Theorem ltanq 9793

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	⊢ (𝐶 ∈ Q → (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.

Step	Hyp	Ref	Expression
1		addnqf 9770	. . 3 ⊢ +Q :(Q × Q)⟶Q
2	1	fdmi 6052	. 2 ⊢ dom +Q = (Q × Q)
3		ltrelnq 9748	. 2 ⊢ <Q ⊆ (Q × Q)
4		0nnq 9746	. 2 ⊢ ¬ ∅ ∈ Q
5		ordpinq 9765	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
6	5	3adant3 1081	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
7		elpqn 9747	. . . . . . 7 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
8	7	3ad2ant3 1084	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
9		elpqn 9747	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
10	9	3ad2ant1 1082	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
11		addpipq2 9758	. . . . . 6 ⊢ ((𝐶 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐶 +pQ 𝐴) = ⟨(((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩)
12	8, 10, 11	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +pQ 𝐴) = ⟨(((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩)
13		elpqn 9747	. . . . . . 7 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
14	13	3ad2ant2 1083	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
15		addpipq2 9758	. . . . . 6 ⊢ ((𝐶 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐶 +pQ 𝐵) = ⟨(((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩)
16	8, 14, 15	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +pQ 𝐵) = ⟨(((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩)
17	12, 16	breq12d 4666	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵) ↔ ⟨(((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩ <pQ ⟨(((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩))
18		addpqnq 9760	. . . . . . . 8 ⊢ ((𝐶 ∈ Q ∧ 𝐴 ∈ Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
19	18	ancoms 469	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
20	19	3adant2 1080	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
21		addpqnq 9760	. . . . . . . 8 ⊢ ((𝐶 ∈ Q ∧ 𝐵 ∈ Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
22	21	ancoms 469	. . . . . . 7 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
23	22	3adant1 1079	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
24	20, 23	breq12d 4666	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵) ↔ ([Q]‘(𝐶 +pQ 𝐴)) <Q ([Q]‘(𝐶 +pQ 𝐵))))
25		lterpq 9792	. . . . 5 ⊢ ((𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵) ↔ ([Q]‘(𝐶 +pQ 𝐴)) <Q ([Q]‘(𝐶 +pQ 𝐵)))
26	24, 25	syl6bbr 278	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵) ↔ (𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵)))
27		xp2nd 7199	. . . . . . . . . 10 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
28	8, 27	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
29		mulclpi 9715	. . . . . . . . 9 ⊢ (((2nd ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
30	28, 28, 29	syl2anc 693	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
31		ltmpi 9726	. . . . . . . 8 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))))
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))))
33		xp2nd 7199	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
34	14, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐵) ∈ N)
35		mulclpi 9715	. . . . . . . . . 10 ⊢ (((2nd ‘𝐶) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐶) ·N (2nd ‘𝐵)) ∈ N)
36	28, 34, 35	syl2anc 693	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐶) ·N (2nd ‘𝐵)) ∈ N)
37		xp1st 7198	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
38	8, 37	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
39		xp2nd 7199	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
40	10, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐴) ∈ N)
41		mulclpi 9715	. . . . . . . . . 10 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ N)
42	38, 40, 41	syl2anc 693	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ N)
43		mulclpi 9715	. . . . . . . . 9 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ∈ N ∧ ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ N) → (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ∈ N)
44	36, 42, 43	syl2anc 693	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ∈ N)
45		ltapi 9725	. . . . . . . 8 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ∈ N → ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) <N ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))))
46	44, 45	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) <N ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))))
47	32, 46	bitrd 268	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) <N ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))))
48		mulcompi 9718	. . . . . . . . . 10 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐶)))
49		fvex 6201	. . . . . . . . . . 11 ⊢ (1st ‘𝐴) ∈ V
50		fvex 6201	. . . . . . . . . . 11 ⊢ (2nd ‘𝐵) ∈ V
51		fvex 6201	. . . . . . . . . . 11 ⊢ (2nd ‘𝐶) ∈ V
52		mulcompi 9718	. . . . . . . . . . 11 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
53		mulasspi 9719	. . . . . . . . . . 11 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
54	49, 50, 51, 52, 53, 51	caov411 6866	. . . . . . . . . 10 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐶))) = (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐶)))
55	48, 54	eqtri 2644	. . . . . . . . 9 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐶)))
56	55	oveq2i 6661	. . . . . . . 8 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) = ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐶))))
57		distrpi 9720	. . . . . . . 8 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N (((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶)))) = ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐶))))
58		mulcompi 9718	. . . . . . . 8 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N (((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶)))) = ((((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵)))
59	56, 57, 58	3eqtr2i 2650	. . . . . . 7 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) = ((((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵)))
60		mulcompi 9718	. . . . . . . . . 10 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) = (((1st ‘𝐶) ·N (2nd ‘𝐴)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵)))
61		fvex 6201	. . . . . . . . . . 11 ⊢ (1st ‘𝐶) ∈ V
62		fvex 6201	. . . . . . . . . . 11 ⊢ (2nd ‘𝐴) ∈ V
63	61, 62, 51, 52, 53, 50	caov411 6866	. . . . . . . . . 10 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐴)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵))) = (((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵)))
64	60, 63	eqtri 2644	. . . . . . . . 9 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) = (((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵)))
65		mulcompi 9718	. . . . . . . . . 10 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐶)))
66		fvex 6201	. . . . . . . . . . 11 ⊢ (1st ‘𝐵) ∈ V
67	66, 62, 51, 52, 53, 51	caov411 6866	. . . . . . . . . 10 ⊢ (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐶))) = (((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶)))
68	65, 67	eqtri 2644	. . . . . . . . 9 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶)))
69	64, 68	oveq12i 6662	. . . . . . . 8 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) = ((((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶))))
70		distrpi 9720	. . . . . . . 8 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N (((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶)))) = ((((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐵))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐶))))
71		mulcompi 9718	. . . . . . . 8 ⊢ (((2nd ‘𝐶) ·N (2nd ‘𝐴)) ·N (((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶)))) = ((((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴)))
72	69, 70, 71	3eqtr2i 2650	. . . . . . 7 ⊢ ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) = ((((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴)))
73	59, 72	breq12i 4662	. . . . . 6 ⊢ (((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))) <N ((((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) +N (((2nd ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))) ↔ ((((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵))) <N ((((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴))))
74	47, 73	syl6bb 276	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ((((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵))) <N ((((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴)))))
75		ordpipq 9764	. . . . 5 ⊢ (⟨(((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩ <pQ ⟨(((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩ ↔ ((((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵))) <N ((((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴))))
76	74, 75	syl6bbr 278	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ⟨(((1st ‘𝐶) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩ <pQ ⟨(((1st ‘𝐶) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐶))), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩))
77	17, 26, 76	3bitr4rd 301	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
78	6, 77	bitrd 268	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
79	2, 3, 4, 78	ndmovord 6824	1 ⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  Ncnpi 9666   +_N cpli 9667   ·_N cmi 9668   <_N clti 9669   +_pQ cplpq 9670   <_pQ cltpq 9672  Qcnq 9674  [Q]cerq 9676   +_Q cplq 9677   <_Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-1nq 9738  df-ltnq 9740

This theorem is referenced by:  ltaddnq  9796  ltbtwnnq  9800  addclpr  9840  distrlem4pr  9848  ltexprlem3  9860  ltexprlem4  9861  ltexprlem6  9863  prlem936  9869