Theorem ltexnqq 7765

Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnqq

Dummy variables 𝑓 𝑔 ℎ 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nqqs 7705	. . 3 ⊢ Q = ((N × N) / ~Q )
2		breq1 4128	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q ↔ 𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ))
3		oveq1 6082	. . . . . 6 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = (𝐴 +Q 𝑥))
4	3	eqeq1d 2247	. . . . 5 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
5	4	rexbidv 2551	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (∃𝑥 ∈ Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
6	2, 5	imbi12d 234	. . 3 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥 ∈ Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ) ↔ (𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q )))
7		breq2 4129	. . . 4 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ↔ 𝐴 <Q 𝐵))
8		eqeq2 2248	. . . . 5 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → ((𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = 𝐵))
9	8	rexbidv 2551	. . . 4 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))
10	7, 9	imbi12d 234	. . 3 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → ((𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ) ↔ (𝐴 <Q 𝐵 → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵)))
11		ordpipqqs 7731	. . . 4 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q ↔ (𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤)))
12		mulclpi 7685	. . . . . . . . 9 ⊢ ((𝑦 ∈ N ∧ 𝑣 ∈ N) → (𝑦 ·N 𝑣) ∈ N)
13		mulclpi 7685	. . . . . . . . 9 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N) → (𝑧 ·N 𝑤) ∈ N)
14	12, 13	anim12i 338	. . . . . . . 8 ⊢ (((𝑦 ∈ N ∧ 𝑣 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
15	14	an42s 597	. . . . . . 7 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
16		ltexpi 7694	. . . . . . 7 ⊢ (((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N) → ((𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤) ↔ ∃𝑢 ∈ N ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
17	15, 16	syl 14	. . . . . 6 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤) ↔ ∃𝑢 ∈ N ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
18		df-rex 2534	. . . . . 6 ⊢ (∃𝑢 ∈ N ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤) ↔ ∃𝑢(𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
19	17, 18	bitrdi 196	. . . . 5 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤) ↔ ∃𝑢(𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))))
20		simpll 531	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → (𝑦 ∈ N ∧ 𝑧 ∈ N))
21		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → 𝑢 ∈ N)
22		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ N ∧ 𝑧 ∈ N) → 𝑧 ∈ N)
23		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ N) → 𝑣 ∈ N)
24	22, 23	anim12i 338	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑣 ∈ N))
25	24	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → (𝑧 ∈ N ∧ 𝑣 ∈ N))
26		mulclpi 7685	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ N ∧ 𝑣 ∈ N) → (𝑧 ·N 𝑣) ∈ N)
27	25, 26	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → (𝑧 ·N 𝑣) ∈ N)
28	20, 21, 27	jca32 310	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → ((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N)))
29	28	adantrr 483	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N)))
30		addpipqqs 7727	. . . . . . . . . 10 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N)) → ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q )
31	29, 30	syl 14	. . . . . . . . 9 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q )
32		simplll 539	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑦 ∈ N)
33		simpllr 540	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑧 ∈ N)
34		simplrr 542	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑣 ∈ N)
35		mulcompig 7688	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
36	35	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
37		mulasspig 7689	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
38	37	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N)) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
39	32, 33, 34, 36, 38	caov12d 6261	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑦 ·N (𝑧 ·N 𝑣)) = (𝑧 ·N (𝑦 ·N 𝑣)))
40	39	oveq1d 6090	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)) = ((𝑧 ·N (𝑦 ·N 𝑣)) +N (𝑧 ·N 𝑢)))
41	32, 34, 12	syl2anc 415	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑦 ·N 𝑣) ∈ N)
42		simprl 535	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑢 ∈ N)
43		distrpig 7690	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ N ∧ (𝑦 ·N 𝑣) ∈ N ∧ 𝑢 ∈ N) → (𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)) = ((𝑧 ·N (𝑦 ·N 𝑣)) +N (𝑧 ·N 𝑢)))
44	33, 41, 42, 43	syl3anc 1278	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)) = ((𝑧 ·N (𝑦 ·N 𝑣)) +N (𝑧 ·N 𝑢)))
45	40, 44	eqtr4d 2274	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)) = (𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)))
46	45	opeq1d 3905	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩ = ⟨(𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩)
47	46	eceq1d 6833	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q = [⟨(𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q )
48		simpllr 540	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → 𝑧 ∈ N)
