Step | Hyp | Ref
| Expression |
1 | | abvpropd.1 |
. . . . 5
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
2 | | abvpropd.2 |
. . . . 5
⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
3 | | abvpropd.3 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
4 | | abvpropd.4 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
5 | 1, 2, 3, 4 | ringpropd 18582 |
. . . 4
⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
6 | 1, 2 | eqtr3d 2658 |
. . . . . 6
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
7 | 6 | feq2d 6031 |
. . . . 5
⊢ (𝜑 → (𝑓:(Base‘𝐾)⟶(0[,)+∞) ↔ 𝑓:(Base‘𝐿)⟶(0[,)+∞))) |
8 | 1, 2, 3 | grpidpropd 17261 |
. . . . . . . . . . 11
⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿)) |
9 | 8 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (0g‘𝐾) = (0g‘𝐿)) |
10 | 9 | eqeq2d 2632 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 = (0g‘𝐾) ↔ 𝑥 = (0g‘𝐿))) |
11 | 10 | bibi2d 332 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ↔ ((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)))) |
12 | 4 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(.r‘𝐾)𝑦)) = (𝑓‘(𝑥(.r‘𝐿)𝑦))) |
13 | 12 | eqeq1d 2624 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ↔ (𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)))) |
14 | 3 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐾)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦))) |
15 | 14 | breq1d 4663 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)) ↔ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) |
16 | 13, 15 | anbi12d 747 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))) |
17 | 16 | anassrs 680 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))) |
18 | 17 | ralbidva 2985 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))) |
19 | 11, 18 | anbi12d 747 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))) |
20 | 19 | ralbidva 2985 |
. . . . . 6
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ ∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))) |
21 | 1 | raleqdv 3144 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))) |
22 | 21 | anbi2d 740 |
. . . . . . 7
⊢ (𝜑 → ((((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))) |
23 | 1, 22 | raleqbidv 3152 |
. . . . . 6
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐾)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))) |
24 | 2 | raleqdv 3144 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))) |
25 | 24 | anbi2d 740 |
. . . . . . 7
⊢ (𝜑 → ((((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))) |
26 | 2, 25 | raleqbidv 3152 |
. . . . . 6
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐿)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))) |
27 | 20, 23, 26 | 3bitr3d 298 |
. . . . 5
⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐾)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐿)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))) |
28 | 7, 27 | anbi12d 747 |
. . . 4
⊢ (𝜑 → ((𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))) ↔ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))))) |
29 | 5, 28 | anbi12d 747 |
. . 3
⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))) ↔ (𝐿 ∈ Ring ∧ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))))) |
30 | | eqid 2622 |
. . . . 5
⊢
(AbsVal‘𝐾) =
(AbsVal‘𝐾) |
31 | 30 | abvrcl 18821 |
. . . 4
⊢ (𝑓 ∈ (AbsVal‘𝐾) → 𝐾 ∈ Ring) |
32 | | eqid 2622 |
. . . . 5
⊢
(Base‘𝐾) =
(Base‘𝐾) |
33 | | eqid 2622 |
. . . . 5
⊢
(+g‘𝐾) = (+g‘𝐾) |
34 | | eqid 2622 |
. . . . 5
⊢
(.r‘𝐾) = (.r‘𝐾) |
35 | | eqid 2622 |
. . . . 5
⊢
(0g‘𝐾) = (0g‘𝐾) |
36 | 30, 32, 33, 34, 35 | isabv 18819 |
. . . 4
⊢ (𝐾 ∈ Ring → (𝑓 ∈ (AbsVal‘𝐾) ↔ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))))) |
37 | 31, 36 | biadan2 674 |
. . 3
⊢ (𝑓 ∈ (AbsVal‘𝐾) ↔ (𝐾 ∈ Ring ∧ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r‘𝐾)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐾)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))))) |
38 | | eqid 2622 |
. . . . 5
⊢
(AbsVal‘𝐿) =
(AbsVal‘𝐿) |
39 | 38 | abvrcl 18821 |
. . . 4
⊢ (𝑓 ∈ (AbsVal‘𝐿) → 𝐿 ∈ Ring) |
40 | | eqid 2622 |
. . . . 5
⊢
(Base‘𝐿) =
(Base‘𝐿) |
41 | | eqid 2622 |
. . . . 5
⊢
(+g‘𝐿) = (+g‘𝐿) |
42 | | eqid 2622 |
. . . . 5
⊢
(.r‘𝐿) = (.r‘𝐿) |
43 | | eqid 2622 |
. . . . 5
⊢
(0g‘𝐿) = (0g‘𝐿) |
44 | 38, 40, 41, 42, 43 | isabv 18819 |
. . . 4
⊢ (𝐿 ∈ Ring → (𝑓 ∈ (AbsVal‘𝐿) ↔ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))))) |
45 | 39, 44 | biadan2 674 |
. . 3
⊢ (𝑓 ∈ (AbsVal‘𝐿) ↔ (𝐿 ∈ Ring ∧ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r‘𝐿)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝐿)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))))) |
46 | 29, 37, 45 | 3bitr4g 303 |
. 2
⊢ (𝜑 → (𝑓 ∈ (AbsVal‘𝐾) ↔ 𝑓 ∈ (AbsVal‘𝐿))) |
47 | 46 | eqrdv 2620 |
1
⊢ (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿)) |