Step | Hyp | Ref
| Expression |
1 | | isassad.1 |
. . 3
β’ (π β π β LMod) |
2 | | isassad.2 |
. . 3
β’ (π β π β Ring) |
3 | 1, 2 | jca 513 |
. 2
β’ (π β (π β LMod β§ π β Ring)) |
4 | | isassad.4 |
. . . . 5
β’ ((π β§ (π β π΅ β§ π₯ β π β§ π¦ β π)) β ((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦))) |
5 | | isassad.5 |
. . . . 5
β’ ((π β§ (π β π΅ β§ π₯ β π β§ π¦ β π)) β (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦))) |
6 | 4, 5 | jca 513 |
. . . 4
β’ ((π β§ (π β π΅ β§ π₯ β π β§ π¦ β π)) β (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦)))) |
7 | 6 | ralrimivvva 3204 |
. . 3
β’ (π β βπ β π΅ βπ₯ β π βπ¦ β π (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦)))) |
8 | | isassad.b |
. . . . 5
β’ (π β π΅ = (BaseβπΉ)) |
9 | | isassad.f |
. . . . . 6
β’ (π β πΉ = (Scalarβπ)) |
10 | 9 | fveq2d 6896 |
. . . . 5
β’ (π β (BaseβπΉ) =
(Baseβ(Scalarβπ))) |
11 | 8, 10 | eqtrd 2773 |
. . . 4
β’ (π β π΅ = (Baseβ(Scalarβπ))) |
12 | | isassad.v |
. . . . 5
β’ (π β π = (Baseβπ)) |
13 | | isassad.t |
. . . . . . . . 9
β’ (π β Γ =
(.rβπ)) |
14 | | isassad.s |
. . . . . . . . . 10
β’ (π β Β· = (
Β·π βπ)) |
15 | 14 | oveqd 7426 |
. . . . . . . . 9
β’ (π β (π Β· π₯) = (π( Β·π
βπ)π₯)) |
16 | | eqidd 2734 |
. . . . . . . . 9
β’ (π β π¦ = π¦) |
17 | 13, 15, 16 | oveq123d 7430 |
. . . . . . . 8
β’ (π β ((π Β· π₯) Γ π¦) = ((π( Β·π
βπ)π₯)(.rβπ)π¦)) |
18 | | eqidd 2734 |
. . . . . . . . 9
β’ (π β π = π) |
19 | 13 | oveqd 7426 |
. . . . . . . . 9
β’ (π β (π₯ Γ π¦) = (π₯(.rβπ)π¦)) |
20 | 14, 18, 19 | oveq123d 7430 |
. . . . . . . 8
β’ (π β (π Β· (π₯ Γ π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦))) |
21 | 17, 20 | eqeq12d 2749 |
. . . . . . 7
β’ (π β (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β ((π( Β·π
βπ)π₯)(.rβπ)π¦) = (π( Β·π
βπ)(π₯(.rβπ)π¦)))) |
22 | | eqidd 2734 |
. . . . . . . . 9
β’ (π β π₯ = π₯) |
23 | 14 | oveqd 7426 |
. . . . . . . . 9
β’ (π β (π Β· π¦) = (π( Β·π
βπ)π¦)) |
24 | 13, 22, 23 | oveq123d 7430 |
. . . . . . . 8
β’ (π β (π₯ Γ (π Β· π¦)) = (π₯(.rβπ)(π( Β·π
βπ)π¦))) |
25 | 24, 20 | eqeq12d 2749 |
. . . . . . 7
β’ (π β ((π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦)) β (π₯(.rβπ)(π( Β·π
βπ)π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦)))) |
26 | 21, 25 | anbi12d 632 |
. . . . . 6
β’ (π β ((((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦))) β (((π( Β·π
βπ)π₯)(.rβπ)π¦) = (π( Β·π
βπ)(π₯(.rβπ)π¦)) β§ (π₯(.rβπ)(π( Β·π
βπ)π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦))))) |
27 | 12, 26 | raleqbidv 3343 |
. . . . 5
β’ (π β (βπ¦ β π (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦))) β βπ¦ β (Baseβπ)(((π( Β·π
βπ)π₯)(.rβπ)π¦) = (π( Β·π
βπ)(π₯(.rβπ)π¦)) β§ (π₯(.rβπ)(π( Β·π
βπ)π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦))))) |
28 | 12, 27 | raleqbidv 3343 |
. . . 4
β’ (π β (βπ₯ β π βπ¦ β π (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦))) β βπ₯ β (Baseβπ)βπ¦ β (Baseβπ)(((π( Β·π
βπ)π₯)(.rβπ)π¦) = (π( Β·π
βπ)(π₯(.rβπ)π¦)) β§ (π₯(.rβπ)(π( Β·π
βπ)π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦))))) |
29 | 11, 28 | raleqbidv 3343 |
. . 3
β’ (π β (βπ β π΅ βπ₯ β π βπ¦ β π (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦))) β βπ β (Baseβ(Scalarβπ))βπ₯ β (Baseβπ)βπ¦ β (Baseβπ)(((π( Β·π
βπ)π₯)(.rβπ)π¦) = (π( Β·π
βπ)(π₯(.rβπ)π¦)) β§ (π₯(.rβπ)(π( Β·π
βπ)π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦))))) |
30 | 7, 29 | mpbid 231 |
. 2
β’ (π β βπ β (Baseβ(Scalarβπ))βπ₯ β (Baseβπ)βπ¦ β (Baseβπ)(((π( Β·π
βπ)π₯)(.rβπ)π¦) = (π( Β·π
βπ)(π₯(.rβπ)π¦)) β§ (π₯(.rβπ)(π( Β·π
βπ)π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦)))) |
31 | | eqid 2733 |
. . 3
β’
(Baseβπ) =
(Baseβπ) |
32 | | eqid 2733 |
. . 3
β’
(Scalarβπ) =
(Scalarβπ) |
33 | | eqid 2733 |
. . 3
β’
(Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) |
34 | | eqid 2733 |
. . 3
β’ (
Β·π βπ) = ( Β·π
βπ) |
35 | | eqid 2733 |
. . 3
β’
(.rβπ) = (.rβπ) |
36 | 31, 32, 33, 34, 35 | isassa 21411 |
. 2
β’ (π β AssAlg β ((π β LMod β§ π β Ring) β§
βπ β
(Baseβ(Scalarβπ))βπ₯ β (Baseβπ)βπ¦ β (Baseβπ)(((π( Β·π
βπ)π₯)(.rβπ)π¦) = (π( Β·π
βπ)(π₯(.rβπ)π¦)) β§ (π₯(.rβπ)(π( Β·π
βπ)π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦))))) |
37 | 3, 30, 36 | sylanbrc 584 |
1
β’ (π β π β AssAlg) |