π
βπ
) |
8 | | eqid 2737 |
. . 3
β’
(Β·πβπ
) =
(Β·πβπ
) |
9 | | eqid 2737 |
. . 3
β’
(TopOpenβπ
) =
(TopOpenβπ
) |
10 | | imasds.e |
. . 3
β’ πΈ = (distβπ
) |
11 | | eqid 2737 |
. . 3
β’
(leβπ
) =
(leβπ
) |
12 | | eqidd 2738 |
. . 3
β’ (π β βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©} = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}) |
13 | | eqidd 2738 |
. . 3
β’ (π β βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©} = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}) |
14 | | eqidd 2738 |
. . 3
β’ (π β βͺ π β π (π β (Baseβ(Scalarβπ
)), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)π))) = βͺ
π β π (π β (Baseβ(Scalarβπ
)), π₯ β {(πΉβπ)} β¦ (πΉβ(π( Β·π
βπ
)π)))) |
15 | | 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©}) |
16 | | eqidd 2738 |
. . 3
β’ (π β ((TopOpenβπ
) qTop πΉ) = ((TopOpenβπ
) qTop πΉ)) |
17 | | eqidd 2738 |
. . 3
β’ (π β (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 (πΈ β π))), β*, < )) = (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 (πΈ β π))), β*, <
))) |
18 | | eqidd 2738 |
. . 3
β’ (π β ((πΉ β (leβπ
)) β β‘πΉ) = ((πΉ β (leβπ
)) β β‘πΉ)) |
19 | | imasbas.f |
. . 3
β’ (π β πΉ:πβontoβπ΅) |
20 | | imasbas.r |
. . 3
β’ (π β π
β π) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20 | imasval 17394 |
. 2
β’ (π β π = (({β¨(Baseβndx), π΅β©,
β¨(+gβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}β©,
β¨(.rβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}β©} βͺ
{β¨(Scalarβndx), (Scalarβπ
)β©, β¨(
Β·π βndx), βͺ π β π (π β (Baseβ(Scalarβπ
)), 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), (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 (πΈ β π))), β*, <
))β©})) |
22 | | eqid 2737 |
. . 3
β’
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}β©,
β¨(.rβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}β©} βͺ
{β¨(Scalarβndx), (Scalarβπ
)β©, β¨(
Β·π βndx), βͺ π β π (π β (Baseβ(Scalarβπ
)), 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), (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 (πΈ β π))), β*, < ))β©}) =
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}β©,
β¨(.rβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}β©} βͺ
{β¨(Scalarβndx), (Scalarβπ
)β©, β¨(
Β·π βndx), βͺ π
β π (π β (Baseβ(Scalarβπ
)), 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), (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 (πΈ β π))), β*, <
))β©}) |
23 | 22 | imasvalstr 17334 |
. 2
β’
(({β¨(Baseβndx), π΅β©, β¨(+gβndx),
βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}β©,
β¨(.rβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}β©} βͺ
{β¨(Scalarβndx), (Scalarβπ
)β©, β¨(
Β·π βndx), βͺ π β π (π β (Baseβ(Scalarβπ
)), 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), (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 (πΈ β π))), β*, < ))β©}) Struct
β¨1, ;12β© |
24 | | dsid 17268 |
. 2
β’ dist =
Slot (distβndx) |
25 | | snsstp3 4779 |
. . 3
β’
{β¨(distβndx), (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 (πΈ β π))), β*, < ))β©}
β {β¨(TopSetβndx), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx), (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 (πΈ β π))), β*, <
))β©} |
26 | | ssun2 4134 |
. . 3
β’
{β¨(TopSetβndx), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx), (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 (πΈ β π))), β*, < ))β©}
β (({β¨(Baseβndx), π΅β©, β¨(+gβndx),
βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}β©,
β¨(.rβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}β©} βͺ
{β¨(Scalarβndx), (Scalarβπ
)β©, β¨(
Β·π βndx), βͺ π β π (π β (Baseβ(Scalarβπ
)), 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), (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 (πΈ β π))), β*, <
))β©}) |
27 | 25, 26 | sstri 3954 |
. 2
β’
{β¨(distβndx), (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 (πΈ β π))), β*, < ))β©}
β (({β¨(Baseβndx), π΅β©, β¨(+gβndx),
βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(+gβπ
)π))β©}β©,
β¨(.rβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (πΉβ(π(.rβπ
)π))β©}β©} βͺ
{β¨(Scalarβndx), (Scalarβπ
)β©, β¨(
Β·π βndx), βͺ π β π (π β (Baseβ(Scalarβπ
)), 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), (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 (πΈ β π))), β*, <
))β©}) |
28 | | fvex 6856 |
. . . . 5
β’
(Baseβπ
)
β V |
29 | 2, 28 | eqeltrdi 2846 |
. . . 4
β’ (π β π β V) |
30 | | focdmex 7889 |
. . . 4
β’ (π β V β (πΉ:πβontoβπ΅ β π΅ β V)) |
31 | 29, 19, 30 | sylc 65 |
. . 3
β’ (π β π΅ β V) |
32 | | mpoexga 8011 |
. . 3
β’ ((π΅ β V β§ π΅ β V) β (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 (πΈ β π))), β*, < )) β
V) |
33 | 31, 31, 32 | syl2anc 585 |
. 2
β’ (π β (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 (πΈ β π))), β*, < )) β
V) |
34 | | imasds.d |
. 2
β’ π· = (distβπ) |
35 | 21, 23, 24, 27, 33, 34 | strfv3 17078 |
1
β’ (π β π· = (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 (πΈ β π))), β*, <
))) |