Step | Hyp | Ref
| Expression |
1 | | xp1st 7577 |
. . . . . 6
⊢ (𝐴 ∈ (N ×
N) → (1st ‘𝐴) ∈ N) |
2 | 1 | 3ad2ant1 1126 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1st ‘𝐴) ∈ N) |
3 | | xp2nd 7578 |
. . . . . 6
⊢ (𝐶 ∈ (N ×
N) → (2nd ‘𝐶) ∈ N) |
4 | 3 | 3ad2ant3 1128 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2nd ‘𝐶) ∈ N) |
5 | | mulclpi 10161 |
. . . . 5
⊢
(((1st ‘𝐴) ∈ N ∧
(2nd ‘𝐶)
∈ N) → ((1st ‘𝐴) ·N
(2nd ‘𝐶))
∈ N) |
6 | 2, 4, 5 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐴) ·N
(2nd ‘𝐶))
∈ N) |
7 | | xp1st 7577 |
. . . . . 6
⊢ (𝐶 ∈ (N ×
N) → (1st ‘𝐶) ∈ N) |
8 | 7 | 3ad2ant3 1128 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1st ‘𝐶) ∈ N) |
9 | | xp2nd 7578 |
. . . . . 6
⊢ (𝐴 ∈ (N ×
N) → (2nd ‘𝐴) ∈ N) |
10 | 9 | 3ad2ant1 1126 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2nd ‘𝐴) ∈ N) |
11 | | mulclpi 10161 |
. . . . 5
⊢
(((1st ‘𝐶) ∈ N ∧
(2nd ‘𝐴)
∈ N) → ((1st ‘𝐶) ·N
(2nd ‘𝐴))
∈ N) |
12 | 8, 10, 11 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐶) ·N
(2nd ‘𝐴))
∈ N) |
13 | | addclpi 10160 |
. . . 4
⊢
((((1st ‘𝐴) ·N
(2nd ‘𝐶))
∈ N ∧ ((1st ‘𝐶) ·N
(2nd ‘𝐴))
∈ N) → (((1st ‘𝐴) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐴)))
∈ N) |
14 | 6, 12, 13 | syl2anc 584 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((1st ‘𝐴) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐴)))
∈ N) |
15 | | mulclpi 10161 |
. . . 4
⊢
(((2nd ‘𝐴) ∈ N ∧
(2nd ‘𝐶)
∈ N) → ((2nd ‘𝐴) ·N
(2nd ‘𝐶))
∈ N) |
16 | 10, 4, 15 | syl2anc 584 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2nd ‘𝐴) ·N
(2nd ‘𝐶))
∈ N) |
17 | | xp1st 7577 |
. . . . . 6
⊢ (𝐵 ∈ (N ×
N) → (1st ‘𝐵) ∈ N) |
18 | 17 | 3ad2ant2 1127 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1st ‘𝐵) ∈ N) |
19 | | mulclpi 10161 |
. . . . 5
⊢
(((1st ‘𝐵) ∈ N ∧
(2nd ‘𝐶)
∈ N) → ((1st ‘𝐵) ·N
(2nd ‘𝐶))
∈ N) |
20 | 18, 4, 19 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐵) ·N
(2nd ‘𝐶))
∈ N) |
21 | | xp2nd 7578 |
. . . . . 6
⊢ (𝐵 ∈ (N ×
N) → (2nd ‘𝐵) ∈ N) |
22 | 21 | 3ad2ant2 1127 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2nd ‘𝐵) ∈ N) |
23 | | mulclpi 10161 |
. . . . 5
⊢
(((1st ‘𝐶) ∈ N ∧
(2nd ‘𝐵)
∈ N) → ((1st ‘𝐶) ·N
(2nd ‘𝐵))
∈ N) |
24 | 8, 22, 23 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐶) ·N
(2nd ‘𝐵))
∈ N) |
25 | | addclpi 10160 |
. . . 4
⊢
((((1st ‘𝐵) ·N
(2nd ‘𝐶))
∈ N ∧ ((1st ‘𝐶) ·N
(2nd ‘𝐵))
∈ N) → (((1st ‘𝐵) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
∈ N) |
26 | 20, 24, 25 | syl2anc 584 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((1st ‘𝐵) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
∈ N) |
27 | | mulclpi 10161 |
. . . 4
⊢
(((2nd ‘𝐵) ∈ N ∧
(2nd ‘𝐶)
∈ N) → ((2nd ‘𝐵) ·N
(2nd ‘𝐶))
∈ N) |
28 | 22, 4, 27 | syl2anc 584 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2nd ‘𝐵) ·N
(2nd ‘𝐶))
∈ N) |
29 | | enqbreq 10187 |
. . 3
⊢
((((((1st ‘𝐴) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐴)))
∈ N ∧ ((2nd ‘𝐴) ·N
(2nd ‘𝐶))
∈ N) ∧ ((((1st ‘𝐵) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
∈ N ∧ ((2nd ‘𝐵) ·N
(2nd ‘𝐶))
∈ N)) → (〈(((1st ‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉 ~Q
〈(((1st ‘𝐵) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐵))), ((2nd ‘𝐵)
·N (2nd ‘𝐶))〉 ↔ ((((1st
‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
(((1st ‘𝐵)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐵)))))) |
30 | 14, 16, 26, 28, 29 | syl22anc 835 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (〈(((1st ‘𝐴) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐴))), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉 ~Q
〈(((1st ‘𝐵) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐵))), ((2nd ‘𝐵)
