Proof of Theorem lmod1lem3
| Step | Hyp | Ref
| Expression |
| 1 | | eqidd 2738 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)) |
| 2 | | simprr 773 |
. . 3
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) ∧ (𝑥 = (𝑞(+g‘(Scalar‘𝑀))𝑟) ∧ 𝑦 = 𝐼)) → 𝑦 = 𝐼) |
| 3 | | simplr 769 |
. . . . . . 7
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring) |
| 4 | | lmod1.m |
. . . . . . . . 9
⊢ 𝑀 = ({〈(Base‘ndx),
{𝐼}〉,
〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉, 〈(Scalar‘ndx),
𝑅〉} ∪ {〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)〉}) |
| 5 | 4 | lmodsca 17372 |
. . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀)) |
| 6 | 5 | fveq2d 6910 |
. . . . . . 7
⊢ (𝑅 ∈ Ring →
(+g‘𝑅) =
(+g‘(Scalar‘𝑀))) |
| 7 | 3, 6 | syl 17 |
. . . . . 6
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (+g‘𝑅) =
(+g‘(Scalar‘𝑀))) |
| 8 | 7 | eqcomd 2743 |
. . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) →
(+g‘(Scalar‘𝑀)) = (+g‘𝑅)) |
| 9 | 8 | oveqd 7448 |
. . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞(+g‘(Scalar‘𝑀))𝑟) = (𝑞(+g‘𝑅)𝑟)) |
| 10 | | simprl 771 |
. . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑞 ∈ (Base‘𝑅)) |
| 11 | | simprr 773 |
. . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑟 ∈ (Base‘𝑅)) |
| 12 | | eqid 2737 |
. . . . . 6
⊢
(Base‘𝑅) =
(Base‘𝑅) |
| 13 | | eqid 2737 |
. . . . . 6
⊢
(+g‘𝑅) = (+g‘𝑅) |
| 14 | 12, 13 | ringacl 20275 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑞(+g‘𝑅)𝑟) ∈ (Base‘𝑅)) |
| 15 | 3, 10, 11, 14 | syl3anc 1373 |
. . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞(+g‘𝑅)𝑟) ∈ (Base‘𝑅)) |
| 16 | 9, 15 | eqeltrd 2841 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞(+g‘(Scalar‘𝑀))𝑟) ∈ (Base‘𝑅)) |
| 17 | | snidg 4660 |
. . . . 5
⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼}) |
| 18 | 17 | adantr 480 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝐼 ∈ {𝐼}) |
| 19 | 18 | adantr 480 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼 ∈ {𝐼}) |
| 20 | | simpl 482 |
. . . 4
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝐼 ∈ 𝑉) |
| 21 | 20 | adantr 480 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼 ∈ 𝑉) |
| 22 | 1, 2, 16, 19, 21 | ovmpod 7585 |
. 2
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)𝐼) = 𝐼) |
| 23 | | fvex 6919 |
. . . . . . 7
⊢
(Base‘𝑅)
∈ V |
| 24 | | snex 5436 |
. . . . . . 7
⊢ {𝐼} ∈ V |
| 25 | 23, 24 | pm3.2i 470 |
. . . . . 6
⊢
((Base‘𝑅)
∈ V ∧ {𝐼} ∈
V) |
| 26 | | mpoexga 8102 |
. . . . . 6
⊢
(((Base‘𝑅)
∈ V ∧ {𝐼} ∈
V) → (𝑥 ∈
(Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) ∈ V) |
| 27 | 25, 26 | mp1i 13 |
. . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) ∈ V) |
| 28 | 4 | lmodvsca 17373 |
. . . . 5
⊢ ((𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) ∈ V → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) = ( ·𝑠
‘𝑀)) |
| 29 | 27, 28 | syl 17 |
. . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) = ( ·𝑠
‘𝑀)) |
| 30 | 29 | eqcomd 2743 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (
·𝑠 ‘𝑀) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)) |
| 31 | 30 | oveqd 7448 |
. 2
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠
‘𝑀)𝐼) = ((𝑞(+g‘(Scalar‘𝑀))𝑟)(𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)𝐼)) |
| 32 | | simprr 773 |
. . . . 5
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) ∧ (𝑥 = 𝑞 ∧ 𝑦 = 𝐼)) → 𝑦 = 𝐼) |
| 33 | 30, 32, 10, 19, 19 | ovmpod 7585 |
. . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞( ·𝑠
‘𝑀)𝐼) = 𝐼) |
| 34 | | simprr 773 |
. . . . 5
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) ∧ (𝑥 = 𝑟 ∧ 𝑦 = 𝐼)) → 𝑦 = 𝐼) |
| 35 | 30, 34, 11, 19, 19 | ovmpod 7585 |
. . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑟( ·𝑠
‘𝑀)𝐼) = 𝐼) |
| 36 | 33, 35 | oveq12d 7449 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞( ·𝑠
‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠
‘𝑀)𝐼)) = (𝐼(+g‘𝑀)𝐼)) |
| 37 | | snex 5436 |
. . . . . 6
⊢
{〈〈𝐼,
𝐼〉, 𝐼〉} ∈ V |
| 38 | 4 | lmodplusg 17371 |
. . . . . 6
⊢
({〈〈𝐼,
𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
| 39 | 37, 38 | mp1i 13 |
. . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
| 40 | 39 | eqcomd 2743 |
. . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (+g‘𝑀) = {〈〈𝐼, 𝐼〉, 𝐼〉}) |
| 41 | 40 | oveqd 7448 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼(+g‘𝑀)𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) |
| 42 | | df-ov 7434 |
. . . 4
⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) |
| 43 | | opex 5469 |
. . . . . . 7
⊢
〈𝐼, 𝐼〉 ∈ V |
| 44 | 20, 43 | jctil 519 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉)) |
| 45 | 44 | adantr 480 |
. . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉)) |
| 46 | | fvsng 7200 |
. . . . 5
⊢
((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
| 47 | 45, 46 | syl 17 |
. . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
| 48 | 42, 47 | eqtrid 2789 |
. . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
| 49 | 36, 41, 48 | 3eqtrd 2781 |
. 2
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞( ·𝑠
‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠
‘𝑀)𝐼)) = 𝐼) |
| 50 | 22, 31, 49 | 3eqtr4d 2787 |
1
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠
‘𝑀)𝐼) = ((𝑞( ·𝑠
‘𝑀)𝐼)(+g‘𝑀)(𝑟( ·𝑠
‘𝑀)𝐼))) |