Step | Hyp | Ref
| Expression |
1 | | elpqn 9626 |
. . . . . . 7
⊢ (𝑥 ∈ Q →
𝑥 ∈ (N
× N)) |
2 | 1 | adantr 480 |
. . . . . 6
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ 𝑥 ∈
(N × N)) |
3 | | xp1st 7089 |
. . . . . 6
⊢ (𝑥 ∈ (N ×
N) → (1^{st} ‘𝑥) ∈ N) |
4 | 2, 3 | syl 17 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (1^{st} ‘𝑥) ∈ N) |
5 | | elpqn 9626 |
. . . . . . 7
⊢ (𝑦 ∈ Q →
𝑦 ∈ (N
× N)) |
6 | 5 | adantl 481 |
. . . . . 6
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ 𝑦 ∈
(N × N)) |
7 | | xp2nd 7090 |
. . . . . 6
⊢ (𝑦 ∈ (N ×
N) → (2^{nd} ‘𝑦) ∈ N) |
8 | 6, 7 | syl 17 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (2^{nd} ‘𝑦) ∈ N) |
9 | | mulclpi 9594 |
. . . . 5
⊢
(((1^{st} ‘𝑥) ∈ N ∧
(2^{nd} ‘𝑦)
∈ N) → ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
∈ N) |
10 | 4, 8, 9 | syl2anc 691 |
. . . 4
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
∈ N) |
11 | | xp1st 7089 |
. . . . . 6
⊢ (𝑦 ∈ (N ×
N) → (1^{st} ‘𝑦) ∈ N) |
12 | 6, 11 | syl 17 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (1^{st} ‘𝑦) ∈ N) |
13 | | xp2nd 7090 |
. . . . . 6
⊢ (𝑥 ∈ (N ×
N) → (2^{nd} ‘𝑥) ∈ N) |
14 | 2, 13 | syl 17 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (2^{nd} ‘𝑥) ∈ N) |
15 | | mulclpi 9594 |
. . . . 5
⊢
(((1^{st} ‘𝑦) ∈ N ∧
(2^{nd} ‘𝑥)
∈ N) → ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
∈ N) |
16 | 12, 14, 15 | syl2anc 691 |
. . . 4
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
∈ N) |
17 | | ltsopi 9589 |
. . . . 5
⊢
<_{N} Or N |
18 | | sotric 4985 |
. . . . 5
⊢ ((
<_{N} Or N ∧ (((1^{st}
‘𝑥)
·_{N} (2^{nd} ‘𝑦)) ∈ N ∧
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) ∈ N)) →
(((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦)) <_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) ↔ ¬ (((1^{st}
‘𝑥)
·_{N} (2^{nd} ‘𝑦)) = ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) ∨ ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) <_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦))))) |
19 | 17, 18 | mpan 702 |
. . . 4
⊢
((((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
∈ N ∧ ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
∈ N) → (((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
<_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
↔ ¬ (((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
= ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
∨ ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
<_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))))) |
20 | 10, 16, 19 | syl2anc 691 |
. . 3
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
<_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
↔ ¬ (((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
= ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
∨ ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
<_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))))) |
21 | | ordpinq 9644 |
. . 3
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑥
<_{Q} 𝑦 ↔ ((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦)) <_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)))) |
22 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (1^{st} ‘𝑥) = (1^{st} ‘𝑦)) |
23 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (2^{nd} ‘𝑥) = (2^{nd} ‘𝑦)) |
24 | 23 | eqcomd 2616 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (2^{nd} ‘𝑦) = (2^{nd} ‘𝑥)) |
25 | 22, 24 | oveq12d 6567 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦)) = ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥))) |
26 | | enqbreq2 9621 |
. . . . . . . 8
⊢ ((𝑥 ∈ (N ×
N) ∧ 𝑦
∈ (N × N)) → (𝑥 ~_{Q} 𝑦 ↔ ((1^{st}
‘𝑥)
·_{N} (2^{nd} ‘𝑦)) = ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)))) |
27 | 1, 5, 26 | syl2an 493 |
. . . . . . 7
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑥
~_{Q} 𝑦 ↔ ((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦)) = ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)))) |
28 | | enqeq 9635 |
. . . . . . . 8
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑥
~_{Q} 𝑦) → 𝑥 = 𝑦) |
29 | 28 | 3expia 1259 |
. . . . . . 7
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑥
~_{Q} 𝑦 → 𝑥 = 𝑦)) |
30 | 27, 29 | sylbird 249 |
. . . . . 6
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
= ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
→ 𝑥 = 𝑦)) |
31 | 25, 30 | impbid2 215 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑥 = 𝑦 ↔ ((1^{st}
‘𝑥)
·_{N} (2^{nd} ‘𝑦)) = ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)))) |
