Step | Hyp | Ref
| Expression |
1 | | fveq2 6103 |
. . . . . 6
⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊)) |
2 | | islmod.v |
. . . . . 6
⊢ 𝑉 = (Base‘𝑊) |
3 | 1, 2 | syl6eqr 2662 |
. . . . 5
⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉) |
4 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑔 = 𝑊 → (+g‘𝑔) = (+g‘𝑊)) |
5 | | islmod.a |
. . . . . . 7
⊢ + =
(+g‘𝑊) |
6 | 4, 5 | syl6eqr 2662 |
. . . . . 6
⊢ (𝑔 = 𝑊 → (+g‘𝑔) = + ) |
7 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑔 = 𝑊 → (Scalar‘𝑔) = (Scalar‘𝑊)) |
8 | | islmod.f |
. . . . . . . 8
⊢ 𝐹 = (Scalar‘𝑊) |
9 | 7, 8 | syl6eqr 2662 |
. . . . . . 7
⊢ (𝑔 = 𝑊 → (Scalar‘𝑔) = 𝐹) |
10 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑔 = 𝑊 → (
·𝑠 ‘𝑔) = ( ·𝑠
‘𝑊)) |
11 | | islmod.s |
. . . . . . . . 9
⊢ · = (
·𝑠 ‘𝑊) |
12 | 10, 11 | syl6eqr 2662 |
. . . . . . . 8
⊢ (𝑔 = 𝑊 → (
·𝑠 ‘𝑔) = · ) |
13 | 12 | sbceq1d 3407 |
. . . . . . 7
⊢ (𝑔 = 𝑊 → ([(
·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))) ↔ [ · / 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))))) |
14 | 9, 13 | sbceqbid 3409 |
. . . . . 6
⊢ (𝑔 = 𝑊 → ([(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))) ↔ [𝐹 / 𝑓][ · / 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))))) |
15 | 6, 14 | sbceqbid 3409 |
. . . . 5
⊢ (𝑔 = 𝑊 → ([(+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))) ↔ [ + / 𝑎][𝐹 / 𝑓][ · / 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))))) |
16 | 3, 15 | sbceqbid 3409 |
. . . 4
⊢ (𝑔 = 𝑊 → ([(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))) ↔ [𝑉 / 𝑣][ + / 𝑎][𝐹 / 𝑓][ · / 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))))) |
17 | | fvex 6113 |
. . . . . . 7
⊢
(Base‘𝑊)
∈ V |
18 | 2, 17 | eqeltri 2684 |
. . . . . 6
⊢ 𝑉 ∈ V |
19 | | fvex 6113 |
. . . . . . 7
⊢
(+g‘𝑊) ∈ V |
20 | 5, 19 | eqeltri 2684 |
. . . . . 6
⊢ + ∈
V |
21 | | fvex 6113 |
. . . . . . 7
⊢
(Scalar‘𝑊)
∈ V |
22 | 8, 21 | eqeltri 2684 |
. . . . . 6
⊢ 𝐹 ∈ V |
23 | | simp3 1056 |
. . . . . . . . . 10
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹) |
24 | 23 | fveq2d 6107 |
. . . . . . . . 9
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹)) |
25 | | islmod.k |
. . . . . . . . 9
⊢ 𝐾 = (Base‘𝐹) |
26 | 24, 25 | syl6eqr 2662 |
. . . . . . . 8
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾) |
27 | 23 | fveq2d 6107 |
. . . . . . . . . 10
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (+g‘𝑓) = (+g‘𝐹)) |
28 | | islmod.p |
. . . . . . . . . 10
⊢ ⨣ =
(+g‘𝐹) |
29 | 27, 28 | syl6eqr 2662 |
. . . . . . . . 9
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (+g‘𝑓) = ⨣ ) |
30 | 23 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (.r‘𝑓) = (.r‘𝐹)) |
31 | | islmod.t |
. . . . . . . . . . . 12
⊢ × =
(.r‘𝐹) |
32 | 30, 31 | syl6eqr 2662 |
. . . . . . . . . . 11
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (.r‘𝑓) = × ) |
33 | 32 | sbceq1d 3407 |
. . . . . . . . . 10
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([(.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))) ↔ [ × / 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))))) |
34 | | fvex 6113 |
. . . . . . . . . . . . 13
⊢
(.r‘𝐹) ∈ V |
35 | 31, 34 | eqeltri 2684 |
. . . . . . . . . . . 12
⊢ × ∈
V |
36 | | oveq 6555 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = × → (𝑞𝑡𝑟) = (𝑞 × 𝑟)) |
37 | 36 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = × → ((𝑞𝑡𝑟)𝑠𝑤) = ((𝑞 × 𝑟)𝑠𝑤)) |
38 | 37 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = × → (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ↔ ((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)))) |
39 | 38 | anbi1d 737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = × → ((((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) |
40 | 39 | anbi2d 736 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = × → ((((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))) |
41 | 40 | 2ralbidv 2972 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = × →
(∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))) |
42 | 41 | 2ralbidv 2972 |
. . . . . . . . . . . . 13
⊢ (𝑡 = × →
(∀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)) ↔ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))) |
43 | 42 | anbi2d 736 |
. . . . . . . . . . . 12
⊢ (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))) ↔ (𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))))) |
44 | 35, 43 | sbcie 3437 |
. . . . . . . . . . 11
⊢ ([
× /
