Step | Hyp | Ref
| Expression |
1 | | cslmd 32340 |
. 2
class
SLMod |
2 | | vf |
. . . . . . . . . . . . 13
setvar π |
3 | 2 | cv 1540 |
. . . . . . . . . . . 12
class π |
4 | | csrg 20008 |
. . . . . . . . . . . 12
class
SRing |
5 | 3, 4 | wcel 2106 |
. . . . . . . . . . 11
wff π β SRing |
6 | | vr |
. . . . . . . . . . . . . . . . . . . 20
setvar π |
7 | 6 | cv 1540 |
. . . . . . . . . . . . . . . . . . 19
class π |
8 | | vw |
. . . . . . . . . . . . . . . . . . . 20
setvar π€ |
9 | 8 | cv 1540 |
. . . . . . . . . . . . . . . . . . 19
class π€ |
10 | | vs |
. . . . . . . . . . . . . . . . . . . 20
setvar π |
11 | 10 | cv 1540 |
. . . . . . . . . . . . . . . . . . 19
class π |
12 | 7, 9, 11 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class (ππ π€) |
13 | | vv |
. . . . . . . . . . . . . . . . . . 19
setvar π£ |
14 | 13 | cv 1540 |
. . . . . . . . . . . . . . . . . 18
class π£ |
15 | 12, 14 | wcel 2106 |
. . . . . . . . . . . . . . . . 17
wff (ππ π€) β π£ |
16 | | vx |
. . . . . . . . . . . . . . . . . . . . 21
setvar π₯ |
17 | 16 | cv 1540 |
. . . . . . . . . . . . . . . . . . . 20
class π₯ |
18 | | va |
. . . . . . . . . . . . . . . . . . . . 21
setvar π |
19 | 18 | cv 1540 |
. . . . . . . . . . . . . . . . . . . 20
class π |
20 | 9, 17, 19 | co 7408 |
. . . . . . . . . . . . . . . . . . 19
class (π€ππ₯) |
21 | 7, 20, 11 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class (ππ (π€ππ₯)) |
22 | 7, 17, 11 | co 7408 |
. . . . . . . . . . . . . . . . . . 19
class (ππ π₯) |
23 | 12, 22, 19 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class ((ππ π€)π(ππ π₯)) |
24 | 21, 23 | wceq 1541 |
. . . . . . . . . . . . . . . . 17
wff (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) |
25 | | vq |
. . . . . . . . . . . . . . . . . . . . 21
setvar π |
26 | 25 | cv 1540 |
. . . . . . . . . . . . . . . . . . . 20
class π |
27 | | vp |
. . . . . . . . . . . . . . . . . . . . 21
setvar π |
28 | 27 | cv 1540 |
. . . . . . . . . . . . . . . . . . . 20
class π |
29 | 26, 7, 28 | co 7408 |
. . . . . . . . . . . . . . . . . . 19
class (πππ) |
30 | 29, 9, 11 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class ((πππ)π π€) |
31 | 26, 9, 11 | co 7408 |
. . . . . . . . . . . . . . . . . . 19
class (ππ π€) |
32 | 31, 12, 19 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class ((ππ π€)π(ππ π€)) |
33 | 30, 32 | wceq 1541 |
. . . . . . . . . . . . . . . . 17
wff ((πππ)π π€) = ((ππ π€)π(ππ π€)) |
34 | 15, 24, 33 | w3a 1087 |
. . . . . . . . . . . . . . . 16
wff ((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) |
35 | | vt |
. . . . . . . . . . . . . . . . . . . . 21
setvar π‘ |
36 | 35 | cv 1540 |
. . . . . . . . . . . . . . . . . . . 20
class π‘ |
37 | 26, 7, 36 | co 7408 |
. . . . . . . . . . . . . . . . . . 19
class (ππ‘π) |
38 | 37, 9, 11 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class ((ππ‘π)π π€) |
39 | 26, 12, 11 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class (ππ (ππ π€)) |
40 | 38, 39 | wceq 1541 |
. . . . . . . . . . . . . . . . 17
wff ((ππ‘π)π π€) = (ππ (ππ π€)) |
41 | | cur 20003 |
. . . . . . . . . . . . . . . . . . . 20
class
1r |
42 | 3, 41 | cfv 6543 |
. . . . . . . . . . . . . . . . . . 19
class
(1rβπ) |
43 | 42, 9, 11 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class
((1rβπ)π π€) |
44 | 43, 9 | wceq 1541 |
. . . . . . . . . . . . . . . . 17
wff
((1rβπ)π π€) = π€ |
45 | | c0g 17384 |
. . . . . . . . . . . . . . . . . . . 20
class
0g |
46 | 3, 45 | cfv 6543 |
. . . . . . . . . . . . . . . . . . 19
class
(0gβπ) |
47 | 46, 9, 11 | co 7408 |
. . . . . . . . . . . . . . . . . 18
