Detailed syntax breakdown of Definition df-slmd
Step | Hyp | Ref
| Expression |
1 | | cslmd 31172 |
. 2
class
SLMod |
2 | | vf |
. . . . . . . . . . . . 13
setvar 𝑓 |
3 | 2 | cv 1542 |
. . . . . . . . . . . 12
class 𝑓 |
4 | | csrg 19520 |
. . . . . . . . . . . 12
class
SRing |
5 | 3, 4 | wcel 2110 |
. . . . . . . . . . 11
wff 𝑓 ∈ SRing |
6 | | vr |
. . . . . . . . . . . . . . . . . . . 20
setvar 𝑟 |
7 | 6 | cv 1542 |
. . . . . . . . . . . . . . . . . . 19
class 𝑟 |
8 | | vw |
. . . . . . . . . . . . . . . . . . . 20
setvar 𝑤 |
9 | 8 | cv 1542 |
. . . . . . . . . . . . . . . . . . 19
class 𝑤 |
10 | | vs |
. . . . . . . . . . . . . . . . . . . 20
setvar 𝑠 |
11 | 10 | cv 1542 |
. . . . . . . . . . . . . . . . . . 19
class 𝑠 |
12 | 7, 9, 11 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class (𝑟𝑠𝑤) |
13 | | vv |
. . . . . . . . . . . . . . . . . . 19
setvar 𝑣 |
14 | 13 | cv 1542 |
. . . . . . . . . . . . . . . . . 18
class 𝑣 |
15 | 12, 14 | wcel 2110 |
. . . . . . . . . . . . . . . . 17
wff (𝑟𝑠𝑤) ∈ 𝑣 |
16 | | vx |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑥 |
17 | 16 | cv 1542 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑥 |
18 | | va |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑎 |
19 | 18 | cv 1542 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑎 |
20 | 9, 17, 19 | co 7213 |
. . . . . . . . . . . . . . . . . . 19
class (𝑤𝑎𝑥) |
21 | 7, 20, 11 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class (𝑟𝑠(𝑤𝑎𝑥)) |
22 | 7, 17, 11 | co 7213 |
. . . . . . . . . . . . . . . . . . 19
class (𝑟𝑠𝑥) |
23 | 12, 22, 19 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) |
24 | 21, 23 | wceq 1543 |
. . . . . . . . . . . . . . . . 17
wff (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) |
25 | | vq |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑞 |
26 | 25 | cv 1542 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑞 |
27 | | vp |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑝 |
28 | 27 | cv 1542 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑝 |
29 | 26, 7, 28 | co 7213 |
. . . . . . . . . . . . . . . . . . 19
class (𝑞𝑝𝑟) |
30 | 29, 9, 11 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class ((𝑞𝑝𝑟)𝑠𝑤) |
31 | 26, 9, 11 | co 7213 |
. . . . . . . . . . . . . . . . . . 19
class (𝑞𝑠𝑤) |
32 | 31, 12, 19 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) |
33 | 30, 32 | wceq 1543 |
. . . . . . . . . . . . . . . . 17
wff ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) |
34 | 15, 24, 33 | w3a 1089 |
. . . . . . . . . . . . . . . 16
wff ((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) |
35 | | vt |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑡 |
36 | 35 | cv 1542 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑡 |
37 | 26, 7, 36 | co 7213 |
. . . . . . . . . . . . . . . . . . 19
class (𝑞𝑡𝑟) |
38 | 37, 9, 11 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class ((𝑞𝑡𝑟)𝑠𝑤) |
39 | 26, 12, 11 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class (𝑞𝑠(𝑟𝑠𝑤)) |
40 | 38, 39 | wceq 1543 |
. . . . . . . . . . . . . . . . 17
wff ((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) |
41 | | cur 19516 |
. . . . . . . . . . . . . . . . . . . 20
class
1r |
42 | 3, 41 | cfv 6380 |
. . . . . . . . . . . . . . . . . . 19
class
(1r‘𝑓) |
43 | 42, 9, 11 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class
((1r‘𝑓)𝑠𝑤) |
