Proof of Theorem isringd
Step | Hyp | Ref
| Expression |
1 | | isringd.g |
. 2
⊢ (𝜑 → 𝑅 ∈ Grp) |
2 | | isringd.b |
. . . 4
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
3 | | eqid 2738 |
. . . . 5
⊢
(mulGrp‘𝑅) =
(mulGrp‘𝑅) |
4 | | eqid 2738 |
. . . . 5
⊢
(Base‘𝑅) =
(Base‘𝑅) |
5 | 3, 4 | mgpbas 19538 |
. . . 4
⊢
(Base‘𝑅) =
(Base‘(mulGrp‘𝑅)) |
6 | 2, 5 | eqtrdi 2795 |
. . 3
⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝑅))) |
7 | | isringd.t |
. . . 4
⊢ (𝜑 → · =
(.r‘𝑅)) |
8 | | eqid 2738 |
. . . . 5
⊢
(.r‘𝑅) = (.r‘𝑅) |
9 | 3, 8 | mgpplusg 19536 |
. . . 4
⊢
(.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
10 | 7, 9 | eqtrdi 2795 |
. . 3
⊢ (𝜑 → · =
(+g‘(mulGrp‘𝑅))) |
11 | | isringd.c |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) |
12 | | isringd.a |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
13 | | isringd.u |
. . 3
⊢ (𝜑 → 1 ∈ 𝐵) |
14 | | isringd.i |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥) |
15 | | isringd.h |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥) |
16 | 6, 10, 11, 12, 13, 14, 15 | ismndd 18223 |
. 2
⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
17 | 2 | eleq2d 2824 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝑅))) |
18 | 2 | eleq2d 2824 |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝑅))) |
19 | 2 | eleq2d 2824 |
. . . . . 6
⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (Base‘𝑅))) |
20 | 17, 18, 19 | 3anbi123d 1438 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ↔ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)))) |
21 | 20 | biimpar 481 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) |
22 | | isringd.d |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
23 | 7 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → · =
(.r‘𝑅)) |
24 | | eqidd 2739 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 = 𝑥) |
25 | | isringd.p |
. . . . . . . 8
⊢ (𝜑 → + =
(+g‘𝑅)) |
26 | 25 | oveqdr 7260 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 + 𝑧) = (𝑦(+g‘𝑅)𝑧)) |
27 | 23, 24, 26 | oveq123d 7253 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = (𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧))) |
28 | 25 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → + =
(+g‘𝑅)) |
29 | 7 | oveqdr 7260 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · 𝑦) = (𝑥(.r‘𝑅)𝑦)) |
30 | 7 | oveqdr 7260 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · 𝑧) = (𝑥(.r‘𝑅)𝑧)) |
31 | 28, 29, 30 | oveq123d 7253 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) + (𝑥 · 𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧))) |
32 | 22, 27, 31 | 3eqtr3d 2786 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧))) |
33 | | isringd.e |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
34 | 25 | oveqdr 7260 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦)) |
35 | | eqidd 2739 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = 𝑧) |
36 | 23, 34, 35 | oveq123d 7253 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧)) |
37 | 7 | oveqdr 7260 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 · 𝑧) = (𝑦(.r‘𝑅)𝑧)) |
38 | 28, 30, 37 | oveq123d 7253 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑧) + (𝑦 · 𝑧)) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))) |
39 | 33, 36, 38 | 3eqtr3d 2786 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))) |
40 | 32, 39 | jca 515 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧)))) |
41 | 21, 40 | syldan 594 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧)))) |
42 | 41 | ralrimivvva 3114 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧)))) |
43 | | eqid 2738 |
. . 3
⊢
(+g‘𝑅) = (+g‘𝑅) |
44 | 4, 3, 43, 8 | isring 19594 |
. 2
⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧
(mulGrp‘𝑅) ∈ Mnd
∧ ∀𝑥 ∈
(Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))))) |
45 | 1, 16, 42, 44 | syl3anbrc 1345 |
1
⊢ (𝜑 → 𝑅 ∈ Ring) |