Proof of Theorem lmod1lem4
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fvex 6919 | . . . . . . 7
⊢
(Base‘𝑅)
∈ V | 
| 2 |  | snex 5436 | . . . . . . 7
⊢ {𝐼} ∈ V | 
| 3 | 1, 2 | pm3.2i 470 | . . . . . 6
⊢
((Base‘𝑅)
∈ V ∧ {𝐼} ∈
V) | 
| 4 | 3 | a1i 11 | . . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((Base‘𝑅) ∈ V ∧ {𝐼} ∈ V)) | 
| 5 |  | mpoexga 8102 | . . . . 5
⊢
(((Base‘𝑅)
∈ V ∧ {𝐼} ∈
V) → (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 17373 | . . . . 5
⊢ ((𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) ∈ V → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) = ( ·𝑠
‘𝑀)) | 
| 8 | 4, 5, 7 | 3syl 18 | . . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦) = ( ·𝑠
‘𝑀)) | 
| 9 | 8 | eqcomd 2743 | . . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (
·𝑠 ‘𝑀) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)) | 
| 10 |  | simprr 773 | . . 3
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) ∧ (𝑥 = 𝑞 ∧ 𝑦 = 𝐼)) → 𝑦 = 𝐼) | 
| 11 |  | simprl 771 | . . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑞 ∈ (Base‘𝑅)) | 
| 12 |  | snidg 4660 | . . . 4
⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼}) | 
| 13 | 12 | ad2antrr 726 | . . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼 ∈ {𝐼}) | 
| 14 | 9, 10, 11, 13, 13 | ovmpod 7585 | . 2
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞( ·𝑠
‘𝑀)𝐼) = 𝐼) | 
| 15 |  | simprr 773 | . . . 4
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) ∧ (𝑥 = 𝑟 ∧ 𝑦 = 𝐼)) → 𝑦 = 𝐼) | 
| 16 |  | simprr 773 | . . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑟 ∈ (Base‘𝑅)) | 
| 17 | 9, 15, 16, 13, 13 | ovmpod 7585 | . . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑟( ·𝑠
‘𝑀)𝐼) = 𝐼) | 
| 18 | 17 | oveq2d 7447 | . 2
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞( ·𝑠
‘𝑀)(𝑟(
·𝑠 ‘𝑀)𝐼)) = (𝑞( ·𝑠
‘𝑀)𝐼)) | 
| 19 |  | simprr 773 | . . 3
⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) ∧ (𝑥 = (𝑞(.r‘(Scalar‘𝑀))𝑟) ∧ 𝑦 = 𝐼)) → 𝑦 = 𝐼) | 
| 20 |  | simplr 769 | . . . . . . 7
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring) | 
| 21 | 6 | lmodsca 17372 | . . . . . . . 8
⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀)) | 
| 22 | 21 | fveq2d 6910 | . . . . . . 7
⊢ (𝑅 ∈ Ring →
(.r‘𝑅) =
(.r‘(Scalar‘𝑀))) | 
| 23 | 20, 22 | syl 17 | . . . . . 6
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (.r‘𝑅) =
(.r‘(Scalar‘𝑀))) | 
| 24 | 23 | eqcomd 2743 | . . . . 5
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) →
(.r‘(Scalar‘𝑀)) = (.r‘𝑅)) | 
| 25 | 24 | oveqd 7448 | . . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞(.r‘(Scalar‘𝑀))𝑟) = (𝑞(.r‘𝑅)𝑟)) | 
| 26 |  | eqid 2737 | . . . . . 6
⊢
(Base‘𝑅) =
(Base‘𝑅) | 
| 27 |  | eqid 2737 | . . . . . 6
⊢
(.r‘𝑅) = (.r‘𝑅) | 
| 28 | 26, 27 | ringcl 20247 | . . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑞(.r‘𝑅)𝑟) ∈ (Base‘𝑅)) | 
| 29 | 20, 11, 16, 28 | syl3anc 1373 | . . . 4
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞(.r‘𝑅)𝑟) ∈ (Base‘𝑅)) | 
| 30 | 25, 29 | eqeltrd 2841 | . . 3
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑞(.r‘(Scalar‘𝑀))𝑟) ∈ (Base‘𝑅)) | 
| 31 | 9, 19, 30, 13, 13 | ovmpod 7585 | . 2
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠
‘𝑀)𝐼) = 𝐼) | 
| 32 | 14, 18, 31 | 3eqtr4rd 2788 | 1
⊢ (((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠
‘𝑀)𝐼) = (𝑞( ·𝑠
‘𝑀)(𝑟(
·𝑠 ‘𝑀)𝐼))) |