Step | Hyp | Ref
| Expression |
1 | | xp1st 7706 |
. . . . 5
⊢ (𝐴 ∈ (N ×
N) → (1^{st} ‘𝐴) ∈ N) |
2 | 1 | 3ad2ant1 1130 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1^{st} ‘𝐴) ∈ N) |
3 | | xp1st 7706 |
. . . . 5
⊢ (𝐶 ∈ (N ×
N) → (1^{st} ‘𝐶) ∈ N) |
4 | 3 | 3ad2ant3 1132 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1^{st} ‘𝐶) ∈ N) |
5 | | mulclpi 10307 |
. . . 4
⊢
(((1^{st} ‘𝐴) ∈ N ∧
(1^{st} ‘𝐶)
∈ N) → ((1^{st} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
∈ N) |
6 | 2, 4, 5 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
∈ N) |
7 | | xp2nd 7707 |
. . . . 5
⊢ (𝐴 ∈ (N ×
N) → (2^{nd} ‘𝐴) ∈ N) |
8 | 7 | 3ad2ant1 1130 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2^{nd} ‘𝐴) ∈ N) |
9 | | xp2nd 7707 |
. . . . 5
⊢ (𝐶 ∈ (N ×
N) → (2^{nd} ‘𝐶) ∈ N) |
10 | 9 | 3ad2ant3 1132 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2^{nd} ‘𝐶) ∈ N) |
11 | | mulclpi 10307 |
. . . 4
⊢
(((2^{nd} ‘𝐴) ∈ N ∧
(2^{nd} ‘𝐶)
∈ N) → ((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
12 | 8, 10, 11 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
13 | | xp1st 7706 |
. . . . 5
⊢ (𝐵 ∈ (N ×
N) → (1^{st} ‘𝐵) ∈ N) |
14 | 13 | 3ad2ant2 1131 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (1^{st} ‘𝐵) ∈ N) |
15 | | mulclpi 10307 |
. . . 4
⊢
(((1^{st} ‘𝐵) ∈ N ∧
(1^{st} ‘𝐶)
∈ N) → ((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶))
∈ N) |
16 | 14, 4, 15 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶))
∈ N) |
17 | | xp2nd 7707 |
. . . . 5
⊢ (𝐵 ∈ (N ×
N) → (2^{nd} ‘𝐵) ∈ N) |
18 | 17 | 3ad2ant2 1131 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (2^{nd} ‘𝐵) ∈ N) |
19 | | mulclpi 10307 |
. . . 4
⊢
(((2^{nd} ‘𝐵) ∈ N ∧
(2^{nd} ‘𝐶)
∈ N) → ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
20 | 18, 10, 19 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
21 | | enqbreq 10333 |
. . 3
⊢
(((((1^{st} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
∈ N ∧ ((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
∈ N) ∧ (((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶))
∈ N ∧ ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))
∈ N)) → (⟨((1^{st} ‘𝐴)
·_{N} (1^{st} ‘𝐶)), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩ ~_{Q}
⟨((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶)),
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩ ↔ (((1^{st}
‘𝐴)
·_{N} (1^{st} ‘𝐶)) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (1^{st} ‘𝐶))))) |
22 | 6, 12, 16, 20, 21 | syl22anc 837 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (⟨((1^{st} ‘𝐴) ·_{N}
(1^{st} ‘𝐶)),
((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩ ~_{Q}
⟨((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶)),
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩ ↔ (((1^{st}
‘𝐴)
·_{N} (1^{st} ‘𝐶)) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (1^{st} ‘𝐶))))) |
23 | | mulpipq2 10353 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐶
∈ (N × N)) → (𝐴 ·_{pQ} 𝐶) = ⟨((1^{st}
‘𝐴)
·_{N} (1^{st} ‘𝐶)), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩) |
24 | 23 | 3adant2 1128 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
·_{pQ} 𝐶) = ⟨((1^{st} ‘𝐴)
·_{N} (1^{st} ‘𝐶)), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩) |
25 | | mulpipq2 10353 |
. . . 4
⊢ ((𝐵 ∈ (N ×
N) ∧ 𝐶
∈ (N × N)) → (𝐵 ·_{pQ} 𝐶) = ⟨((1^{st}
‘𝐵)
·_{N} (1^{st} ‘𝐶)), ((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩) |
26 | 25 | 3adant1 1127 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐵
·_{pQ} 𝐶) = ⟨((1^{st} ‘𝐵)
·_{N} (1^{st} ‘𝐶)), ((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩) |
27 | 24, 26 | breq12d 5044 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((𝐴 ·_{pQ} 𝐶) ~_{Q}
(𝐵
·_{pQ} 𝐶) ↔ ⟨((1^{st} ‘𝐴)
·_{N} (1^{st} ‘𝐶)), ((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶))⟩ ~_{Q}
