Step | Hyp | Ref
| Expression |
1 | | fvexd 6862 |
. . . 4
β’ (π€ = π β (Scalarβπ€) β V) |
2 | | fveq2 6847 |
. . . . 5
β’ (π€ = π β (Scalarβπ€) = (Scalarβπ)) |
3 | | isassa.f |
. . . . 5
β’ πΉ = (Scalarβπ) |
4 | 2, 3 | eqtr4di 2795 |
. . . 4
β’ (π€ = π β (Scalarβπ€) = πΉ) |
5 | | simpr 486 |
. . . . . 6
β’ ((π€ = π β§ π = πΉ) β π = πΉ) |
6 | 5 | eleq1d 2823 |
. . . . 5
β’ ((π€ = π β§ π = πΉ) β (π β CRing β πΉ β CRing)) |
7 | 5 | fveq2d 6851 |
. . . . . . 7
β’ ((π€ = π β§ π = πΉ) β (Baseβπ) = (BaseβπΉ)) |
8 | | isassa.b |
. . . . . . 7
β’ π΅ = (BaseβπΉ) |
9 | 7, 8 | eqtr4di 2795 |
. . . . . 6
β’ ((π€ = π β§ π = πΉ) β (Baseβπ) = π΅) |
10 | | fveq2 6847 |
. . . . . . . . 9
β’ (π€ = π β (Baseβπ€) = (Baseβπ)) |
11 | | isassa.v |
. . . . . . . . 9
β’ π = (Baseβπ) |
12 | 10, 11 | eqtr4di 2795 |
. . . . . . . 8
β’ (π€ = π β (Baseβπ€) = π) |
13 | | fvexd 6862 |
. . . . . . . . . 10
β’ (π€ = π β (
Β·π βπ€) β V) |
14 | | fvexd 6862 |
. . . . . . . . . . 11
β’ ((π€ = π β§ π = ( Β·π
βπ€)) β
(.rβπ€)
β V) |
15 | | simpr 486 |
. . . . . . . . . . . . . . 15
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β π‘ = (.rβπ€)) |
16 | | fveq2 6847 |
. . . . . . . . . . . . . . . . 17
β’ (π€ = π β (.rβπ€) = (.rβπ)) |
17 | 16 | ad2antrr 725 |
. . . . . . . . . . . . . . . 16
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β
(.rβπ€) =
(.rβπ)) |
18 | | isassa.t |
. . . . . . . . . . . . . . . 16
β’ Γ =
(.rβπ) |
19 | 17, 18 | eqtr4di 2795 |
. . . . . . . . . . . . . . 15
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β
(.rβπ€) =
Γ
) |
20 | 15, 19 | eqtrd 2777 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β π‘ = Γ ) |
21 | | simplr 768 |
. . . . . . . . . . . . . . . 16
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β π = ( Β·π
βπ€)) |
22 | | fveq2 6847 |
. . . . . . . . . . . . . . . . . 18
β’ (π€ = π β (
Β·π βπ€) = ( Β·π
βπ)) |
23 | 22 | ad2antrr 725 |
. . . . . . . . . . . . . . . . 17
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β (
Β·π βπ€) = ( Β·π
βπ)) |
24 | | isassa.s |
. . . . . . . . . . . . . . . . 17
β’ Β· = (
Β·π βπ) |
25 | 23, 24 | eqtr4di 2795 |
. . . . . . . . . . . . . . . 16
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β (
Β·π βπ€) = Β· ) |
26 | 21, 25 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β π = Β· ) |
27 | 26 | oveqd 7379 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β (ππ π₯) = (π Β· π₯)) |
28 | | eqidd 2738 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β π¦ = π¦) |
29 | 20, 27, 28 | oveq123d 7383 |
. . . . . . . . . . . . 13
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β ((ππ π₯)π‘π¦) = ((π Β· π₯) Γ π¦)) |
30 | | eqidd 2738 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β π = π) |
31 | 20 | oveqd 7379 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β (π₯π‘π¦) = (π₯ Γ π¦)) |
32 | 26, 30, 31 | oveq123d 7383 |
. . . . . . . . . . . . 13
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β (ππ (π₯π‘π¦)) = (π Β· (π₯ Γ π¦))) |
33 | 29, 32 | eqeq12d 2753 |
. . . . . . . . . . . 12
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β (((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β ((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)))) |
34 | | eqidd 2738 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β π₯ = π₯) |
35 | 26 | oveqd 7379 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β (ππ π¦) = (π Β· π¦)) |
36 | 20, 34, 35 | oveq123d 7383 |
. . . . . . . . . . . . 13
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β (π₯π‘(ππ π¦)) = (π₯ Γ (π Β· π¦))) |
37 | 36, 32 | eqeq12d 2753 |
. . . . . . . . . . . 12