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))) ↔ (𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))) |
45 | 23 | eleq1d 2672 |
. . . . . . . . . . . 12
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑓 ∈ Ring ↔ 𝐹 ∈ Ring)) |
46 | | simp1 1054 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑣 = 𝑉) |
47 | 46 | eleq2d 2673 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠𝑤) ∈ 𝑣 ↔ (𝑟𝑠𝑤) ∈ 𝑉)) |
48 | | simp2 1055 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → 𝑎 = + ) |
49 | 48 | oveqd 6566 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑤𝑎𝑥) = (𝑤 + 𝑥)) |
50 | 49 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (𝑟𝑠(𝑤𝑎𝑥)) = (𝑟𝑠(𝑤 + 𝑥))) |
51 | 48 | oveqd 6566 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥))) |
52 | 50, 51 | eqeq12d 2625 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ↔ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)))) |
53 | 48 | oveqd 6566 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) |
54 | 53 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤)) ↔ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)))) |
55 | 47, 52, 54 | 3anbi123d 1391 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ↔ ((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))))) |
56 | 23 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (1r‘𝑓) = (1r‘𝐹)) |
57 | | islmod.u |
. . . . . . . . . . . . . . . . . . . 20
⊢ 1 =
(1r‘𝐹) |
58 | 56, 57 | syl6eqr 2662 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (1r‘𝑓) = 1 ) |
59 | 58 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((1r‘𝑓)𝑠𝑤) = ( 1 𝑠𝑤)) |
60 | 59 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (((1r‘𝑓)𝑠𝑤) = 𝑤 ↔ ( 1 𝑠𝑤) = 𝑤)) |
61 | 60 | anbi2d 736 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))) |
62 | 55, 61 | anbi12d 743 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))) |
63 | 46, 62 | raleqbidv 3129 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))) |
64 | 46, 63 | raleqbidv 3129 |
. . . . . . . . . . . . 13
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))) |
65 | 64 | 2ralbidv 2972 |
. . . . . . . . . . . 12
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → (∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)) ↔ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))) |
66 | 45, 65 | anbi12d 743 |
. . . . . . . . . . 11
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ((𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))) |
67 | 44, 66 | syl5bb 271 |
. . . . . . . . . 10
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 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))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))) |
68 | 33, 67 | bitrd 267 |
. . . . . . . . 9
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([(.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))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))) |
69 | 29, 68 | sbceqbid 3409 |
. . . . . . . 8
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([(+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))) ↔ [ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))) |
70 | 26, 69 | sbceqbid 3409 |
. . . . . . 7
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([(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))) ↔ [𝐾 / 𝑘][ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))) |
71 | 70 | sbcbidv 3457 |
. . . . . 6
⊢ ((𝑣 = 𝑉 ∧ 𝑎 = + ∧ 𝑓 = 𝐹) → ([ · / 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))) ↔ [ · / 𝑠][𝐾 / 𝑘][ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))))) |
72 | 18, 20, 22, 71 | sbc3ie 3474 |
. . . . 5
⊢
([𝑉 / 𝑣][ + / 𝑎][𝐹 / 𝑓][ · / 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))) ↔ [ · / 𝑠][𝐾 / 𝑘][ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)))) |
73 | | fvex 6113 |
. . . . . . 7
⊢ (
·𝑠 ‘𝑊) ∈ V |
74 | 11, 73 | eqeltri 2684 |
. . . . . 6
⊢ · ∈
V |
75 | | fvex 6113 |
. . . . . . 7
⊢
(Base‘𝐹)
∈ V |
76 | 25, 75 | eqeltri 2684 |
. . . . . 6
⊢ 𝐾 ∈ V |
77 | | fvex 6113 |
. . . . . . 7
⊢
(+g‘𝐹) ∈ V |
78 | 28, 77 | eqeltri 2684 |
. . . . . 6
⊢ ⨣ ∈
V |
79 | | simp2 1055 |
. . . . . . . 8
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑘 = 𝐾) |
80 | | simp1 1054 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑠 = · ) |
81 | 80 | oveqd 6566 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠𝑤) = (𝑟 · 𝑤)) |
82 | 81 | eleq1d 2672 |
. . . . . . . . . . . 12
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠𝑤) ∈ 𝑉 ↔ (𝑟 · 𝑤) ∈ 𝑉)) |
