Proof of Theorem lmod1lem2
Step | Hyp | Ref
| Expression |
1 | | fvex 6787 |
. . . . . . 7
⊢
(Base‘𝑅)
∈ V |
2 | | snex 5354 |
. . . . . . 7
⊢ {𝐼} ∈ V |
3 | 1, 2 | pm3.2i 471 |
. . . . . 6
⊢
((Base‘𝑅)
∈ V ∧ {𝐼} ∈
V) |
4 | | mpoexga 7918 |
. . . . . 6
⊢
(((Base‘𝑅)
∈ V ∧ {𝐼} ∈
V) → (𝑥 ∈
(Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) ∈ V) |
5 | 3, 4 | mp1i 13 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) ∈ V) |
6 | | lmod1.m |
. . . . . 6
⊢ 𝑀 = ({〈(Base‘ndx),
{𝐼}〉,
〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx),
𝑅〉} ∪ {〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)〉}) |
7 | 6 | lmodvsca 17039 |
. . . . 5
⊢ ((𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) ∈ V → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) = ( ·𝑠
‘𝑀)) |
8 | 5, 7 | syl 17 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) = ( ·𝑠
‘𝑀)) |
9 | 8 | eqcomd 2744 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (
·𝑠 ‘𝑀) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)) |
10 | | simprr 770 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) ∧ (𝑥 = 𝑟 ∧ 𝑦 = 𝐼)) → 𝑦 = 𝐼) |
11 | | simp3 1137 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → 𝑟 ∈ (Base‘𝑅)) |
12 | | snidg 4595 |
. . . 4
⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼}) |
13 | 12 | 3ad2ant1 1132 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → 𝐼 ∈ {𝐼}) |
14 | 9, 10, 11, 13, 13 | ovmpod 7425 |
. 2
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠
‘𝑀)𝐼) = 𝐼) |
15 | | snex 5354 |
. . . . . . 7
⊢
{〈〈𝐼,
𝐼〉, 𝐼〉} ∈ V |
16 | 6 | lmodplusg 17037 |
. . . . . . 7
⊢
({〈〈𝐼,
𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
17 | 15, 16 | mp1i 13 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
18 | 17 | eqcomd 2744 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (+g‘𝑀) = {〈〈𝐼, 𝐼〉, 𝐼〉}) |
19 | 18 | oveqd 7292 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝐼(+g‘𝑀)𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) |
20 | | df-ov 7278 |
. . . . 5
⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) |
21 | | opex 5379 |
. . . . . 6
⊢
〈𝐼, 𝐼〉 ∈ V |
22 | | simp1 1135 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → 𝐼 ∈ 𝑉) |
23 | | fvsng 7052 |
. . . . . 6
⊢
((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
24 | 21, 22, 23 | sylancr 587 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
25 | 20, 24 | eqtrid 2790 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
26 | 19, 25 | eqtrd 2778 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝐼(+g‘𝑀)𝐼) = 𝐼) |
27 | 26 | oveq2d 7291 |
. 2
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠
‘𝑀)(𝐼(+g‘𝑀)𝐼)) = (𝑟( ·𝑠
‘𝑀)𝐼)) |
28 | 2 | a1i 11 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → {𝐼} ∈ V) |
29 | 1, 28, 4 | sylancr 587 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) ∈ V) |
30 | 29, 7 | syl 17 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) = ( ·𝑠
‘𝑀)) |
31 | 30 | eqcomd 2744 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (
·𝑠 ‘𝑀) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)) |
32 | 31, 10, 11, 13, 13 | ovmpod 7425 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠
‘𝑀)𝐼) = 𝐼) |
33 | 32, 32 | oveq12d 7293 |
. . 3
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((𝑟( ·𝑠
‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠
‘𝑀)𝐼)) = (𝐼(+g‘𝑀)𝐼)) |
34 | 33, 26 | eqtrd 2778 |
. 2
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → ((𝑟( ·𝑠
‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠
‘𝑀)𝐼)) = 𝐼) |
35 | 14, 27, 34 | 3eqtr4d 2788 |
1
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠
‘𝑀)(𝐼(+g‘𝑀)𝐼)) = ((𝑟( ·𝑠
‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠
‘𝑀)𝐼))) |