·N (2nd ‘𝐶))〉 ↔ ((((1st
‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
(((1st ‘𝐵)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐵)))))) |
31 | | addpipq2 10204 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐶
∈ (N × N)) → (𝐴 +pQ 𝐶) = 〈(((1st
‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉) |
32 | 31 | 3adant2 1124 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
+pQ 𝐶) = 〈(((1st ‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉) |
33 | | addpipq2 10204 |
. . . 4
⊢ ((𝐵 ∈ (N ×
N) ∧ 𝐶
∈ (N × N)) → (𝐵 +pQ 𝐶) = 〈(((1st
‘𝐵)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐵))), ((2nd ‘𝐵)
·N (2nd ‘𝐶))〉) |
34 | 33 | 3adant1 1123 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐵
+pQ 𝐶) = 〈(((1st ‘𝐵)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐵))), ((2nd ‘𝐵)
·N (2nd ‘𝐶))〉) |
35 | 32, 34 | breq12d 4975 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((𝐴 +pQ 𝐶) ~Q
(𝐵
+pQ 𝐶) ↔ 〈(((1st
‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉 ~Q
〈(((1st ‘𝐵) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐵))), ((2nd ‘𝐵)
·N (2nd ‘𝐶))〉)) |
36 | | enqbreq2 10188 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st
‘𝐴)
·N (2nd ‘𝐵)) = ((1st ‘𝐵)
·N (2nd ‘𝐴)))) |
37 | 36 | 3adant3 1125 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~Q 𝐵 ↔ ((1st ‘𝐴)
·N (2nd ‘𝐵)) = ((1st ‘𝐵)
·N (2nd ‘𝐴)))) |
38 | | mulclpi 10161 |
. . . . 5
⊢
(((2nd ‘𝐶) ∈ N ∧
(2nd ‘𝐶)
∈ N) → ((2nd ‘𝐶) ·N
(2nd ‘𝐶))
∈ N) |
39 | 4, 4, 38 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2nd ‘𝐶) ·N
(2nd ‘𝐶))
∈ N) |
40 | | mulclpi 10161 |
. . . . 5
⊢
(((1st ‘𝐴) ∈ N ∧
(2nd ‘𝐵)
∈ N) → ((1st ‘𝐴) ·N
(2nd ‘𝐵))
∈ N) |
41 | 2, 22, 40 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐴) ·N
(2nd ‘𝐵))
∈ N) |
42 | | mulcanpi 10168 |
. . . 4
⊢
((((2nd ‘𝐶) ·N
(2nd ‘𝐶))
∈ N ∧ ((1st ‘𝐴) ·N
(2nd ‘𝐵))
∈ N) → ((((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
↔ ((1st ‘𝐴) ·N
(2nd ‘𝐵))
= ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) |
43 | 39, 41, 42 | syl2anc 584 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
↔ ((1st ‘𝐴) ·N
(2nd ‘𝐵))
= ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) |
44 | | mulcompi 10164 |
. . . . . . . 8
⊢
(((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐴) ·N
(2nd ‘𝐵))
·N ((2nd ‘𝐶) ·N
(2nd ‘𝐶))) |
45 | | fvex 6551 |
. . . . . . . . 9
⊢
(1st ‘𝐴) ∈ V |
46 | | fvex 6551 |
. . . . . . . . 9
⊢
(2nd ‘𝐵) ∈ V |
47 | | fvex 6551 |
. . . . . . . . 9
⊢
(2nd ‘𝐶) ∈ V |
48 | | mulcompi 10164 |
. . . . . . . . 9
⊢ (𝑥
·N 𝑦) = (𝑦 ·N 𝑥) |
49 | | mulasspi 10165 |
. . . . . . . . 9
⊢ ((𝑥
·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦
·N 𝑧)) |
50 | 45, 46, 47, 48, 49, 47 | caov4 7235 |
. . . . . . . 8
⊢
(((1st ‘𝐴) ·N
(2nd ‘𝐵))
·N ((2nd ‘𝐶) ·N
(2nd ‘𝐶)))
= (((1st ‘𝐴) ·N
(2nd ‘𝐶))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶))) |
51 | 44, 50 | eqtri 2819 |
. . . . . . 7
⊢
(((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐴) ·N
(2nd ‘𝐶))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶))) |
52 | | fvex 6551 |
. . . . . . . . 9
⊢
(2nd ‘𝐴) ∈ V |
53 | | fvex 6551 |
. . . . . . . . 9
⊢
(1st ‘𝐶) ∈ V |
54 | 52, 47, 53, 48, 49, 46 | caov4 7235 |
. . . . . . . 8
⊢
(((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
= (((2nd ‘𝐴) ·N
(1st ‘𝐶))
·N ((2nd ‘𝐶) ·N
(2nd ‘𝐵))) |
55 | | mulcompi 10164 |
. . . . . . . . 9
⊢
((2nd ‘𝐴) ·N
(1st ‘𝐶))
= ((1st ‘𝐶) ·N
(2nd ‘𝐴)) |
56 | | mulcompi 10164 |
. . . . . . . . 9
⊢
((2nd ‘𝐶) ·N
(2nd ‘𝐵))
= ((2nd ‘𝐵) ·N
(2nd ‘𝐶)) |
57 | 55, 56 | oveq12i 7028 |
. . . . . . . 8
⊢
(((2nd ‘𝐴) ·N