83 | 80 | oveqd 6566 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠(𝑤 + 𝑥)) = (𝑟 · (𝑤 + 𝑥))) |
84 | 80 | oveqd 6566 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑟𝑠𝑥) = (𝑟 · 𝑥)) |
85 | 81, 84 | oveq12d 6567 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥))) |
86 | 83, 85 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ↔ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)))) |
87 | | simp3 1056 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → 𝑝 = ⨣ ) |
88 | 87 | oveqd 6566 |
. . . . . . . . . . . . . . 15
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑝𝑟) = (𝑞 ⨣ 𝑟)) |
89 | 88 | oveq1d 6564 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟)𝑠𝑤)) |
90 | 80 | oveqd 6566 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞 ⨣ 𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟) · 𝑤)) |
91 | 89, 90 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞 ⨣ 𝑟) · 𝑤)) |
92 | 80 | oveqd 6566 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠𝑤) = (𝑞 · 𝑤)) |
93 | 92, 81 | oveq12d 6567 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) |
94 | 91, 93 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤)) ↔ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤)))) |
95 | 82, 86, 94 | 3anbi123d 1391 |
. . . . . . . . . . 11
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ↔ ((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))))) |
96 | 80 | oveqd 6566 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝑞 × 𝑟)𝑠𝑤) = ((𝑞 × 𝑟) · 𝑤)) |
97 | 81 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟𝑠𝑤)) = (𝑞𝑠(𝑟 · 𝑤))) |
98 | 80 | oveqd 6566 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟 · 𝑤)) = (𝑞 · (𝑟 · 𝑤))) |
99 | 97, 98 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (𝑞𝑠(𝑟𝑠𝑤)) = (𝑞 · (𝑟 · 𝑤))) |
100 | 96, 99 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ↔ ((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)))) |
101 | 80 | oveqd 6566 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ( 1 𝑠𝑤) = ( 1 · 𝑤)) |
102 | 101 | eqeq1d 2612 |
. . . . . . . . . . . 12
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → (( 1 𝑠𝑤) = 𝑤 ↔ ( 1 · 𝑤) = 𝑤)) |
103 | 100, 102 | anbi12d 743 |
. . . . . . . . . . 11
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤) ↔ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))) |
104 | 95, 103 | anbi12d 743 |
. . . . . . . . . 10
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)) ↔ (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))) |
105 | 104 | 2ralbidv 2972 |
. . . . . . . . 9
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) →
(∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)) ↔ ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))) |
106 | 79, 105 | raleqbidv 3129 |
. . . . . . . 8
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) →
(∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)) ↔ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))) |
107 | 79, 106 | raleqbidv 3129 |
. . . . . . 7
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) →
(∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤)) ↔ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))) |
108 | 107 | anbi2d 736 |
. . . . . 6
⊢ ((𝑠 = · ∧ 𝑘 = 𝐾 ∧ 𝑝 = ⨣ ) → ((𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))) |
109 | 74, 76, 78, 108 | sbc3ie 3474 |
. . . . 5
⊢ ([
·
/ 𝑠][𝐾 / 𝑘][ ⨣ / 𝑝](𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟𝑠𝑤) ∈ 𝑉 ∧ (𝑟𝑠(𝑤 + 𝑥)) = ((𝑟𝑠𝑤) + (𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤) + (𝑟𝑠𝑤))) ∧ (((𝑞 × 𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ( 1 𝑠𝑤) = 𝑤))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))) |
110 | 72, 109 | bitri 263 |
. . . 4
⊢
([𝑉 / 𝑣][ + / 𝑎][𝐹 / 𝑓][ · / 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))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))) |
111 | 16, 110 | syl6bb 275 |
. . 3
⊢ (𝑔 = 𝑊 → ([(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))) ↔ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))) |
112 | | df-lmod 18688 |
. . 3
⊢ 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)))} |
113 | 111, 112 | elrab2 3333 |
. 2
⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))) |
114 | | 3anass 1035 |
. 2
⊢ ((𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))) ↔ (𝑊 ∈ Grp ∧ (𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))) |
115 | 113, 114 | bitr4i 266 |
1
⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤)))) |