Detailed syntax breakdown of Definition df-lmod
Step | Hyp | Ref
| Expression |
1 | | clmod 20132 |
. 2
class
LMod |
2 | | vf |
. . . . . . . . . . . . 13
setvar 𝑓 |
3 | 2 | cv 1538 |
. . . . . . . . . . . 12
class 𝑓 |
4 | | crg 19792 |
. . . . . . . . . . . 12
class
Ring |
5 | 3, 4 | wcel 2107 |
. . . . . . . . . . 11
wff 𝑓 ∈ Ring |
6 | | vr |
. . . . . . . . . . . . . . . . . . . 20
setvar 𝑟 |
7 | 6 | cv 1538 |
. . . . . . . . . . . . . . . . . . 19
class 𝑟 |
8 | | vw |
. . . . . . . . . . . . . . . . . . . 20
setvar 𝑤 |
9 | 8 | cv 1538 |
. . . . . . . . . . . . . . . . . . 19
class 𝑤 |
10 | | vs |
. . . . . . . . . . . . . . . . . . . 20
setvar 𝑠 |
11 | 10 | cv 1538 |
. . . . . . . . . . . . . . . . . . 19
class 𝑠 |
12 | 7, 9, 11 | co 7284 |
. . . . . . . . . . . . . . . . . 18
class (𝑟𝑠𝑤) |
13 | | vv |
. . . . . . . . . . . . . . . . . . 19
setvar 𝑣 |
14 | 13 | cv 1538 |
. . . . . . . . . . . . . . . . . 18
class 𝑣 |
15 | 12, 14 | wcel 2107 |
. . . . . . . . . . . . . . . . 17
wff (𝑟𝑠𝑤) ∈ 𝑣 |
16 | | vx |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑥 |
17 | 16 | cv 1538 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑥 |
18 | | va |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑎 |
19 | 18 | cv 1538 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑎 |
20 | 9, 17, 19 | co 7284 |
. . . . . . . . . . . . . . . . . . 19
class (𝑤𝑎𝑥) |
21 | 7, 20, 11 | co 7284 |
. . . . . . . . . . . . . . . . . 18
class (𝑟𝑠(𝑤𝑎𝑥)) |
22 | 7, 17, 11 | co 7284 |
. . . . . . . . . . . . . . . . . . 19
class (𝑟𝑠𝑥) |
23 | 12, 22, 19 | co 7284 |
. . . . . . . . . . . . . . . . . 18
class ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) |
24 | 21, 23 | wceq 1539 |
. . . . . . . . . . . . . . . . 17
wff (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) |
25 | | vq |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑞 |
26 | 25 | cv 1538 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑞 |
27 | | vp |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑝 |
28 | 27 | cv 1538 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑝 |
29 | 26, 7, 28 | co 7284 |
. . . . . . . . . . . . . . . . . . 19
class (𝑞𝑝𝑟) |
30 | 29, 9, 11 | co 7284 |
. . . . . . . . . . . . . . . . . 18
class ((𝑞𝑝𝑟)𝑠𝑤) |
31 | 26, 9, 11 | co 7284 |
. . . . . . . . . . . . . . . . . . 19
class (𝑞𝑠𝑤) |
32 | 31, 12, 19 | co 7284 |
. . . . . . . . . . . . . . . . . 18
class ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) |
33 | 30, 32 | wceq 1539 |
. . . . . . . . . . . . . . . . 17
wff ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) |
34 | 15, 24, 33 | w3a 1086 |
. . . . . . . . . . . . . . . 16
wff ((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) |
35 | | vt |
. . . . . . . . . . . . . . . . . . . . 21
setvar 𝑡 |
36 | 35 | cv 1538 |
. . . . . . . . . . . . . . . . . . . 20
class 𝑡 |
37 | 26, 7, 36 | co 7284 |
. . . . . . . . . . . . . . . . . . 19
class (𝑞𝑡𝑟) |
38 | 37, 9, 11 | co 7284 |
. . . . . . . . . . . . . . . . . 18
class ((𝑞𝑡𝑟)𝑠𝑤) |
39 | 26, 12, 11 | co 7284 |
. . . . . . . . . . . . . . . . . 18
class (𝑞𝑠(𝑟𝑠𝑤)) |
40 | 38, 39 | wceq 1539 |
. . . . . . . . . . . . . . . . 17
wff ((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) |
41 | | cur 19746 |
. . . . . . . . . . . . . . . . . . . 20
class
1r |
42 | 3, 41 | cfv 6437 |
. . . . . . . . . . . . . . . . . . 19
class
(1r‘𝑓) |
43 | 42, 9, 11 | co 7284 |
. . . . . . . . . . . . . . . . . 18
class
((1r‘𝑓)𝑠𝑤) |
44 | 43, 9 | wceq 1539 |
