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 | | isassad.f |
. . . 4
β’ (π β πΉ = (Scalarβπ)) |
4 | | isassad.3 |
. . . 4
β’ (π β πΉ β CRing) |
5 | 3, 4 | eqeltrrd 2835 |
. . 3
β’ (π β (Scalarβπ) β CRing) |
6 | 1, 2, 5 | 3jca 1129 |
. 2
β’ (π β (π β LMod β§ π β Ring β§ (Scalarβπ) β
CRing)) |
7 | | 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¦))) |
8 | | 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¦))) |
9 | 7, 8 | 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¦)))) |
10 | 9 | ralrimivvva 3197 |
. . 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¦)))) |
11 | | isassad.b |
. . . . 5
β’ (π β π΅ = (BaseβπΉ)) |
12 | 3 | fveq2d 6847 |
. . . . 5
β’ (π β (BaseβπΉ) =
(Baseβ(Scalarβπ))) |
13 | 11, 12 | eqtrd 2773 |
. . . 4
β’ (π β π΅ = (Baseβ(Scalarβπ))) |
14 | | isassad.v |
. . . . 5
β’ (π β π = (Baseβπ)) |
15 | | isassad.t |
. . . . . . . . 9
β’ (π β Γ =
(.rβπ)) |
16 | | isassad.s |
. . . . . . . . . 10
β’ (π β Β· = (
Β·π βπ)) |
17 | 16 | oveqd 7375 |
. . . . . . . . 9
β’ (π β (π Β· π₯) = (π( Β·π
βπ)π₯)) |
18 | | eqidd 2734 |
. . . . . . . . 9
β’ (π β π¦ = π¦) |
19 | 15, 17, 18 | oveq123d 7379 |
. . . . . . . 8
β’ (π β ((π Β· π₯) Γ π¦) = ((π( Β·π
βπ)π₯)(.rβπ)π¦)) |
20 | | eqidd 2734 |
. . . . . . . . 9
β’ (π β π = π) |
21 | 15 | oveqd 7375 |
. . . . . . . . 9
β’ (π β (π₯ Γ π¦) = (π₯(.rβπ)π¦)) |
22 | 16, 20, 21 | oveq123d 7379 |
. . . . . . . 8
β’ (π β (π Β· (π₯ Γ π¦)) = (π( Β·π
βπ)(π₯(.rβπ)π¦))) |
23 | 19, 22 | 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¦)))) |
24 | | eqidd 2734 |
. . . . . . . . 9
β’ (π β π₯ = π₯) |
25 | 16 | oveqd 7375 |
. . . . . . . . 9
β’ (π β (π Β· π¦) = (π( Β·π
βπ)π¦)) |
26 | 15, 24, 25 | oveq123d 7379 |
. . . . . . . 8
β’ (π β (π₯ Γ (π Β· π¦)) = (π₯(.rβπ)(π( Β·π
βπ)π¦))) |
27 | 26, 22 | 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¦)))) |
28 | 23, 27 | 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¦))))) |
29 | 14, 28 | raleqbidv 3318 |
. . . . 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¦))))) |
30 | 14, 29 | raleqbidv 3318 |
. . . 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¦))))) |
31 | 13, 30 | raleqbidv 3318 |
. . 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¦))))) |
32 | 10, 31 | 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¦)))) |
33 | | eqid 2733 |
. . 3
β’
(Baseβπ) =
(Baseβπ) |
34 | | eqid 2733 |
. . 3
β’
(Scalarβπ) =
(Scalarβπ) |
35 | | eqid 2733 |
. . 3
β’
(Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) |
36 | | eqid 2733 |
. . 3
β’ (
Β·π βπ) = ( Β·π
βπ) |
37 | | eqid 2733 |
. . 3
β’
(.rβπ) = (.rβπ) |
38 | 33, 34, 35, 36, 37 | isassa 21278 |
. 2
β’ (π β AssAlg β ((π β LMod β§ π β Ring β§
(Scalarβπ) β
CRing) 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¦))))) |
39 | 6, 32, 38 | sylanbrc 584 |
1
β’ (π β π β AssAlg) |