Step | Hyp | Ref
| Expression |
1 | | ringi.1 |
. . . . 5
⊢ 𝐺 = (1st ‘𝑅) |
2 | | ringi.2 |
. . . . 5
⊢ 𝐻 = (2nd ‘𝑅) |
3 | | ringi.3 |
. . . . 5
⊢ 𝑋 = ran 𝐺 |
4 | 1, 2, 3 | rngoi 32868 |
. . . 4
⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))) |
5 | 4 | simprd 478 |
. . 3
⊢ (𝑅 ∈ RingOps →
(∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) |
6 | 5 | simpld 474 |
. 2
⊢ (𝑅 ∈ RingOps →
∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))) |
7 | | simp2 1055 |
. . . . . 6
⊢ ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))) |
8 | 7 | ralimi 2936 |
. . . . 5
⊢
(∀𝑧 ∈
𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑧 ∈ 𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))) |
9 | 8 | ralimi 2936 |
. . . 4
⊢
(∀𝑦 ∈
𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))) |
10 | 9 | ralimi 2936 |
. . 3
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))) |
11 | | oveq1 6556 |
. . . . 5
⊢ (𝑥 = 𝐴 → (𝑥𝐻(𝑦𝐺𝑧)) = (𝐴𝐻(𝑦𝐺𝑧))) |
12 | | oveq1 6556 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦)) |
13 | | oveq1 6556 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧)) |
14 | 12, 13 | oveq12d 6567 |
. . . . 5
⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧))) |
15 | 11, 14 | eqeq12d 2625 |
. . . 4
⊢ (𝑥 = 𝐴 → ((𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ↔ (𝐴𝐻(𝑦𝐺𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)))) |
16 | | oveq1 6556 |
. . . . . 6
⊢ (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧)) |
17 | 16 | oveq2d 6565 |
. . . . 5
⊢ (𝑦 = 𝐵 → (𝐴𝐻(𝑦𝐺𝑧)) = (𝐴𝐻(𝐵𝐺𝑧))) |
18 | | oveq2 6557 |
. . . . . 6
⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵)) |
19 | 18 | oveq1d 6564 |
. . . . 5
⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧))) |
20 | 17, 19 | eqeq12d 2625 |
. . . 4
⊢ (𝑦 = 𝐵 → ((𝐴𝐻(𝑦𝐺𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)) ↔ (𝐴𝐻(𝐵𝐺𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)))) |
21 | | oveq2 6557 |
. . . . . 6
⊢ (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶)) |
22 | 21 | oveq2d 6565 |
. . . . 5
⊢ (𝑧 = 𝐶 → (𝐴𝐻(𝐵𝐺𝑧)) = (𝐴𝐻(𝐵𝐺𝐶))) |
23 | | oveq2 6557 |
. . . . . 6
⊢ (𝑧 = 𝐶 → (𝐴𝐻𝑧) = (𝐴𝐻𝐶)) |
24 | 23 | oveq2d 6565 |
. . . . 5
⊢ (𝑧 = 𝐶 → ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))) |
25 | 22, 24 | eqeq12d 2625 |
. . . 4
⊢ (𝑧 = 𝐶 → ((𝐴𝐻(𝐵𝐺𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)) ↔ (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))) |
26 | 15, 20, 25 | rspc3v 3296 |
. . 3
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))) |
27 | 10, 26 | syl5 33 |
. 2
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))) |
28 | 6, 27 | mpan9 485 |
1
⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))) |