class
((0gβπ)π π€) |
48 | | vg |
. . . . . . . . . . . . . . . . . . . 20
setvar π |
49 | 48 | cv 1540 |
. . . . . . . . . . . . . . . . . . 19
class π |
50 | 49, 45 | cfv 6543 |
. . . . . . . . . . . . . . . . . 18
class
(0gβπ) |
51 | 47, 50 | wceq 1541 |
. . . . . . . . . . . . . . . . 17
wff
((0gβπ)π π€) = (0gβπ) |
52 | 40, 44, 51 | w3a 1087 |
. . . . . . . . . . . . . . . 16
wff (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)) |
53 | 34, 52 | wa 396 |
. . . . . . . . . . . . . . 15
wff (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ))) |
54 | 53, 8, 14 | wral 3061 |
. . . . . . . . . . . . . 14
wff
βπ€ β
π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ))) |
55 | 54, 16, 14 | wral 3061 |
. . . . . . . . . . . . 13
wff
βπ₯ β
π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ))) |
56 | | vk |
. . . . . . . . . . . . . 14
setvar π |
57 | 56 | cv 1540 |
. . . . . . . . . . . . 13
class π |
58 | 55, 6, 57 | wral 3061 |
. . . . . . . . . . . 12
wff
βπ β
π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ))) |
59 | 58, 25, 57 | wral 3061 |
. . . . . . . . . . 11
wff
βπ β
π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ))) |
60 | 5, 59 | wa 396 |
. . . . . . . . . 10
wff (π β SRing β§
βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)))) |
61 | | cmulr 17197 |
. . . . . . . . . . 11
class
.r |
62 | 3, 61 | cfv 6543 |
. . . . . . . . . 10
class
(.rβπ) |
63 | 60, 35, 62 | wsbc 3777 |
. . . . . . . . 9
wff
[(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)))) |
64 | | cplusg 17196 |
. . . . . . . . . 10
class
+g |
65 | 3, 64 | cfv 6543 |
. . . . . . . . 9
class
(+gβπ) |
66 | 63, 27, 65 | wsbc 3777 |
. . . . . . . 8
wff
[(+gβπ) / π][(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)))) |
67 | | cbs 17143 |
. . . . . . . . 9
class
Base |
68 | 3, 67 | cfv 6543 |
. . . . . . . 8
class
(Baseβπ) |
69 | 66, 56, 68 | wsbc 3777 |
. . . . . . 7
wff
[(Baseβπ) / π][(+gβπ) / π][(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)))) |
70 | | csca 17199 |
. . . . . . . 8
class
Scalar |
71 | 49, 70 | cfv 6543 |
. . . . . . 7
class
(Scalarβπ) |
72 | 69, 2, 71 | wsbc 3777 |
. . . . . 6
wff
[(Scalarβπ) / π][(Baseβπ) / π][(+gβπ) / π][(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)))) |
73 | | cvsca 17200 |
. . . . . . 7
class
Β·π |
74 | 49, 73 | cfv 6543 |
. . . . . 6
class (
Β·π βπ) |
75 | 72, 10, 74 | wsbc 3777 |
. . . . 5
wff [(
Β·π βπ) / π ][(Scalarβπ) / π][(Baseβπ) / π][(+gβπ) / π][(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)))) |
76 | 49, 64 | cfv 6543 |
. . . . 5
class
(+gβπ) |
77 | 75, 18, 76 | wsbc 3777 |
. . . 4
wff
[(+gβπ) / π][(
Β·π βπ) / π ][(Scalarβπ) / π][(Baseβπ) / π][(+gβπ) / π][(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)))) |
78 | 49, 67 | cfv 6543 |
. . . 4
class
(Baseβπ) |
79 | 77, 13, 78 | wsbc 3777 |
. . 3
wff
[(Baseβπ) / π£][(+gβπ) / π][(
Β·π βπ) / π ][(Scalarβπ) / π][(Baseβπ) / π][(+gβπ) / π][(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ)))) |
80 | | ccmn 19647 |
. . 3
class
CMnd |
81 | 79, 48, 80 | crab 3432 |
. 2
class {π β CMnd β£
[(Baseβπ) /
π£][(+gβπ) / π][(
Β·π βπ) / π ][(Scalarβπ) / π][(Baseβπ) / π][(+gβπ) / π][(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ))))} |
82 | 1, 81 | wceq 1541 |
1
wff SLMod =
{π β CMnd β£
[(Baseβπ) /
π£][(+gβπ) / π][(
Β·π βπ) / π ][(Scalarβπ) / π][(Baseβπ) / π][(+gβπ) / π][(.rβπ) / π‘](π β SRing β§ βπ β π βπ β π βπ₯ β π£ βπ€ β π£ (((ππ π€) β π£ β§ (ππ (π€ππ₯)) = ((ππ π€)π(ππ π₯)) β§ ((πππ)π π€) = ((ππ π€)π(ππ π€))) β§ (((ππ‘π)π π€) = (ππ (ππ π€)) β§ ((1rβπ)π π€) = π€ β§ ((0gβπ)π π€) = (0gβπ))))} |