Step | Hyp | Ref
| Expression |
1 | | xp1st 7242 |
. . . . . 6
⊢ (𝐴 ∈ (N ×
N) → (1^{st} ‘𝐴) ∈ N) |
2 | 1 | 3ad2ant1 1102 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1^{st} ‘𝐴) ∈ N) |
3 | | xp2nd 7243 |
. . . . . 6
⊢ (𝐶 ∈ (N ×
N) → (2^{nd} ‘𝐶) ∈ N) |
4 | 3 | 3ad2ant3 1104 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2^{nd} ‘𝐶) ∈ N) |
5 | | mulclpi 9753 |
. . . . 5
⊢
(((1^{st} ‘𝐴) ∈ N ∧
(2^{nd} ‘𝐶)
∈ N) → ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
6 | 2, 4, 5 | syl2anc 694 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
7 | | xp1st 7242 |
. . . . . 6
⊢ (𝐶 ∈ (N ×
N) → (1^{st} ‘𝐶) ∈ N) |
8 | 7 | 3ad2ant3 1104 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1^{st} ‘𝐶) ∈ N) |
9 | | xp2nd 7243 |
. . . . . 6
⊢ (𝐴 ∈ (N ×
N) → (2^{nd} ‘𝐴) ∈ N) |
10 | 9 | 3ad2ant1 1102 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2^{nd} ‘𝐴) ∈ N) |
11 | | mulclpi 9753 |
. . . . 5
⊢
(((1^{st} ‘𝐶) ∈ N ∧
(2^{nd} ‘𝐴)
∈ N) → ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴))
∈ N) |
12 | 8, 10, 11 | syl2anc 694 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴))
∈ N) |
13 | | addclpi 9752 |
. . . 4
⊢
((((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N ∧ ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴))
∈ N) → (((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴)))
∈ N) |
14 | 6, 12, 13 | syl2anc 694 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴)))
∈ N) |
15 | | mulclpi 9753 |
. . . 4
⊢
(((2^{nd} ‘𝐴) ∈ N ∧
(2^{nd} ‘𝐶)
∈ N) → ((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
16 | 10, 4, 15 | syl2anc 694 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
17 | | xp1st 7242 |
. . . . . 6
⊢ (𝐵 ∈ (N ×
N) → (1^{st} ‘𝐵) ∈ N) |
18 | 17 | 3ad2ant2 1103 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1^{st} ‘𝐵) ∈ N) |
19 | | mulclpi 9753 |
. . . . 5
⊢
(((1^{st} ‘𝐵) ∈ N ∧
(2^{nd} ‘𝐶)
∈ N) → ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
20 | 18, 4, 19 | syl2anc 694 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
21 | | xp2nd 7243 |
. . . . . 6
⊢ (𝐵 ∈ (N ×
N) → (2^{nd} ‘𝐵) ∈ N) |
22 | 21 | 3ad2ant2 1103 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2^{nd} ‘𝐵) ∈ N) |
23 | | mulclpi 9753 |
. . . . 5
⊢
(((1^{st} ‘𝐶) ∈ N ∧
(2^{nd} ‘𝐵)
∈ N) → ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))
∈ N) |
24 | 8, 22, 23 | syl2anc 694 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))
∈ N) |
25 | | addclpi 9752 |
. . . 4
⊢
((((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N ∧ ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))
∈ N) → (((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
∈ N) |
26 | 20, 24, 25 | syl2anc 694 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
∈ N) |
27 | | mulclpi 9753 |
. . . 4
⊢
(((2^{nd} ‘𝐵) ∈ N ∧
(2^{nd} ‘𝐶)
∈ N) → ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
28 | 22, 4, 27 | syl2anc 694 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
29 | | enqbreq 9779 |
. . 3
⊢
((((((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴)))
∈ N ∧ ((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N) ∧ ((((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
∈ N ∧ ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N)) → (⟨(((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩ ~_{Q}
⟨(((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))), ((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩ ↔ ((((1^{st}
‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
(((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵)))))) |
30 | 14, 16, 26, 28, 29 | syl22anc 1367 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (⟨(((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴))), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩ ~_{Q}
⟨(((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))), ((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩ ↔ ((((1^{st}
‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
(((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵)))))) |
31 | | addpipq2 9796 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐶
∈ (N × N)) → (𝐴 +_{pQ} 𝐶) = ⟨(((1^{st}
‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩) |
32 | 31 | 3adant2 1100 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
+_{pQ} 𝐶) = ⟨(((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩) |
33 | | addpipq2 9796 |
. . . 4
⊢ ((𝐵 ∈ (N ×
N) ∧ 𝐶
∈ (N × N)) → (𝐵 +_{pQ} 𝐶) = ⟨(((1^{st}
‘𝐵)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵))), ((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩) |
34 | 33 | 3adant1 1099 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐵
+_{pQ} 𝐶) = ⟨(((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵))), ((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩) |
35 | 32, 34 | breq12d 4698 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((𝐴 +_{pQ} 𝐶) ~_{Q}
(𝐵
+_{pQ} 𝐶) ↔ ⟨(((1^{st}
‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩ ~_{Q}
⟨(((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))), ((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩)) |
36 | | enqbreq2 9780 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N)) → (𝐴 ~_{Q} 𝐵 ↔ ((1^{st}
‘𝐴)
