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 | | imasip.i |
. . 3
β’ , =
(Β·πβπ
) |
9 | | eqid 2737 |
. . 3
β’
(TopOpenβπ
) =
(TopOpenβπ
) |
10 | | eqid 2737 |
. . 3
β’
(distβπ
) =
(distβπ
) |
11 | | eqid 2737 |
. . 3
β’
(leβπ
) =
(leβπ
) |
12 | | imasbas.f |
. . . 4
β’ (π β πΉ:πβontoβπ΅) |
13 | | imasbas.r |
. . . 4
β’ (π β π
β π) |
14 | | eqid 2737 |
. . . 4
β’
(+gβπ) = (+gβπ) |
15 | 1, 2, 12, 13, 3, 14 | 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©}) |
16 | | eqid 2737 |
. . . 4
β’
(.rβπ) = (.rβπ) |
17 | 1, 2, 12, 13, 4, 16 | 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©}) |
18 | | eqid 2737 |
. . . 4
β’ (
Β·π βπ) = ( Β·π
βπ) |
19 | 1, 2, 12, 13, 5, 6, 7, 18 | imasvsca 17403 |
. . 3
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 |
. . 3
β’ (π β βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}) |
21 | | eqidd 2738 |
. . 3
β’ (π β ((TopOpenβπ
) qTop πΉ) = ((TopOpenβπ
) qTop πΉ)) |
22 | | eqid 2737 |
. . . 4
β’
(distβπ) =
(distβπ) |
23 | 1, 2, 12, 13, 10, 22 | 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*,
< ))) |
24 | | eqidd 2738 |
. . 3
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 |
. 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), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}β©}) βͺ
{β¨(TopSetβndx), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©})) |
26 | | 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), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , 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), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}β©}) βͺ
{β¨(TopSetβndx), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
27 | 26 | 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), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , 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© |
28 | | ipid 17213 |
. 2
β’
Β·π = Slot
(Β·πβndx) |
29 | | snsstp3 4779 |
. . . 4
β’
{β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}β©} β
{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 , π)β©}β©} |
30 | | ssun2 4134 |
. . . 4
β’
{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 , 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), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}β©}) |
31 | 29, 30 | sstri 3954 |
. . 3
β’
{β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , 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), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}β©}) |
32 | | ssun1 4133 |
. . 3
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 , 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), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}β©}) βͺ
{β¨(TopSetβndx), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
33 | 31, 32 | sstri 3954 |
. 2
β’
{β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , 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), (
Β·π βπ)β©,
β¨(Β·πβndx), βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}β©}) βͺ
{β¨(TopSetβndx), ((TopOpenβπ
) qTop πΉ)β©, β¨(leβndx), ((πΉ β (leβπ
)) β β‘πΉ)β©, β¨(distβndx),
(distβπ)β©}) |
34 | | fvex 6856 |
. . . 4
β’
(Baseβπ
)
β V |
35 | 2, 34 | eqeltrdi 2846 |
. . 3
β’ (π β π β V) |
36 | | snex 5389 |
. . . . . 6
β’
{β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V |
37 | 36 | rgenw 3069 |
. . . . 5
β’
βπ β
π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V |
38 | | iunexg 7897 |
. . . . 5
β’ ((π β V β§ βπ β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V) β βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V) |
39 | 35, 37, 38 | sylancl 587 |
. . . 4
β’ (π β βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V) |
40 | 39 | ralrimivw 3148 |
. . 3
β’ (π β βπ β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V) |
41 | | iunexg 7897 |
. . 3
β’ ((π β V β§ βπ β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V) β βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V) |
42 | 35, 40, 41 | syl2anc 585 |
. 2
β’ (π β βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©} β V) |
43 | | imasip.w |
. 2
β’ πΌ =
(Β·πβπ) |
44 | 25, 27, 28, 33, 42, 43 | strfv3 17078 |
1
β’ (π β πΌ = βͺ π β π βͺ π β π {β¨β¨(πΉβπ), (πΉβπ)β©, (π , π)β©}) |