32 | | ordpinq 9644 |
. . . . . 6
⊢ ((𝑦 ∈ Q ∧
𝑥 ∈ Q)
→ (𝑦
<_{Q} 𝑥 ↔ ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) <_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦)))) |
33 | 32 | ancoms 468 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑦
<_{Q} 𝑥 ↔ ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) <_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦)))) |
34 | 31, 33 | orbi12d 742 |
. . . 4
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ ((𝑥 = 𝑦 ∨ 𝑦 <_{Q} 𝑥) ↔ (((1^{st}
‘𝑥)
·_{N} (2^{nd} ‘𝑦)) = ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) ∨ ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) <_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦))))) |
35 | 34 | notbid 307 |
. . 3
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (¬ (𝑥 = 𝑦 ∨ 𝑦 <_{Q} 𝑥) ↔ ¬ (((1^{st}
‘𝑥)
·_{N} (2^{nd} ‘𝑦)) = ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) ∨ ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) <_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦))))) |
36 | 20, 21, 35 | 3bitr4d 299 |
. 2
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
→ (𝑥
<_{Q} 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <_{Q} 𝑥))) |
37 | 21 | 3adant3 1074 |
. . . . . 6
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (𝑥
<_{Q} 𝑦 ↔ ((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦)) <_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)))) |
38 | | elpqn 9626 |
. . . . . . . 8
⊢ (𝑧 ∈ Q →
𝑧 ∈ (N
× N)) |
39 | 38 | 3ad2ant3 1077 |
. . . . . . 7
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → 𝑧
∈ (N × N)) |
40 | | xp2nd 7090 |
. . . . . . 7
⊢ (𝑧 ∈ (N ×
N) → (2^{nd} ‘𝑧) ∈ N) |
41 | | ltmpi 9605 |
. . . . . . 7
⊢
((2^{nd} ‘𝑧) ∈ N →
(((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦)) <_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥)) ↔ ((2^{nd} ‘𝑧)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦)))
<_{N} ((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥))))) |
42 | 39, 40, 41 | 3syl 18 |
. . . . . 6
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦))
<_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))
↔ ((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦))) <_{N}
((2^{nd} ‘𝑧)
·_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))))) |
43 | 37, 42 | bitrd 267 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (𝑥
<_{Q} 𝑦 ↔ ((2^{nd} ‘𝑧)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦)))
<_{N} ((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥))))) |
44 | | ordpinq 9644 |
. . . . . . 7
⊢ ((𝑦 ∈ Q ∧
𝑧 ∈ Q)
→ (𝑦
<_{Q} 𝑧 ↔ ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑧)) <_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑦)))) |
45 | 44 | 3adant1 1072 |
. . . . . 6
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (𝑦
<_{Q} 𝑧 ↔ ((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑧)) <_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑦)))) |
46 | 1 | 3ad2ant1 1075 |
. . . . . . 7
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → 𝑥
∈ (N × N)) |
47 | | ltmpi 9605 |
. . . . . . 7
⊢
((2^{nd} ‘𝑥) ∈ N →
(((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑧)) <_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑦)) ↔ ((2^{nd} ‘𝑥)
·_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑧)))
<_{N} ((2^{nd} ‘𝑥) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑦))))) |
48 | 46, 13, 47 | 3syl 18 |
. . . . . 6
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑧))
<_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑦))
↔ ((2^{nd} ‘𝑥) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑧))) <_{N}
((2^{nd} ‘𝑥)
·_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑦))))) |
49 | 45, 48 | bitrd 267 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (𝑦
<_{Q} 𝑧 ↔ ((2^{nd} ‘𝑥)
·_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑧)))
<_{N} ((2^{nd} ‘𝑥) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑦))))) |
50 | 43, 49 | anbi12d 743 |
. . . 4
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → ((𝑥
<_{Q} 𝑦 ∧ 𝑦 <_{Q} 𝑧) ↔ (((2^{nd}
‘𝑧)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦)))
<_{N} ((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥))) ∧ ((2^{nd} ‘𝑥)
·_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑧)))
<_{N} ((2^{nd} ‘𝑥) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑦)))))) |
51 | | fvex 6113 |
. . . . . . 7
⊢
(2^{nd} ‘𝑥) ∈ V |
52 | | fvex 6113 |
. . . . . . 7
⊢
(1^{st} ‘𝑦) ∈ V |