44 | 43, 9 | wceq 1543 |
. . . . . . . . . . . . . . . . 17
wff
((1r‘𝑓)𝑠𝑤) = 𝑤 |
45 | | c0g 16944 |
. . . . . . . . . . . . . . . . . . . 20
class
0g |
46 | 3, 45 | cfv 6380 |
. . . . . . . . . . . . . . . . . . 19
class
(0g‘𝑓) |
47 | 46, 9, 11 | co 7213 |
. . . . . . . . . . . . . . . . . 18
class
((0g‘𝑓)𝑠𝑤) |
48 | | vg |
. . . . . . . . . . . . . . . . . . . 20
setvar 𝑔 |
49 | 48 | cv 1542 |
. . . . . . . . . . . . . . . . . . 19
class 𝑔 |
50 | 49, 45 | cfv 6380 |
. . . . . . . . . . . . . . . . . 18
class
(0g‘𝑔) |
51 | 47, 50 | wceq 1543 |
. . . . . . . . . . . . . . . . 17
wff
((0g‘𝑓)𝑠𝑤) = (0g‘𝑔) |
52 | 40, 44, 51 | w3a 1089 |
. . . . . . . . . . . . . . . 16
wff (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)) |
53 | 34, 52 | wa 399 |
. . . . . . . . . . . . . . 15
wff (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))) |
54 | 53, 8, 14 | wral 3061 |
. . . . . . . . . . . . . 14
wff
∀𝑤 ∈
𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))) |
55 | 54, 16, 14 | wral 3061 |
. . . . . . . . . . . . 13
wff
∀𝑥 ∈
𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))) |
56 | | vk |
. . . . . . . . . . . . . 14
setvar 𝑘 |
57 | 56 | cv 1542 |
. . . . . . . . . . . . 13
class 𝑘 |
58 | 55, 6, 57 | wral 3061 |
. . . . . . . . . . . 12
wff
∀𝑟 ∈
𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))) |
59 | 58, 25, 57 | wral 3061 |
. . . . . . . . . . 11
wff
∀𝑞 ∈
𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))) |
60 | 5, 59 | wa 399 |
. . . . . . . . . 10
wff (𝑓 ∈ SRing ∧
∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)))) |
61 | | cmulr 16803 |
. . . . . . . . . . 11
class
.r |
62 | 3, 61 | cfv 6380 |
. . . . . . . . . 10
class
(.r‘𝑓) |
63 | 60, 35, 62 | wsbc 3694 |
. . . . . . . . 9
wff
[(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)))) |
64 | | cplusg 16802 |
. . . . . . . . . 10
class
+g |
65 | 3, 64 | cfv 6380 |
. . . . . . . . 9
class
(+g‘𝑓) |
66 | 63, 27, 65 | wsbc 3694 |
. . . . . . . 8
wff
[(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)))) |
67 | | cbs 16760 |
. . . . . . . . 9
class
Base |
68 | 3, 67 | cfv 6380 |
. . . . . . . 8
class
(Base‘𝑓) |
69 | 66, 56, 68 | wsbc 3694 |
. . . . . . 7
wff
[(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)))) |
70 | | csca 16805 |
. . . . . . . 8
class
Scalar |
71 | 49, 70 | cfv 6380 |
. . . . . . 7
class
(Scalar‘𝑔) |
72 | 69, 2, 71 | wsbc 3694 |
. . . . . 6
wff
[(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)))) |
73 | | cvsca 16806 |
. . . . . . 7
class
·𝑠 |
74 | 49, 73 | cfv 6380 |
. . . . . 6
class (
·𝑠 ‘𝑔) |
75 | 72, 10, 74 | wsbc 3694 |
. . . . 5
wff [(
·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)))) |
76 | 49, 64 | cfv 6380 |
. . . . 5
class
(+g‘𝑔) |
77 | 75, 18, 76 | wsbc 3694 |
. . . 4
wff
[(+g‘𝑔) / 𝑎][(
·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)))) |
78 | 49, 67 | cfv 6380 |
. . . 4
class
(Base‘𝑔) |
79 | 77, 13, 78 | wsbc 3694 |
. . 3
wff
[(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(
·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔)))) |
80 | | ccmn 19170 |
. . 3
class
CMnd |
81 | 79, 48, 80 | crab 3065 |
. 2
class {𝑔 ∈ CMnd ∣
[(Base‘𝑔) /
𝑣][(+g‘𝑔) / 𝑎][(
·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))))} |
82 | 1, 81 | wceq 1543 |
1
wff SLMod =
{𝑔 ∈ CMnd ∣
[(Base‘𝑔) /
𝑣][(+g‘𝑔) / 𝑎][(
·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))))} |