(1st ‘𝐶))
·N ((2nd ‘𝐶) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐶) ·N
(2nd ‘𝐴))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶))) |
58 | 54, 57 | eqtri 2819 |
. . . . . . 7
⊢
(((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐶) ·N
(2nd ‘𝐴))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶))) |
59 | 51, 58 | oveq12i 7028 |
. . . . . 6
⊢
((((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))) = ((((1st ‘𝐴)
·N (2nd ‘𝐶)) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) +N
(((1st ‘𝐶)
·N (2nd ‘𝐴)) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶)))) |
60 | | addcompi 10162 |
. . . . . 6
⊢
((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))) = ((((2nd ‘𝐶)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐴)
·N (2nd ‘𝐵))) +N
(((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐶)
·N (2nd ‘𝐵)))) |
61 | | ovex 7048 |
. . . . . . 7
⊢
((1st ‘𝐴) ·N
(2nd ‘𝐶))
∈ V |
62 | | ovex 7048 |
. . . . . . 7
⊢
((1st ‘𝐶) ·N
(2nd ‘𝐴))
∈ V |
63 | | ovex 7048 |
. . . . . . 7
⊢
((2nd ‘𝐵) ·N
(2nd ‘𝐶))
∈ V |
64 | | distrpi 10166 |
. . . . . . 7
⊢ (𝑥
·N (𝑦 +N 𝑧)) = ((𝑥 ·N 𝑦) +N
(𝑥
·N 𝑧)) |
65 | 61, 62, 63, 48, 64 | caovdir 7238 |
. . . . . 6
⊢
((((1st ‘𝐴) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐴)))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶)))
= ((((1st ‘𝐴) ·N
(2nd ‘𝐶))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶)))
+N (((1st ‘𝐶) ·N
(2nd ‘𝐴))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶)))) |
66 | 59, 60, 65 | 3eqtr4i 2829 |
. . . . 5
⊢
((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))) = ((((1st ‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) |
67 | | addcompi 10162 |
. . . . . 6
⊢
((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐶)))) = ((((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐶))) +N
(((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐶)
·N (2nd ‘𝐵)))) |
68 | | mulasspi 10165 |
. . . . . . . 8
⊢
(((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
= ((2nd ‘𝐶) ·N
((2nd ‘𝐶)
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) |
69 | | mulcompi 10164 |
. . . . . . . . . 10
⊢
((2nd ‘𝐶) ·N
((2nd ‘𝐶)
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) = (((2nd ‘𝐶)
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
·N (2nd ‘𝐶)) |
70 | | mulasspi 10165 |
. . . . . . . . . . . 12
⊢
(((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N (1st ‘𝐵)) = ((2nd ‘𝐴)
·N ((2nd ‘𝐶) ·N
(1st ‘𝐵))) |
71 | | mulcompi 10164 |
. . . . . . . . . . . 12
⊢
((2nd ‘𝐴) ·N
((2nd ‘𝐶)
·N (1st ‘𝐵))) = (((2nd ‘𝐶)
·N (1st ‘𝐵)) ·N
(2nd ‘𝐴)) |
72 | | mulasspi 10165 |
. . . . . . . . . . . 12
⊢
(((2nd ‘𝐶) ·N
(1st ‘𝐵))
·N (2nd ‘𝐴)) = ((2nd ‘𝐶)
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴))) |
73 | 70, 71, 72 | 3eqtrri 2824 |
. . . . . . . . . . 11
⊢
((2nd ‘𝐶) ·N
((1st ‘𝐵)
·N (2nd ‘𝐴))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
(1st ‘𝐵)) |
74 | 73 | oveq1i 7026 |
. . . . . . . . . 10
⊢
(((2nd ‘𝐶) ·N
((1st ‘𝐵)
·N (2nd ‘𝐴))) ·N
(2nd ‘𝐶))
= ((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N (1st ‘𝐵)) ·N
(2nd ‘𝐶)) |
75 | 69, 74 | eqtri 2819 |
. . . . . . . . 9
⊢
((2nd ‘𝐶) ·N
((2nd ‘𝐶)
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) = ((((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
(1st ‘𝐵))
·N (2nd ‘𝐶)) |
76 | | mulasspi 10165 |
. . . . . . . . 9
⊢
((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N (1st ‘𝐵)) ·N
(2nd ‘𝐶))
= (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐶))) |
77 | 75, 76 | eqtri 2819 |
. . . . . . . 8
⊢
((2nd ‘𝐶) ·N
((2nd ‘𝐶)
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐶))) |