⟨((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶)),
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))⟩)) |
28 | | enqbreq2 10334 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N)) → (𝐴 ~_{Q} 𝐵 ↔ ((1^{st}
‘𝐴)
·_{N} (2^{nd} ‘𝐵)) = ((1^{st} ‘𝐵)
·_{N} (2^{nd} ‘𝐴)))) |
29 | 28 | 3adant3 1129 |
. . 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)))) |
30 | | mulclpi 10307 |
. . . . 5
⊢
(((1^{st} ‘𝐶) ∈ N ∧
(2^{nd} ‘𝐶)
∈ N) → ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
31 | 4, 10, 30 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
∈ N) |
32 | | mulclpi 10307 |
. . . . 5
⊢
(((1^{st} ‘𝐴) ∈ N ∧
(2^{nd} ‘𝐵)
∈ N) → ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
∈ N) |
33 | 2, 18, 32 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
∈ N) |
34 | | mulcanpi 10314 |
. . . 4
⊢
((((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
∈ N ∧ ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
∈ N) → ((((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘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)))) |
35 | 31, 33, 34 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘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)))) |
36 | | mulcompi 10310 |
. . . . . 6
⊢
(((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))) |
37 | | fvex 6659 |
. . . . . . 7
⊢
(1^{st} ‘𝐴) ∈ V |
38 | | fvex 6659 |
. . . . . . 7
⊢
(2^{nd} ‘𝐵) ∈ V |
39 | | fvex 6659 |
. . . . . . 7
⊢
(1^{st} ‘𝐶) ∈ V |
40 | | mulcompi 10310 |
. . . . . . 7
⊢ (𝑥
·_{N} 𝑦) = (𝑦 ·_{N} 𝑥) |
41 | | mulasspi 10311 |
. . . . . . 7
⊢ ((𝑥
·_{N} 𝑦) ·_{N} 𝑧) = (𝑥 ·_{N} (𝑦
·_{N} 𝑧)) |
42 | | fvex 6659 |
. . . . . . 7
⊢
(2^{nd} ‘𝐶) ∈ V |
43 | 37, 38, 39, 40, 41, 42 | caov4 7361 |
. . . . . 6
⊢
(((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶)))
= (((1^{st} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))) |
44 | 36, 43 | eqtri 2821 |
. . . . 5
⊢
(((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶))) |
45 | | mulcompi 10310 |
. . . . . 6
⊢
(((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
= (((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))) |
46 | | fvex 6659 |
. . . . . . 7
⊢
(1^{st} ‘𝐵) ∈ V |
47 | | fvex 6659 |
. . . . . . 7
⊢
(2^{nd} ‘𝐴) ∈ V |
48 | 46, 47, 39, 40, 41, 42 | caov4 7361 |
. . . . . 6
⊢
(((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴))
·_{N} ((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶)))
= (((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶))
·_{N} ((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))) |
49 | | mulcompi 10310 |
. . . . . 6
⊢
(((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶))
·_{N} ((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶)))
= (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶))) |
50 | 45, 48, 49 | 3eqtri 2825 |
. . . . 5
⊢
(((1^{st} ‘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}
(1^{st} ‘𝐶))) |
51 | 44, 50 | eqeq12i 2813 |
. . . 4
⊢
((((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
↔ (((1^{st} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶)))
= (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶)))) |
52 | 51 | a1i 11 |
. . 3
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → ((((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐴) ·_{N}
(2^{nd} ‘𝐵)))
= (((1^{st} ‘𝐶) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(2^{nd} ‘𝐴)))
↔ (((1^{st} ‘𝐴) ·_{N}
(1^{st} ‘𝐶))
·_{N} ((2^{nd} ‘𝐵) ·_{N}
(2^{nd} ‘𝐶)))
= (((2^{nd} ‘𝐴) ·_{N}
(2^{nd} ‘𝐶))
·_{N} ((1^{st} ‘𝐵) ·_{N}
(1^{st} ‘𝐶))))) |
53 | 29, 35, 52 | 3bitr2d 310 |
. 2
⊢ ((𝐴 ∈ (N ×
N) ∧ 𝐵
∈ (N × N) ∧ 𝐶 ∈ (N ×
N)) → (𝐴
~_{Q} 𝐵 ↔ (((1^{st} ‘𝐴)
·_{N} (1^{st} ‘𝐶)) ·_{N}
((2^{nd} ‘𝐵)
·_{N} (2^{nd} ‘𝐶))) = (((2^{nd} ‘𝐴)
·_{N} (2^{nd} ‘𝐶)) ·_{N}
((1^{st} ‘𝐵)
·_{N} (1^{st} ‘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))) |