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Theorem ltanq 9672
 Description: Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltanq (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))

Proof of Theorem ltanq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addnqf 9649 . . 3 +Q :(Q × Q)⟶Q
21fdmi 5965 . 2 dom +Q = (Q × Q)
3 ltrelnq 9627 . 2 <Q ⊆ (Q × Q)
4 0nnq 9625 . 2 ¬ ∅ ∈ Q
5 ordpinq 9644 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
653adant3 1074 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
7 elpqn 9626 . . . . . . 7 (𝐶Q𝐶 ∈ (N × N))
873ad2ant3 1077 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
9 elpqn 9626 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
1093ad2ant1 1075 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
11 addpipq2 9637 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐶 +pQ 𝐴) = ⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩)
128, 10, 11syl2anc 691 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 +pQ 𝐴) = ⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩)
13 elpqn 9626 . . . . . . 7 (𝐵Q𝐵 ∈ (N × N))
14133ad2ant2 1076 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
15 addpipq2 9637 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐶 +pQ 𝐵) = ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩)
168, 14, 15syl2anc 691 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 +pQ 𝐵) = ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩)
1712, 16breq12d 4596 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵) ↔ ⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩))
18 addpqnq 9639 . . . . . . . 8 ((𝐶Q𝐴Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
1918ancoms 468 . . . . . . 7 ((𝐴Q𝐶Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
20193adant2 1073 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
21 addpqnq 9639 . . . . . . . 8 ((𝐶Q𝐵Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
2221ancoms 468 . . . . . . 7 ((𝐵Q𝐶Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
23223adant1 1072 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
2420, 23breq12d 4596 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵) ↔ ([Q]‘(𝐶 +pQ 𝐴)) <Q ([Q]‘(𝐶 +pQ 𝐵))))
25 lterpq 9671 . . . . 5 ((𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵) ↔ ([Q]‘(𝐶 +pQ 𝐴)) <Q ([Q]‘(𝐶 +pQ 𝐵)))
2624, 25syl6bbr 277 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵) ↔ (𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵)))
27 xp2nd 7090 . . . . . . . . . 10 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
288, 27syl 17 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
29 mulclpi 9594 . . . . . . . . 9 (((2nd𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐶) ·N (2nd𝐶)) ∈ N)
3028, 28, 29syl2anc 691 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐶) ·N (2nd𝐶)) ∈ N)
31 ltmpi 9605 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐶)) ∈ N → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
3230, 31syl 17 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
33 xp2nd 7090 . . . . . . . . . . 11 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
3414, 33syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐵) ∈ N)
35 mulclpi 9594 . . . . . . . . . 10 (((2nd𝐶) ∈ N ∧ (2nd𝐵) ∈ N) → ((2nd𝐶) ·N (2nd𝐵)) ∈ N)
3628, 34, 35syl2anc 691 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((2nd𝐶) ·N (2nd𝐵)) ∈ N)
37 xp1st 7089 . . . . . . . . . . 11 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
388, 37syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
39 xp2nd 7090 . . . . . . . . . . 11 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
4010, 39syl 17 . . . . . . . . . 10 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐴) ∈ N)
41 mulclpi 9594 . . . . . . . . . 10 (((1st𝐶) ∈ N ∧ (2nd𝐴) ∈ N) → ((1st𝐶) ·N (2nd𝐴)) ∈ N)
4238, 40, 41syl2anc 691 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐴)) ∈ N)
43 mulclpi 9594 . . . . . . . . 9 ((((2nd𝐶) ·N (2nd𝐵)) ∈ N ∧ ((1st𝐶) ·N (2nd𝐴)) ∈ N) → (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) ∈ N)
4436, 42, 43syl2anc 691 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) ∈ N)
45 ltapi 9604 . . . . . . . 8 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) ∈ N → ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) <N ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))))))
4644, 45syl 17 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) <N ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))))))
4732, 46bitrd 267 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) <N ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))))))
48 mulcompi 9597 . . . . . . . . . 10 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶)))
49 fvex 6113 . . . . . . . . . . 11 (1st𝐴) ∈ V
50 fvex 6113 . . . . . . . . . . 11 (2nd𝐵) ∈ V
51 fvex 6113 . . . . . . . . . . 11 (2nd𝐶) ∈ V
52 mulcompi 9597 . . . . . . . . . . 11 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
53 mulasspi 9598 . . . . . . . . . . 11 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
5449, 50, 51, 52, 53, 51caov411 6764 . . . . . . . . . 10 (((1st𝐴) ·N (2nd𝐵)) ·N ((2nd𝐶) ·N (2nd𝐶))) = (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐴) ·N (2nd𝐶)))
5548, 54eqtri 2632 . . . . . . . . 9 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐴) ·N (2nd𝐶)))
5655oveq2i 6560 . . . . . . . 8 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐴) ·N (2nd𝐶))))
57 distrpi 9599 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐵)) ·N (((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶)))) = ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐴) ·N (2nd𝐶))))
58 mulcompi 9597 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐵)) ·N (((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶)))) = ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵)))
5956, 57, 583eqtr2i 2638 . . . . . . 7 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) = ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵)))
60 mulcompi 9597 . . . . . . . . . 10 (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) = (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵)))
61 fvex 6113 . . . . . . . . . . 11 (1st𝐶) ∈ V
62 fvex 6113 . . . . . . . . . . 11 (2nd𝐴) ∈ V
6361, 62, 51, 52, 53, 50caov411 6764 . . . . . . . . . 10 (((1st𝐶) ·N (2nd𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵))) = (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
6460, 63eqtri 2632 . . . . . . . . 9 (((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) = (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐵)))
65 mulcompi 9597 . . . . . . . . . 10 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) ·N ((2nd𝐶) ·N (2nd𝐶)))
66 fvex 6113 . . . . . . . . . . 11 (1st𝐵) ∈ V
6766, 62, 51, 52, 53, 51caov411 6764 . . . . . . . . . 10 (((1st𝐵) ·N (2nd𝐴)) ·N ((2nd𝐶) ·N (2nd𝐶))) = (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
6865, 67eqtri 2632 . . . . . . . . 9 (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd𝐶)))
6964, 68oveq12i 6561 . . . . . . . 8 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd𝐶))))
70 distrpi 9599 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐴)) ·N (((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶)))) = ((((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐵))) +N (((2nd𝐶) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd𝐶))))
71 mulcompi 9597 . . . . . . . 8 (((2nd𝐶) ·N (2nd𝐴)) ·N (((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶)))) = ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴)))
7269, 70, 713eqtr2i 2638 . . . . . . 7 ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) = ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴)))
7359, 72breq12i 4592 . . . . . 6 (((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵)))) <N ((((2nd𝐶) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐴))) +N (((2nd𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))) ↔ ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵))) <N ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴))))
7447, 73syl6bb 275 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵))) <N ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴)))))
75 ordpipq 9643 . . . . 5 (⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩ ↔ ((((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐵))) <N ((((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))) ·N ((2nd𝐶) ·N (2nd𝐴))))
7674, 75syl6bbr 277 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ⟨(((1st𝐶) ·N (2nd𝐴)) +N ((1st𝐴) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨(((1st𝐶) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐶))), ((2nd𝐶) ·N (2nd𝐵))⟩))
7717, 26, 763bitr4rd 300 . . 3 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
786, 77bitrd 267 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
792, 3, 4, 78ndmovord 6722 1 (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Ncnpi 9545   +N cpli 9546   ·N cmi 9547
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