53 | | fvex 6113 |
. . . . . . 7
⊢
(2^{nd} ‘𝑧) ∈ V |
54 | | mulcompi 9597 |
. . . . . . 7
⊢ (𝑟
·_{N} 𝑠) = (𝑠 ·_{N} 𝑟) |
55 | | mulasspi 9598 |
. . . . . . 7
⊢ ((𝑟
·_{N} 𝑠) ·_{N} 𝑡) = (𝑟 ·_{N} (𝑠
·_{N} 𝑡)) |
56 | 51, 52, 53, 54, 55 | caov13 6762 |
. . . . . 6
⊢
((2^{nd} ‘𝑥) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑧))) = ((2^{nd} ‘𝑧)
·_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥))) |
57 | | fvex 6113 |
. . . . . . 7
⊢
(1^{st} ‘𝑧) ∈ V |
58 | | fvex 6113 |
. . . . . . 7
⊢
(2^{nd} ‘𝑦) ∈ V |
59 | 51, 57, 58, 54, 55 | caov13 6762 |
. . . . . 6
⊢
((2^{nd} ‘𝑥) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑦))) = ((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑥))) |
60 | 56, 59 | breq12i 4592 |
. . . . 5
⊢
(((2^{nd} ‘𝑥) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑧))) <_{N}
((2^{nd} ‘𝑥)
·_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑦)))
↔ ((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥))) <_{N}
((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑥)))) |
61 | | fvex 6113 |
. . . . . . 7
⊢
(1^{st} ‘𝑥) ∈ V |
62 | 53, 61, 58, 54, 55 | caov13 6762 |
. . . . . 6
⊢
((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦))) = ((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑧))) |
63 | | ltrelpi 9590 |
. . . . . . 7
⊢
<_{N} ⊆ (N ×
N) |
64 | 17, 63 | sotri 5442 |
. . . . . 6
⊢
((((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦))) <_{N}
((2^{nd} ‘𝑧)
·_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥)))
∧ ((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥))) <_{N}
((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑥)))) → ((2^{nd} ‘𝑧)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑦)))
<_{N} ((2^{nd} ‘𝑦) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥)))) |
65 | 62, 64 | syl5eqbrr 4619 |
. . . . 5
⊢
((((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦))) <_{N}
((2^{nd} ‘𝑧)
·_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥)))
∧ ((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑥))) <_{N}
((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑥)))) → ((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑧)))
<_{N} ((2^{nd} ‘𝑦) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥)))) |
66 | 60, 65 | sylan2b 491 |
. . . 4
⊢
((((2^{nd} ‘𝑧) ·_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑦))) <_{N}
((2^{nd} ‘𝑧)
·_{N} ((1^{st} ‘𝑦) ·_{N}
(2^{nd} ‘𝑥)))
∧ ((2^{nd} ‘𝑥) ·_{N}
((1^{st} ‘𝑦)
·_{N} (2^{nd} ‘𝑧))) <_{N}
((2^{nd} ‘𝑥)
·_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑦)))) → ((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑧)))
<_{N} ((2^{nd} ‘𝑦) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥)))) |
67 | 50, 66 | syl6bi 242 |
. . 3
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → ((𝑥
<_{Q} 𝑦 ∧ 𝑦 <_{Q} 𝑧) → ((2^{nd}
‘𝑦)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑧)))
<_{N} ((2^{nd} ‘𝑦) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥))))) |
68 | | ordpinq 9644 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑧 ∈ Q)
→ (𝑥
<_{Q} 𝑧 ↔ ((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑧)) <_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥)))) |
69 | 68 | 3adant2 1073 |
. . . 4
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (𝑥
<_{Q} 𝑧 ↔ ((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑧)) <_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥)))) |
70 | 5 | 3ad2ant2 1076 |
. . . . 5
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → 𝑦
∈ (N × N)) |
71 | | ltmpi 9605 |
. . . . 5
⊢
((2^{nd} ‘𝑦) ∈ N →
(((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑧)) <_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥)) ↔ ((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑧)))
<_{N} ((2^{nd} ‘𝑦) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥))))) |
72 | 70, 7, 71 | 3syl 18 |
. . . 4
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑧))
<_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑥))
↔ ((2^{nd} ‘𝑦) ·_{N}
((1^{st} ‘𝑥)
·_{N} (2^{nd} ‘𝑧))) <_{N}
((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑧) ·_{N}
(2^{nd} ‘𝑥))))) |
73 | 69, 72 | bitrd 267 |
. . 3
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → (𝑥
<_{Q} 𝑧 ↔ ((2^{nd} ‘𝑦)
·_{N} ((1^{st} ‘𝑥) ·_{N}
(2^{nd} ‘𝑧)))
<_{N} ((2^{nd} ‘𝑦) ·_{N}
((1^{st} ‘𝑧)
·_{N} (2^{nd} ‘𝑥))))) |
74 | 67, 73 | sylibrd 248 |
. 2
⊢ ((𝑥 ∈ Q ∧
𝑦 ∈ Q
∧ 𝑧 ∈
Q) → ((𝑥
<_{Q} 𝑦 ∧ 𝑦 <_{Q} 𝑧) → 𝑥 <_{Q} 𝑧)) |
75 | 36, 74 | isso2i 4991 |
1
⊢
<_{Q} Or Q |