. . . . . . . . . . . . . . . . 17
wff
((1r‘𝑓)𝑠𝑤) = 𝑤 |
45 | 40, 44 | wa 396 |
. . . . . . . . . . . . . . . 16
wff (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤) |
46 | 34, 45 | wa 396 |
. . . . . . . . . . . . . . 15
wff (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) |
47 | 46, 8, 14 | wral 3065 |
. . . . . . . . . . . . . 14
wff
∀𝑤 ∈
𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) |
48 | 47, 16, 14 | wral 3065 |
. . . . . . . . . . . . 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)) |
49 | | vk |
. . . . . . . . . . . . . 14
setvar 𝑘 |
50 | 49 | cv 1538 |
. . . . . . . . . . . . 13
class 𝑘 |
51 | 48, 6, 50 | wral 3065 |
. . . . . . . . . . . 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)) |
52 | 51, 25, 50 | wral 3065 |
. . . . . . . . . . 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)) |
53 | 5, 52 | wa 396 |
. . . . . . . . . 10
wff (𝑓 ∈ Ring ∧ ∀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))) |
54 | | cmulr 16972 |
. . . . . . . . . . 11
class
.r |
55 | 3, 54 | cfv 6437 |
. . . . . . . . . 10
class
(.r‘𝑓) |
56 | 53, 35, 55 | wsbc 3717 |
. . . . . . . . 9
wff
[(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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))) |
57 | | cplusg 16971 |
. . . . . . . . . 10
class
+g |
58 | 3, 57 | cfv 6437 |
. . . . . . . . 9
class
(+g‘𝑓) |
59 | 56, 27, 58 | wsbc 3717 |
. . . . . . . 8
wff
[(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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))) |
60 | | cbs 16921 |
. . . . . . . . 9
class
Base |
61 | 3, 60 | cfv 6437 |
. . . . . . . 8
class
(Base‘𝑓) |
62 | 59, 49, 61 | wsbc 3717 |
. . . . . . 7
wff
[(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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))) |
63 | | vg |
. . . . . . . . 9
setvar 𝑔 |
64 | 63 | cv 1538 |
. . . . . . . 8
class 𝑔 |
65 | | cvsca 16975 |
. . . . . . . 8
class
·𝑠 |
66 | 64, 65 | cfv 6437 |
. . . . . . 7
class (
·𝑠 ‘𝑔) |
67 | 62, 10, 66 | wsbc 3717 |
. . . . . 6
wff [(
·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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))) |
68 | | csca 16974 |
. . . . . . 7
class
Scalar |
69 | 64, 68 | cfv 6437 |
. . . . . 6
class
(Scalar‘𝑔) |
70 | 67, 2, 69 | wsbc 3717 |
. . . . 5
wff
[(Scalar‘𝑔) / 𝑓][(
·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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))) |
71 | 64, 57 | cfv 6437 |
. . . . 5
class
(+g‘𝑔) |
72 | 70, 18, 71 | wsbc 3717 |
. . . 4
wff
[(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][(
·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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))) |
73 | 64, 60 | cfv 6437 |
. . . 4
class
(Base‘𝑔) |
74 | 72, 13, 73 | wsbc 3717 |
. . 3
wff
[(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][(
·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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))) |
75 | | cgrp 18586 |
. . 3
class
Grp |
76 | 74, 63, 75 | crab 3069 |
. 2
class {𝑔 ∈ Grp ∣
[(Base‘𝑔) /
𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][(
·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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)))} |
77 | 1, 76 | wceq 1539 |
1
wff LMod =
{𝑔 ∈ Grp ∣
[(Base‘𝑔) /
𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][(
·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀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)))} |