Theorem ltanq 10985

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 10962	. . 3 ⊢ +Q :(Q × Q)⟶Q
2	1	fdmi 6717	. 2 ⊢ dom +Q = (Q × Q)
3		ltrelnq 10940	. 2 ⊢ <Q ⊆ (Q × Q)
4		0nnq 10938	. 2 ⊢ ¬ ∅ ∈ Q
5		ordpinq 10957	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
6	5	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
7		elpqn 10939	. . . . . . 7 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
8	7	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
9		elpqn 10939	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
10	9	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
11		addpipq2 10950	. . . . . 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 584	. . . . 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 10939	. . . . . . 7 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
14	13	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
15		addpipq2 10950	. . . . . 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 584	. . . . 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 5132	. . . 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 10952	. . . . . . . 8 ⊢ ((𝐶 ∈ Q ∧ 𝐴 ∈ Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
19	18	ancoms 458	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
20	19	3adant2 1131	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) = ([Q]‘(𝐶 +pQ 𝐴)))
21		addpqnq 10952	. . . . . . . 8 ⊢ ((𝐶 ∈ Q ∧ 𝐵 ∈ Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
22	21	ancoms 458	. . . . . . 7 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
23	22	3adant1 1130	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐵) = ([Q]‘(𝐶 +pQ 𝐵)))
24	20, 23	breq12d 5132	. . . . 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 10984	. . . . 5 ⊢ ((𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵) ↔ ([Q]‘(𝐶 +pQ 𝐴)) <Q ([Q]‘(𝐶 +pQ 𝐵)))
26	24, 25	bitr4di 289	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵) ↔ (𝐶 +pQ 𝐴) <pQ (𝐶 +pQ 𝐵)))
27		xp2nd 8021	. . . . . . . . . 10 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
28	8, 27	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
29		mulclpi 10907	. . . . . . . . 9 ⊢ (((2nd ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
30	28, 28, 29	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
31		ltmpi 10918	. . . . . . . 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 8021	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
34	14, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐵) ∈ N)
35		mulclpi 10907	. . . . . . . . . 10 ⊢ (((2nd ‘𝐶) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐶) ·N (2nd ‘𝐵)) ∈ N)
36	28, 34, 35	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((2nd ‘𝐶) ·N (2nd ‘𝐵)) ∈ N)
37		xp1st 8020	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
38	8, 37	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
39		xp2nd 8021	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
40	10, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐴) ∈ N)
41		mulclpi 10907	. . . . . . . . . 10 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ N)
42	38, 40, 41	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐶) ·N (2nd ‘𝐴)) ∈ N)
43		mulclpi 10907	. . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((2nd ‘𝐶) ·N (2nd ‘𝐵)) ·N ((1st ‘𝐶) ·N (2nd ‘𝐴))) ∈ N)
45		ltapi 10917	. . . . . . . 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 279	. . . . . 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 10910	. . . . . . . . . 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 6889	. . . . . . . . . . 11 ⊢ (1st ‘𝐴) ∈ V
50		fvex 6889	. . . . . . . . . . 11 ⊢ (2nd ‘𝐵) ∈ V
51		fvex 6889	. . . . . . . . . . 11 ⊢ (2nd ‘𝐶) ∈ V
52		mulcompi 10910	. . . . . . . . . . 11 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
53		mulasspi 10911	. . . . . . . . . . 11 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
54	49, 50, 51, 52, 53, 51	caov411 7639	. . . . . . . . . 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 2758	. . . . . . . . 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 7416	. . . . . . . 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 10912	. . . . . . . 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 10910	. . . . . . . 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 2764	. . . . . . 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 10910	. . . . . . . . . 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 6889	. . . . . . . . . . 11 ⊢ (1st ‘𝐶) ∈ V
62		fvex 6889	. . . . . . . . . . 11 ⊢ (2nd ‘𝐴) ∈ V
63	61, 62, 51, 52, 53, 50	caov411 7639	. . . . . . . . . 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 2758	. . . . . . . . 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 10910	. . . . . . . . . 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 6889	. . . . . . . . . . 11 ⊢ (1st ‘𝐵) ∈ V
67	66, 62, 51, 52, 53, 51	caov411 7639	. . . . . . . . . 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 2758	. . . . . . . . 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 7417	. . . . . . . 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 10912	. . . . . . . 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 10910	. . . . . . . 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 2764	. . . . . . 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 5128	. . . . . 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	bitrdi 287	. . . . 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 10956	. . . . 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	bitr4di 289	. . . 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 312	. . 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 279	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
79	2, 3, 4, 78	ndmovord 7597	1 ⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ⟨cop 4607   class class class wbr 5119   × cxp 5652  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  Ncnpi 10858   +_N cpli 10859   ·_N cmi 10860   <_N clti 10861   +_pQ cplpq 10862   <_pQ cltpq 10864  Qcnq 10866  [Q]cerq 10868   +_Q cplq 10869   <_Q cltq 10872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8719  df-ni 10886  df-pli 10887  df-mi 10888  df-lti 10889  df-plpq 10922  df-ltpq 10924  df-enq 10925  df-nq 10926  df-erq 10927  df-plq 10928  df-1nq 10930  df-ltnq 10932

This theorem is referenced by:  ltaddnq  10988  ltbtwnnq  10992  addclpr  11032  distrlem4pr  11040  ltexprlem3  11052  ltexprlem4  11053  ltexprlem6  11055  prlem936  11061