·_{N} (2^{nd} ‘𝐵)) = ((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴)))) |
37 | 36 | 3adant3 1101 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~_{Q} 𝐵 ↔ ((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐵)) = ((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴)))) |
38 | | mulclpi 9753 |
. . . . 5
⊢
(((2^{nd} ‘𝐶) ∈ N ∧
(2^{nd} ‘𝐶)
∈ N) → ((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
39 | 4, 4, 38 | syl2anc 694 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
40 | | mulclpi 9753 |
. . . . 5
⊢
(((1^{st} ‘𝐴) ∈ N ∧
(2^{nd} ‘𝐵)
∈ N) → ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
∈ N) |
41 | 2, 22, 40 | syl2anc 694 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
∈ N) |
42 | | mulcanpi 9760 |
. . . 4
⊢
((((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
∈ N ∧ ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
∈ N) → ((((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
↔ ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
= ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))) |
43 | 39, 41, 42 | syl2anc 694 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
↔ ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
= ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))) |
44 | | mulcompi 9756 |
. . . . . . . 8
⊢
(((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
·_{N} ((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))) |
45 | | fvex 6239 |
. . . . . . . . 9
⊢
(1^{st} ‘𝐴) ∈ V |
46 | | fvex 6239 |
. . . . . . . . 9
⊢
(2^{nd} ‘𝐵) ∈ V |
47 | | fvex 6239 |
. . . . . . . . 9
⊢
(2^{nd} ‘𝐶) ∈ V |
48 | | mulcompi 9756 |
. . . . . . . . 9
⊢ (𝑥
·_{N} 𝑦) = (𝑦 ·_{N} 𝑥) |
49 | | mulasspi 9757 |
. . . . . . . . 9
⊢ ((𝑥
·_{N} 𝑦) ·_{N} 𝑧) = (𝑥 ·_{N} (𝑦
·_{N} 𝑧)) |
50 | 45, 46, 47, 48, 49, 47 | caov4 6907 |
. . . . . . . 8
⊢
(((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
·_{N} ((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶)))
= (((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))) |
51 | 44, 50 | eqtri 2673 |
. . . . . . 7
⊢
(((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))) |
52 | | fvex 6239 |
. . . . . . . . 9
⊢
(2^{nd} ‘𝐴) ∈ V |
53 | | fvex 6239 |
. . . . . . . . 9
⊢
(1^{st} ‘𝐶) ∈ V |
54 | 52, 47, 53, 48, 49, 46 | caov4 6907 |
. . . . . . . 8
⊢
(((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
= (((2^{nd} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
·_{N} ((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))) |
55 | | mulcompi 9756 |
. . . . . . . . 9
⊢
((2^{nd} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
= ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴)) |
56 | | mulcompi 9756 |
. . . . . . . . 9
⊢
((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))
= ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶)) |
57 | 55, 56 | oveq12i 6702 |
. . . . . . . 8
⊢
(((2^{nd} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
·_{N} ((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))) |
58 | 54, 57 | eqtri 2673 |
. . . . . . 7
⊢
(((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))) |
59 | 51, 58 | oveq12i 6702 |
. . . . . 6
⊢
((((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))) = ((((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) +_{N}
(((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴)) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶)))) |
60 | | addcompi 9754 |
. . . . . 6
⊢
((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))) = ((((2^{nd} ‘𝐶)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐵))) +_{N}
(((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵)))) |
61 | | ovex 6718 |
. . . . . . 7
⊢
((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ V |
62 | | ovex 6718 |
. . . . . . 7
⊢
((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴))
∈ V |
63 | | ovex 6718 |
. . . . . . 7
⊢
((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ V |
64 | | distrpi 9758 |
. . . . . . 7
⊢ (𝑥
·_{N} (𝑦 +_{N} 𝑧)) = ((𝑥 ·_{N} 𝑦) +_{N}
(𝑥
·_{N} 𝑧)) |
65 | 61, 62, 63, 48, 64 | caovdir 6910 |
. . . . . 6
⊢
((((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴)))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶)))
= ((((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶)))
+_{N} (((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶)))) |
66 | 59, 60, 65 | 3eqtr4i 2683 |
. . . . 5
⊢
((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))) = ((((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) |
67 | | addcompi 9754 |
. . . . . 6
⊢
((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶)))) = ((((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) +_{N}
(((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵)))) |
68 | | mulasspi 9757 |
. . . . . . . 8
⊢
(((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
= ((2^{nd} ‘𝐶) ·_{N}
((2^{nd} ‘𝐶)
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))) |
69 | | mulcompi 9756 |
. . . . . . . . . 10
⊢
((2^{nd} ‘𝐶) ·_{N}
((2^{nd} ‘𝐶)
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))) = (((2^{nd} ‘𝐶)
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
·_{N} (2^{nd} ‘𝐶)) |
70 | | mulasspi 9757 |
. . . . . . . . . . . 12
⊢
