| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | rrgval.e | . 2
⊢ 𝐸 = (RLReg‘𝑅) | 
| 2 |  | fveq2 6905 | . . . . . 6
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | 
| 3 |  | rrgval.b | . . . . . 6
⊢ 𝐵 = (Base‘𝑅) | 
| 4 | 2, 3 | eqtr4di 2794 | . . . . 5
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) | 
| 5 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | 
| 6 |  | rrgval.t | . . . . . . . . . 10
⊢  · =
(.r‘𝑅) | 
| 7 | 5, 6 | eqtr4di 2794 | . . . . . . . . 9
⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) | 
| 8 | 7 | oveqd 7449 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦)) | 
| 9 |  | fveq2 6905 | . . . . . . . . 9
⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | 
| 10 |  | rrgval.z | . . . . . . . . 9
⊢  0 =
(0g‘𝑅) | 
| 11 | 9, 10 | eqtr4di 2794 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) | 
| 12 | 8, 11 | eqeq12d 2752 | . . . . . . 7
⊢ (𝑟 = 𝑅 → ((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) ↔ (𝑥 · 𝑦) = 0 )) | 
| 13 | 11 | eqeq2d 2747 | . . . . . . 7
⊢ (𝑟 = 𝑅 → (𝑦 = (0g‘𝑟) ↔ 𝑦 = 0 )) | 
| 14 | 12, 13 | imbi12d 344 | . . . . . 6
⊢ (𝑟 = 𝑅 → (((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟)) ↔ ((𝑥 · 𝑦) = 0 → 𝑦 = 0 ))) | 
| 15 | 4, 14 | raleqbidv 3345 | . . . . 5
⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟)) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 ))) | 
| 16 | 4, 15 | rabeqbidv 3454 | . . . 4
⊢ (𝑟 = 𝑅 → {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}) | 
| 17 |  | df-rlreg 20695 | . . . 4
⊢ RLReg =
(𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))}) | 
| 18 | 3 | fvexi 6919 | . . . . 5
⊢ 𝐵 ∈ V | 
| 19 | 18 | rabex 5338 | . . . 4
⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} ∈
V | 
| 20 | 16, 17, 19 | fvmpt 7015 | . . 3
⊢ (𝑅 ∈ V →
(RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}) | 
| 21 |  | fvprc 6897 | . . . 4
⊢ (¬
𝑅 ∈ V →
(RLReg‘𝑅) =
∅) | 
| 22 |  | fvprc 6897 | . . . . . . 7
⊢ (¬
𝑅 ∈ V →
(Base‘𝑅) =
∅) | 
| 23 | 3, 22 | eqtrid 2788 | . . . . . 6
⊢ (¬
𝑅 ∈ V → 𝐵 = ∅) | 
| 24 | 23 | rabeqdv 3451 | . . . . 5
⊢ (¬
𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}) | 
| 25 |  | rab0 4385 | . . . . 5
⊢ {𝑥 ∈ ∅ ∣
∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} =
∅ | 
| 26 | 24, 25 | eqtrdi 2792 | . . . 4
⊢ (¬
𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} =
∅) | 
| 27 | 21, 26 | eqtr4d 2779 | . . 3
⊢ (¬
𝑅 ∈ V →
(RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}) | 
| 28 | 20, 27 | pm2.61i 182 | . 2
⊢
(RLReg‘𝑅) =
{𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} | 
| 29 | 1, 28 | eqtri 2764 | 1
⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} |