Step | Hyp | Ref
| Expression |
1 | | xp1st 7793 |
. . . . 5
⊢ (𝐴 ∈ (N ×
N) → (1st ‘𝐴) ∈ N) |
2 | 1 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1st ‘𝐴) ∈ N) |
3 | | xp1st 7793 |
. . . . 5
⊢ (𝐶 ∈ (N ×
N) → (1st ‘𝐶) ∈ N) |
4 | 3 | 3ad2ant3 1137 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1st ‘𝐶) ∈ N) |
5 | | mulclpi 10507 |
. . . 4
⊢
(((1st ‘𝐴) ∈ N ∧
(1st ‘𝐶)
∈ N) → ((1st ‘𝐴) ·N
(1st ‘𝐶))
∈ N) |
6 | 2, 4, 5 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐴) ·N
(1st ‘𝐶))
∈ N) |
7 | | xp2nd 7794 |
. . . . 5
⊢ (𝐴 ∈ (N ×
N) → (2nd ‘𝐴) ∈ N) |
8 | 7 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2nd ‘𝐴) ∈ N) |
9 | | xp2nd 7794 |
. . . . 5
⊢ (𝐶 ∈ (N ×
N) → (2nd ‘𝐶) ∈ N) |
10 | 9 | 3ad2ant3 1137 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2nd ‘𝐶) ∈ N) |
11 | | mulclpi 10507 |
. . . 4
⊢
(((2nd ‘𝐴) ∈ N ∧
(2nd ‘𝐶)
∈ N) → ((2nd ‘𝐴) ·N
(2nd ‘𝐶))
∈ N) |
12 | 8, 10, 11 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2nd ‘𝐴) ·N
(2nd ‘𝐶))
∈ N) |
13 | | xp1st 7793 |
. . . . 5
⊢ (𝐵 ∈ (N ×
N) → (1st ‘𝐵) ∈ N) |
14 | 13 | 3ad2ant2 1136 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1st ‘𝐵) ∈ N) |
15 | | mulclpi 10507 |
. . . 4
⊢
(((1st ‘𝐵) ∈ N ∧
(1st ‘𝐶)
∈ N) → ((1st ‘𝐵) ·N
(1st ‘𝐶))
∈ N) |
16 | 14, 4, 15 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐵) ·N
(1st ‘𝐶))
∈ N) |
17 | | xp2nd 7794 |
. . . . 5
⊢ (𝐵 ∈ (N ×
N) → (2nd ‘𝐵) ∈ N) |
18 | 17 | 3ad2ant2 1136 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2nd ‘𝐵) ∈ N) |
19 | | mulclpi 10507 |
. . . 4
⊢
(((2nd ‘𝐵) ∈ N ∧
(2nd ‘𝐶)
∈ N) → ((2nd ‘𝐵) ·N
(2nd ‘𝐶))
∈ N) |
20 | 18, 10, 19 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2nd ‘𝐵) ·N
(2nd ‘𝐶))
∈ N) |
21 | | enqbreq 10533 |
. . 3
⊢
(((((1st ‘𝐴) ·N
(1st ‘𝐶))
∈ N ∧ ((2nd ‘𝐴) ·N
(2nd ‘𝐶))
∈ N) ∧ (((1st ‘𝐵) ·N
(1st ‘𝐶))
∈ N ∧ ((2nd ‘𝐵) ·N
(2nd ‘𝐶))
∈ N)) → (〈((1st ‘𝐴)
·N (1st ‘𝐶)), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉 ~Q
〈((1st ‘𝐵) ·N
(1st ‘𝐶)),
((2nd ‘𝐵)
·N (2nd ‘𝐶))〉 ↔ (((1st
‘𝐴)
·N (1st ‘𝐶)) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (1st ‘𝐶))))) |
22 | 6, 12, 16, 20, 21 | syl22anc 839 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (〈((1st ‘𝐴) ·N
(1st ‘𝐶)),
((2nd ‘𝐴)
·N (2nd ‘𝐶))〉 ~Q
〈((1st ‘𝐵) ·N
(1st ‘𝐶)),
((2nd ‘𝐵)
·N (2nd ‘𝐶))〉 ↔ (((1st
‘𝐴)
·N (1st ‘𝐶)) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (1st ‘𝐶))))) |
23 | | mulpipq2 10553 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐶
∈ (N × N)) → (𝐴 ·pQ 𝐶) = 〈((1st
‘𝐴)
·N (1st ‘𝐶)), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉) |
24 | 23 | 3adant2 1133 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
·pQ 𝐶) = 〈((1st ‘𝐴)
·N (1st ‘𝐶)), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉) |
25 | | mulpipq2 10553 |
. . . 4
⊢ ((𝐵 ∈ (N ×
N) ∧ 𝐶
∈ (N × N)) → (𝐵 ·pQ 𝐶) = 〈((1st
‘𝐵)
·N (1st ‘𝐶)), ((2nd ‘𝐵)
·N (2nd ‘𝐶))〉) |
26 | 25 | 3adant1 1132 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐵
·pQ 𝐶) = 〈((1st ‘𝐵)
·N (1st ‘𝐶)), ((2nd ‘𝐵)
·N (2nd ‘𝐶))〉) |
27 | 24, 26 | breq12d 5066 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((𝐴 ·pQ 𝐶) ~Q
(𝐵
·pQ 𝐶) ↔ 〈((1st ‘𝐴)
·N (1st ‘𝐶)), ((2nd ‘𝐴)
·N (2nd ‘𝐶))〉 ~Q
〈((1st ‘𝐵) ·N
(1st ‘𝐶)),
((2nd ‘𝐵)
·N (2nd ‘𝐶))〉)) |
28 | | enqbreq2 10534 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st
‘𝐴)
·N (2nd ‘𝐵)) = ((1st ‘𝐵)
·N (2nd ‘𝐴)))) |
29 | 28 | 3adant3 1134 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~Q 𝐵 ↔ ((1st ‘𝐴)