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β ((π₯π‘(ππ π¦)) = (ππ (π₯π‘π¦)) β (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦)))) |
38 | 33, 37 | anbi12d 632 |
. . . . . . . . . . 11
β’ (((π€ = π β§ π = ( Β·π
βπ€)) β§ π‘ = (.rβπ€)) β ((((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘π¦))) β (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦))))) |
39 | 14, 38 | sbcied 3789 |
. . . . . . . . . 10
β’ ((π€ = π β§ π = ( Β·π
βπ€)) β
([(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘π¦))) β (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦))))) |
40 | 13, 39 | sbcied 3789 |
. . . . . . . . 9
β’ (π€ = π β ([(
Β·π βπ€) / π ][(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘π¦))) β (((π Β· π₯) Γ π¦) = (π Β· (π₯ Γ π¦)) β§ (π₯ Γ (π Β· π¦)) = (π Β· (π₯ Γ π¦))))) |
41 | 12, 40 | raleqbidv 3322 |
. . . . . . . 8
β’ (π€ = π β (βπ¦ β (Baseβπ€)[(
Β·π βπ€) / π ][(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘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¦))))) |
42 | 12, 41 | raleqbidv 3322 |
. . . . . . 7
β’ (π€ = π β (βπ₯ β (Baseβπ€)βπ¦ β (Baseβπ€)[(
Β·π βπ€) / π ][(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘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¦))))) |
43 | 42 | adantr 482 |
. . . . . 6
β’ ((π€ = π β§ π = πΉ) β (βπ₯ β (Baseβπ€)βπ¦ β (Baseβπ€)[(
Β·π βπ€) / π ][(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘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¦))))) |
44 | 9, 43 | raleqbidv 3322 |
. . . . 5
β’ ((π€ = π β§ π = πΉ) β (βπ β (Baseβπ)βπ₯ β (Baseβπ€)βπ¦ β (Baseβπ€)[(
Β·π βπ€) / π ][(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘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¦))))) |
45 | 6, 44 | anbi12d 632 |
. . . 4
β’ ((π€ = π β§ π = πΉ) β ((π β CRing β§ βπ β (Baseβπ)βπ₯ β (Baseβπ€)βπ¦ β (Baseβπ€)[(
Β·π βπ€) / π ][(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘π¦)))) β (πΉ β CRing 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¦)))))) |
46 | 1, 4, 45 | sbcied2 3791 |
. . 3
β’ (π€ = π β ([(Scalarβπ€) / π](π β CRing β§ βπ β (Baseβπ)βπ₯ β (Baseβπ€)βπ¦ β (Baseβπ€)[(
Β·π βπ€) / π ][(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘π¦)))) β (πΉ β CRing 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¦)))))) |
47 | | df-assa 21275 |
. . 3
β’ AssAlg =
{π€ β (LMod β© Ring)
β£ [(Scalarβπ€) / π](π β CRing β§ βπ β (Baseβπ)βπ₯ β (Baseβπ€)βπ¦ β (Baseβπ€)[(
Β·π βπ€) / π ][(.rβπ€) / π‘](((ππ π₯)π‘π¦) = (ππ (π₯π‘π¦)) β§ (π₯π‘(ππ π¦)) = (ππ (π₯π‘π¦))))} |
48 | 46, 47 | elrab2 3653 |
. 2
β’ (π β AssAlg β (π β (LMod β© Ring) β§
(πΉ β CRing 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¦)))))) |
49 | | anass 470 |
. 2
β’ (((π β (LMod β© Ring) β§
πΉ β CRing) 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 (π β (LMod β© Ring) β§ (πΉ β CRing 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¦)))))) |
50 | | elin 3931 |
. . . . 5
β’ (π β (LMod β© Ring) β
(π β LMod β§ π β Ring)) |
51 | 50 | anbi1i 625 |
. . . 4
β’ ((π β (LMod β© Ring) β§
πΉ β CRing) β
((π β LMod β§ π β Ring) β§ πΉ β CRing)) |
52 | | df-3an 1090 |
. . . 4
β’ ((π β LMod β§ π β Ring β§ πΉ β CRing) β ((π β LMod β§ π β Ring) β§ πΉ β CRing)) |
53 | 51, 52 | bitr4i 278 |
. . 3
β’ ((π β (LMod β© Ring) β§
πΉ β CRing) β
(π β LMod β§ π β Ring β§ πΉ β CRing)) |
54 | 53 | anbi1i 625 |
. 2
β’ (((π β (LMod β© Ring) β§
πΉ β CRing) 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 ((π β LMod β§ π β Ring β§ πΉ β CRing) 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¦))))) |
55 | 48, 49, 54 | 3bitr2i 299 |
1
β’ (π β AssAlg β ((π β LMod β§ π β Ring β§ πΉ β CRing) 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¦))))) |