(((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} (1^{st} ‘𝐵)) = ((2^{nd} ‘𝐴)
·_{N} ((2^{nd} ‘𝐶) ·_{N}
(1^{st} ‘𝐵))) |
71 | | mulcompi 9756 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘𝐴) ·_{N}
((2^{nd} ‘𝐶)
·_{N} (1^{st} ‘𝐵))) = (((2^{nd} ‘𝐶)
·_{N} (1^{st} ‘𝐵)) ·_{N}
(2^{nd} ‘𝐴)) |
72 | | mulasspi 9757 |
. . . . . . . . . . . 12
⊢
(((2^{nd} ‘𝐶) ·_{N}
(1^{st} ‘𝐵))
·_{N} (2^{nd} ‘𝐴)) = ((2^{nd} ‘𝐶)
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴))) |
73 | 70, 71, 72 | 3eqtrri 2678 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘𝐶) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
(1^{st} ‘𝐵)) |
74 | 73 | oveq1i 6700 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘𝐶) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴))) ·_{N}
(2^{nd} ‘𝐶))
= ((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} (1^{st} ‘𝐵)) ·_{N}
(2^{nd} ‘𝐶)) |
75 | 69, 74 | eqtri 2673 |
. . . . . . . . 9
⊢
((2^{nd} ‘𝐶) ·_{N}
((2^{nd} ‘𝐶)
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))) = ((((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
(1^{st} ‘𝐵))
·_{N} (2^{nd} ‘𝐶)) |
76 | | mulasspi 9757 |
. . . . . . . . 9
⊢
((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} (1^{st} ‘𝐵)) ·_{N}
(2^{nd} ‘𝐶))
= (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))) |
77 | 75, 76 | eqtri 2673 |
. . . . . . . 8
⊢
((2^{nd} ‘𝐶) ·_{N}
((2^{nd} ‘𝐶)
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) |
78 | 68, 77 | eqtri 2673 |
. . . . . . 7
⊢
(((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
= (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))) |
79 | 78 | oveq2i 6701 |
. . . . . 6
⊢
((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))) = ((((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵))) +_{N}
(((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶)))) |
80 | | distrpi 9758 |
. . . . . 6
⊢
(((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} (((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))) = ((((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) +_{N}
(((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵)))) |
81 | 67, 79, 80 | 3eqtr4i 2683 |
. . . . 5
⊢
((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
(((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵)))) |
82 | 66, 81 | eqeq12i 2665 |
. . . 4
⊢
(((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))) = ((((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵))) +_{N}
(((2^{nd} ‘𝐶)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴)))) ↔ ((((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
(((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵))))) |
83 | | mulclpi 9753 |
. . . . . 6
⊢
((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N ∧ ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵))
∈ N) → (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
∈ N) |
84 | 16, 24, 83 | syl2anc 694 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
∈ N) |
85 | | mulclpi 9753 |
. . . . . 6
⊢
((((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
∈ N ∧ ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
∈ N) → (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
∈ N) |
86 | 39, 41, 85 | syl2anc 694 |
. . . . 5
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
∈ N) |
87 | | addcanpi 9759 |
. . . . 5
⊢
(((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
∈ N ∧ (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
∈ N) → (((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))) = ((((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵))) +_{N}
(((2^{nd} ‘𝐶)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴)))) ↔ (((2^{nd} ‘𝐶)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐵))) = (((2^{nd} ‘𝐶)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴))))) |
88 | 84, 86, 87 | syl2anc 694 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (((((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))
+_{N} (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))) = ((((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵))) +_{N}
(((2^{nd} ‘𝐶)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴)))) ↔ (((2^{nd} ‘𝐶)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐵))) = (((2^{nd} ‘𝐶)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴))))) |
89 | 82, 88 | syl5rbbr 275 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((2^{nd} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
↔ ((((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐴)))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶)))
= (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} (((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
+_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐵)))))) |
90 | 37, 43, 89 | 3bitr2d 296 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~_{Q} 𝐵 ↔ ((((1^{st} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐴))) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
(((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐶)) +_{N}
((1^{st} ‘𝐶)
·_{N} (2^{nd} ‘𝐵)))))) |
91 | 30, 35, 90 | 3bitr4rd 301 |
1
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~_{Q} 𝐵 ↔ (𝐴 +_{pQ} 𝐶) ~_{Q}
(𝐵
+_{pQ} 𝐶))) |