·N (2nd ‘𝐵)) = ((1st ‘𝐵)
·N (2nd ‘𝐴)))) |
30 | | mulclpi 10507 |
. . . . 5
⊢
(((1st ‘𝐶) ∈ N ∧
(2nd ‘𝐶)
∈ N) → ((1st ‘𝐶) ·N
(2nd ‘𝐶))
∈ N) |
31 | 4, 10, 30 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐶) ·N
(2nd ‘𝐶))
∈ N) |
32 | | mulclpi 10507 |
. . . . 5
⊢
(((1st ‘𝐴) ∈ N ∧
(2nd ‘𝐵)
∈ N) → ((1st ‘𝐴) ·N
(2nd ‘𝐵))
∈ N) |
33 | 2, 18, 32 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1st ‘𝐴) ·N
(2nd ‘𝐵))
∈ N) |
34 | | mulcanpi 10514 |
. . . 4
⊢
((((1st ‘𝐶) ·N
(2nd ‘𝐶))
∈ N ∧ ((1st ‘𝐴) ·N
(2nd ‘𝐵))
∈ N) → ((((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
↔ ((1st ‘𝐴) ·N
(2nd ‘𝐵))
= ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) |
35 | 31, 33, 34 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
↔ ((1st ‘𝐴) ·N
(2nd ‘𝐵))
= ((1st ‘𝐵) ·N
(2nd ‘𝐴)))) |
36 | | mulcompi 10510 |
. . . . . 6
⊢
(((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐴) ·N
(2nd ‘𝐵))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐶))) |
37 | | fvex 6730 |
. . . . . . 7
⊢
(1st ‘𝐴) ∈ V |
38 | | fvex 6730 |
. . . . . . 7
⊢
(2nd ‘𝐵) ∈ V |
39 | | fvex 6730 |
. . . . . . 7
⊢
(1st ‘𝐶) ∈ V |
40 | | mulcompi 10510 |
. . . . . . 7
⊢ (𝑥
·N 𝑦) = (𝑦 ·N 𝑥) |
41 | | mulasspi 10511 |
. . . . . . 7
⊢ ((𝑥
·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦
·N 𝑧)) |
42 | | fvex 6730 |
. . . . . . 7
⊢
(2nd ‘𝐶) ∈ V |
43 | 37, 38, 39, 40, 41, 42 | caov4 7439 |
. . . . . 6
⊢
(((1st ‘𝐴) ·N
(2nd ‘𝐵))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐶)))
= (((1st ‘𝐴) ·N
(1st ‘𝐶))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶))) |
44 | 36, 43 | eqtri 2765 |
. . . . 5
⊢
(((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐴) ·N
(1st ‘𝐶))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶))) |
45 | | mulcompi 10510 |
. . . . . 6
⊢
(((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
= (((1st ‘𝐵) ·N
(2nd ‘𝐴))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐶))) |
46 | | fvex 6730 |
. . . . . . 7
⊢
(1st ‘𝐵) ∈ V |
47 | | fvex 6730 |
. . . . . . 7
⊢
(2nd ‘𝐴) ∈ V |
48 | 46, 47, 39, 40, 41, 42 | caov4 7439 |
. . . . . 6
⊢
(((1st ‘𝐵) ·N
(2nd ‘𝐴))
·N ((1st ‘𝐶) ·N
(2nd ‘𝐶)))
= (((1st ‘𝐵) ·N
(1st ‘𝐶))
·N ((2nd ‘𝐴) ·N
(2nd ‘𝐶))) |
49 | | mulcompi 10510 |
. . . . . 6
⊢
(((1st ‘𝐵) ·N
(1st ‘𝐶))
·N ((2nd ‘𝐴) ·N
(2nd ‘𝐶)))
= (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(1st ‘𝐶))) |
50 | 45, 48, 49 | 3eqtri 2769 |
. . . . 5
⊢
(((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
= (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(1st ‘𝐶))) |
51 | 44, 50 | eqeq12i 2755 |
. . . 4
⊢
((((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
↔ (((1st ‘𝐴) ·N
(1st ‘𝐶))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶)))
= (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(1st ‘𝐶)))) |
52 | 51 | a1i 11 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐴) ·N
(2nd ‘𝐵)))
= (((1st ‘𝐶) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(2nd ‘𝐴)))
↔ (((1st ‘𝐴) ·N
(1st ‘𝐶))
·N ((2nd ‘𝐵) ·N
(2nd ‘𝐶)))
= (((2nd ‘𝐴) ·N
(2nd ‘𝐶))
·N ((1st ‘𝐵) ·N
(1st ‘𝐶))))) |
53 | 29, 35, 52 | 3bitr2d 310 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~Q 𝐵 ↔ (((1st ‘𝐴)
·N (1st ‘𝐶)) ·N
((2nd ‘𝐵)
·N (2nd ‘𝐶))) = (((2nd ‘𝐴)
·N (2nd ‘𝐶)) ·N
((1st ‘𝐵)
·N (1st ‘𝐶))))) |
54 | 22, 27, 53 | 3bitr4rd 315 |
1
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~Q 𝐵 ↔ (𝐴 ·pQ 𝐶) ~Q
(𝐵
·pQ 𝐶))) |