78 | 68, 77 | eqtri 2819 |
. . . . . . 7
⊢
(((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
= (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐶))) |
79 | 78 | oveq2i 7027 |
. . . . . 6
⊢
((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) = ((((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐶)
·N (2nd ‘𝐵))) +N
(((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐶)))) |
80 | | distrpi 10166 |
. . . . . 6
⊢
(((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N (((1st ‘𝐵) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))) = ((((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐶))) +N
(((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐶)
·N (2nd ‘𝐵)))) |
81 | 67, 79, 80 | 3eqtr4i 2829 |
. . . . 5
⊢
((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
(((1st ‘𝐵)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐵)))) |
82 | 66, 81 | eqeq12i 2809 |
. . . 4
⊢
(((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))) = ((((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐶)
·N (2nd ‘𝐵))) +N
(((2nd ‘𝐶)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐴)))) ↔ ((((1st ‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
(((1st ‘𝐵)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐵))))) |
83 | | mulclpi 10161 |
. . . . . 6
⊢
((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
∈ N ∧ ((1st ‘𝐶) ·N
(2nd ‘𝐵))
∈ N) → (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
∈ N) |
84 | 16, 24, 83 | syl2anc 584 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
∈ N) |
85 | | mulclpi 10161 |
. . . . . 6
⊢
((((2nd ‘𝐶) ·N
(2nd ‘𝐶))
∈ N ∧ ((1st ‘𝐴) ·N
(2nd ‘𝐵))
∈ N) → (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
∈ N) |
86 | 39, 41, 85 | syl2anc 584 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
∈ N) |
87 | | addcanpi 10167 |
. . . . 5
⊢
(((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
∈ N ∧ (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
∈ N) → (((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))) = ((((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐶)
·N (2nd ‘𝐵))) +N
(((2nd ‘𝐶)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐴)))) ↔ (((2nd ‘𝐶)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐴)
·N (2nd ‘𝐵))) = (((2nd ‘𝐶)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐴))))) |
88 | 84, 86, 87 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))
+N (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))) = ((((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐶)
·N (2nd ‘𝐵))) +N
(((2nd ‘𝐶)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐴)))) ↔ (((2nd ‘𝐶)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐴)
·N (2nd ‘𝐵))) = (((2nd ‘𝐶)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (2nd ‘𝐴))))) |
89 | 82, 88 | syl5rbbr 287 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((2nd ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
↔ ((((1st ‘𝐴) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐴)))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶)))
= (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N (((1st ‘𝐵) ·N
(2nd ‘𝐶))
+N ((1st ‘𝐶) ·N
(2nd ‘𝐵)))))) |
90 | 37, 43, 89 | 3bitr2d 308 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~Q 𝐵 ↔ ((((1st ‘𝐴)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐴))) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
(((1st ‘𝐵)
·N (2nd ‘𝐶)) +N
((1st ‘𝐶)
·N (2nd ‘𝐵)))))) |
91 | 30, 35, 90 | 3bitr4rd 313 |
1
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~Q 𝐵 ↔ (𝐴 +pQ 𝐶) ~Q
(𝐵
+pQ 𝐶))) |