| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-nr 11096 | . 2
⊢
R = ((P × P) /
~R ) | 
| 2 |  | oveq1 7438 | . . 3
⊢
([〈𝑥, 𝑦〉]
~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R
·R 0R) = (𝐴
·R
0R)) | 
| 3 | 2 | eqeq1d 2739 | . 2
⊢
([〈𝑥, 𝑦〉]
~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R
·R 0R) =
0R ↔ (𝐴 ·R
0R) = 0R)) | 
| 4 |  | 1pr 11055 | . . . . 5
⊢
1P ∈ P | 
| 5 |  | mulsrpr 11116 | . . . . 5
⊢ (((𝑥 ∈ P ∧
𝑦 ∈ P)
∧ (1P ∈ P ∧
1P ∈ P)) → ([〈𝑥, 𝑦〉] ~R
·R [〈1P,
1P〉] ~R ) =
[〈((𝑥
·P 1P)
+P (𝑦 ·P
1P)), ((𝑥 ·P
1P) +P (𝑦 ·P
1P))〉] ~R
) | 
| 6 | 4, 4, 5 | mpanr12 705 | . . . 4
⊢ ((𝑥 ∈ P ∧
𝑦 ∈ P)
→ ([〈𝑥, 𝑦〉]
~R ·R
[〈1P, 1P〉]
~R ) = [〈((𝑥 ·P
1P) +P (𝑦 ·P
1P)), ((𝑥 ·P
1P) +P (𝑦 ·P
1P))〉] ~R
) | 
| 7 |  | mulclpr 11060 | . . . . . . . . . 10
⊢ ((𝑥 ∈ P ∧
1P ∈ P) → (𝑥 ·P
1P) ∈ P) | 
| 8 | 4, 7 | mpan2 691 | . . . . . . . . 9
⊢ (𝑥 ∈ P →
(𝑥
·P 1P) ∈
P) | 
| 9 |  | mulclpr 11060 | . . . . . . . . . 10
⊢ ((𝑦 ∈ P ∧
1P ∈ P) → (𝑦 ·P
1P) ∈ P) | 
| 10 | 4, 9 | mpan2 691 | . . . . . . . . 9
⊢ (𝑦 ∈ P →
(𝑦
·P 1P) ∈
P) | 
| 11 |  | addclpr 11058 | . . . . . . . . 9
⊢ (((𝑥
·P 1P) ∈
P ∧ (𝑦
·P 1P) ∈
P) → ((𝑥
·P 1P)
+P (𝑦 ·P
1P)) ∈ P) | 
| 12 | 8, 10, 11 | syl2an 596 | . . . . . . . 8
⊢ ((𝑥 ∈ P ∧
𝑦 ∈ P)
→ ((𝑥
·P 1P)
+P (𝑦 ·P
1P)) ∈ P) | 
| 13 | 12, 12 | anim12i 613 | . . . . . . 7
⊢ (((𝑥 ∈ P ∧
𝑦 ∈ P)
∧ (𝑥 ∈
P ∧ 𝑦
∈ P)) → (((𝑥 ·P
1P) +P (𝑦 ·P
1P)) ∈ P ∧ ((𝑥 ·P
1P) +P (𝑦 ·P
1P)) ∈ P)) | 
| 14 |  | eqid 2737 | . . . . . . . 8
⊢ (((𝑥
·P 1P)
+P (𝑦 ·P
1P)) +P
1P) = (((𝑥 ·P
1P) +P (𝑦 ·P
1P)) +P
1P) | 
| 15 |  | enreceq 11106 | . . . . . . . 8
⊢
(((((𝑥
·P 1P)
+P (𝑦 ·P
1P)) ∈ P ∧ ((𝑥 ·P
1P) +P (𝑦 ·P
1P)) ∈ P) ∧
(1P ∈ P ∧
1P ∈ P)) → ([〈((𝑥
·P 1P)
+P (𝑦 ·P
1P)), ((𝑥 ·P
1P) +P (𝑦 ·P
1P))〉] ~R =
[〈1P, 1P〉]
~R ↔ (((𝑥 ·P
1P) +P (𝑦 ·P
1P)) +P
1P) = (((𝑥 ·P
1P) +P (𝑦 ·P
1P)) +P
1P))) | 
| 16 | 14, 15 | mpbiri 258 | . . . . . . 7
⊢
(((((𝑥
·P 1P)
+P (𝑦 ·P
1P)) ∈ P ∧ ((𝑥 ·P
1P) +P (𝑦 ·P
1P)) ∈ P) ∧
(1P ∈ P ∧
1P ∈ P)) → [〈((𝑥
·P 1P)
+P (𝑦 ·P
1P)), ((𝑥 ·P
1P) +P (𝑦 ·P
1P))〉] ~R =
[〈1P, 1P〉]
~R ) | 
| 17 | 13, 16 | sylan 580 | . . . . . 6
⊢ ((((𝑥 ∈ P ∧
𝑦 ∈ P)
∧ (𝑥 ∈
P ∧ 𝑦
∈ P)) ∧ (1P ∈
P ∧ 1P ∈ P))
→ [〈((𝑥
·P 1P)
+P (𝑦 ·P
1P)), ((𝑥 ·P
1P) +P (𝑦 ·P
1P))〉] ~R =
[〈1P, 1P〉]
~R ) | 
| 18 | 4, 4, 17 | mpanr12 705 | . . . . 5
⊢ (((𝑥 ∈ P ∧
𝑦 ∈ P)
∧ (𝑥 ∈
P ∧ 𝑦
∈ P)) → [〈((𝑥 ·P
1P) +P (𝑦 ·P
1P)), ((𝑥 ·P
1P) +P (𝑦 ·P
1P))〉] ~R =
[〈1P, 1P〉]
~R ) | 
| 19 | 18 | anidms 566 | . . . 4
⊢ ((𝑥 ∈ P ∧
𝑦 ∈ P)
→ [〈((𝑥
·P 1P)
+P (𝑦 ·P
1P)), ((𝑥 ·P
1P) +P (𝑦 ·P
1P))〉] ~R =
[〈1P, 1P〉]
~R ) | 
| 20 | 6, 19 | eqtrd 2777 | . . 3
⊢ ((𝑥 ∈ P ∧
𝑦 ∈ P)
→ ([〈𝑥, 𝑦〉]
~R ·R
[〈1P, 1P〉]
~R ) = [〈1P,
1P〉] ~R
) | 
| 21 |  | df-0r 11100 | . . . 4
⊢
0R = [〈1P,
1P〉] ~R | 
| 22 | 21 | oveq2i 7442 | . . 3
⊢
([〈𝑥, 𝑦〉]
~R ·R
0R) = ([〈𝑥, 𝑦〉] ~R
·R [〈1P,
1P〉] ~R
) | 
| 23 | 20, 22, 21 | 3eqtr4g 2802 | . 2
⊢ ((𝑥 ∈ P ∧
𝑦 ∈ P)
→ ([〈𝑥, 𝑦〉]
~R ·R
0R) = 0R) | 
| 24 | 1, 3, 23 | ecoptocl 8847 | 1
⊢ (𝐴 ∈ R →
(𝐴
·R 0R) =
0R) |