Theorem ltexnq 9676

Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltexnq	⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnq

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 9627	. . . 4 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5090	. . 3 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3		ordpinq 9644	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
4		elpqn 9626	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ (N × N))
6		xp1st 7089	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐴) ∈ N)
8		elpqn 9626	. . . . . . . . 9 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ (N × N))
10		xp2nd 7090	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
11	9, 10	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐵) ∈ N)
12		mulclpi 9594	. . . . . . 7 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
13	7, 11, 12	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
14		xp1st 7089	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
15	9, 14	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐵) ∈ N)
16		xp2nd 7090	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
17	5, 16	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐴) ∈ N)
18		mulclpi 9594	. . . . . . 7 ⊢ (((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
19	15, 17, 18	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
20		ltexpi 9603	. . . . . 6 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N ∧ ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
21	13, 19, 20	syl2anc 691	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
22		relxp 5150	. . . . . . . . . . . 12 ⊢ Rel (N × N)
23	4	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 ∈ (N × N))
24		1st2nd 7105	. . . . . . . . . . . 12 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
25	22, 23, 24	sylancr 694	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25	oveq1d 6564	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
27	7	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐴) ∈ N)
28	17	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐴) ∈ N)
29		simpr 476	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝑦 ∈ N)
30		mulclpi 9594	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
31	17, 11, 30	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
32	31	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33		addpipq 9638	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ (𝑦 ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
34	27, 28, 29, 32, 33	syl22anc 1319	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
35	26, 34	eqtrd 2644	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
36		oveq2 6557	. . . . . . . . . . . 12 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
37		distrpi 9599	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦))
38		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐴) ∈ V
39		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝐴) ∈ V
40		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐵) ∈ V
41		mulcompi 9597	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
42		mulasspi 9598	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
43	38, 39, 40, 41, 42	caov12 6760	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = ((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
44		mulcompi 9597	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N 𝑦) = (𝑦 ·N (2nd ‘𝐴))
45	43, 44	oveq12i 6561	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦)) = (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴)))
46	37, 45	eqtr2i 2633	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦))
47		mulasspi 9598	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵)))
48		mulcompi 9597	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))
49	48	oveq2i 6560	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵))) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
50	47, 49	eqtri 2632	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
51	36, 46, 50	3eqtr4g 2669	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)))
52		mulasspi 9598	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
53	52	eqcomi 2619	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)))
55	51, 54	opeq12d 4348	. . . . . . . . . 10 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩)
56	55	eqeq2d 2620	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ ↔ (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
57	35, 56	syl5ibcom 234	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
58		fveq2 6103	. . . . . . . . 9 ⊢ ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
59		adderpq 9657	. . . . . . . . . . 11 ⊢ (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
60		nqerid 9634	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
61	60	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐴) = 𝐴)
62	61	oveq1d 6564	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
63	59, 62	syl5eqr 2658	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
64		mulclpi 9594	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
65	17, 17, 64	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
66	65	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
67	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐵) ∈ N)
68	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐵) ∈ N)
69		mulcanenq 9661	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
70	66, 67, 68, 69	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
71	8	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 ∈ (N × N))
72		1st2nd 7105	. . . . . . . . . . . . . 14 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
73	22, 71, 72	sylancr 694	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
74	70, 73	breqtrrd 4611	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵)
75		mulclpi 9594	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
76	66, 67, 75	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
77		mulclpi 9594	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (2nd ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
78	66, 68, 77	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
79		opelxpi 5072	. . . . . . . . . . . . . 14 ⊢ (((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N ∧ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N))
80	76, 78, 79	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N))
81		nqereq 9636	. . . . . . . . . . . . 13 ⊢ ((⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
82	80, 71, 81	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
83	74, 82	mpbid 221	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵))
84		nqerid 9634	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Q → ([Q]‘𝐵) = 𝐵)
85	84	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐵) = 𝐵)
86	83, 85	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = 𝐵)
87	63, 86	eqeq12d 2625	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
88	58, 87	syl5ib 233	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
89	57, 88	syld 46	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
90		fvex 6113	. . . . . . . 8 ⊢ ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) ∈ V
91		oveq2 6557	. . . . . . . . 9 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → (𝐴 +Q 𝑥) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
92	91	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → ((𝐴 +Q 𝑥) = 𝐵 ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
93	90, 92	spcev 3273	. . . . . . 7 ⊢ ((𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
94	89, 93	syl6 34	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
95	94	rexlimdva 3013	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
96	21, 95	sylbid 229	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
97	3, 96	sylbid 229	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
98	2, 97	mpcom 37	. 2 ⊢ (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
99		eleq1 2676	. . . . . . 7 ⊢ ((𝐴 +Q 𝑥) = 𝐵 → ((𝐴 +Q 𝑥) ∈ Q ↔ 𝐵 ∈ Q))
100	99	biimparc 503	. . . . . 6 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) ∈ Q)
101		addnqf 9649	. . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q
102	101	fdmi 5965	. . . . . . 7 ⊢ dom +Q = (Q × Q)
103		0nnq 9625	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
104	102, 103	ndmovrcl 6718	. . . . . 6 ⊢ ((𝐴 +Q 𝑥) ∈ Q → (𝐴 ∈ Q ∧ 𝑥 ∈ Q))
105		ltaddnq 9675	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
106	100, 104, 105	3syl 18	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q (𝐴 +Q 𝑥))
107		simpr 476	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) = 𝐵)
108	106, 107	breqtrd 4609	. . . 4 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q 𝐵)
109	108	ex 449	. . 3 ⊢ (𝐵 ∈ Q → ((𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
110	109	exlimdv 1848	. 2 ⊢ (𝐵 ∈ Q → (∃𝑥(𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
111	98, 110	impbid2 215	1 ⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131   class class class wbr 4583   × cxp 5036  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  Ncnpi 9545   +_N cpli 9546   ·_N cmi 9547   <_N clti 9548   +_pQ cplpq 9549   ~_Q ceq 9552  Qcnq 9553  [Q]cerq 9555   +_Q cplq 9556   <_Q cltq 9559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-ltnq 9619

This theorem is referenced by:  ltbtwnnq  9679  prnmadd  9698  ltexprlem4  9740  ltexprlem7  9743  prlem936  9748