49	12	ad2ant2rl 515	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑦 ·N 𝑣) ∈ N)
50		addclpi 7684	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ·N 𝑣) ∈ N ∧ 𝑢 ∈ N) → ((𝑦 ·N 𝑣) +N 𝑢) ∈ N)
51	49, 50	sylan 283	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → ((𝑦 ·N 𝑣) +N 𝑢) ∈ N)
52	48, 51, 27	3jca 1208	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → (𝑧 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) ∈ N ∧ (𝑧 ·N 𝑣) ∈ N))
53	52	adantrr 483	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑧 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) ∈ N ∧ (𝑧 ·N 𝑣) ∈ N))
54		mulcanenqec 7743	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) ∈ N ∧ (𝑧 ·N 𝑣) ∈ N) → [⟨(𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q = [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q )
55	53, 54	syl 14	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨(𝑧 ·N ((𝑦 ·N 𝑣) +N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q = [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q )
56	47, 55	eqtrd 2271	. . . . . . . . 9 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨((𝑦 ·N (𝑧 ·N 𝑣)) +N (𝑧 ·N 𝑢)), (𝑧 ·N (𝑧 ·N 𝑣))⟩] ~Q = [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q )
57		3anass 1013	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) ↔ (𝑧 ∈ N ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
58	57	biimpri 133	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ N ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N))
59	58	adantll 480	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N))
60	59	anim1i 340	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)) → ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
61	60	adantrl 482	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)))
62		opeq1 3899	. . . . . . . . . . . 12 ⊢ (((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤) → ⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩ = ⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩)
63	62	eceq1d 6833	. . . . . . . . . . 11 ⊢ (((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤) → [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q = [⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩] ~Q )
64		mulcanenqec 7743	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) → [⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
65	63, 64	sylan9eqr 2293	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)) → [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
66	61, 65	syl 14	. . . . . . . . 9 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨((𝑦 ·N 𝑣) +N 𝑢), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
67	31, 56, 66	3eqtrd 2275	. . . . . . . 8 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨𝑤, 𝑣⟩] ~Q )
68	33, 34, 26	syl2anc 415	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (𝑧 ·N 𝑣) ∈ N)
69		opelxpi 4801	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N) → ⟨𝑢, (𝑧 ·N 𝑣)⟩ ∈ (N × N))
70		enqex 7717	. . . . . . . . . . . . 13 ⊢ ~Q ∈ V
71	70	ecelqsi 6853	. . . . . . . . . . . 12 ⊢ (⟨𝑢, (𝑧 ·N 𝑣)⟩ ∈ (N × N) → [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ∈ ((N × N) / ~Q ))
72	69, 71	syl 14	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N) → [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ∈ ((N × N) / ~Q ))
73	42, 68, 72	syl2anc 415	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ∈ ((N × N) / ~Q ))
74	73, 1	eleqtrrdi 2332	. . . . . . . . 9 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ∈ Q)
75		oveq2 6083	. . . . . . . . . . 11 ⊢ (𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ))
76	75	eqeq1d 2247	. . . . . . . . . 10 ⊢ (𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨𝑤, 𝑣⟩] ~Q ))
77	76	adantl 277	. . . . . . . . 9 ⊢ (((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) ∧ 𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨𝑤, 𝑣⟩] ~Q ))
78	74, 77	rspcedv 2933	. . . . . . . 8 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → (([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ) = [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥 ∈ Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
79	67, 78	mpd 13	. . . . . . 7 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → ∃𝑥 ∈ Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q )
80	79	ex 115	. . . . . 6 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)) → ∃𝑥 ∈ Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
81	80	exlimdv 1872	. . . . 5 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (∃𝑢(𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤)) → ∃𝑥 ∈ Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
82	19, 81	sylbid 150	. . . 4 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑦 ·N 𝑣) <N (𝑧 ·N 𝑤) → ∃𝑥 ∈ Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
83	11, 82	sylbid 150	. . 3 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q → ∃𝑥 ∈ Q ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
84	1, 6, 10, 83	2ecoptocl 6887	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))
85		ltaddnq 7764	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
86		breq2 4129	. . . . 5 ⊢ ((𝐴 +Q 𝑥) = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑥) ↔ 𝐴 <Q 𝐵))
87	85, 86	syl5ibcom 155	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → ((𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
88	87	rexlimdva 2668	. . 3 ⊢ (𝐴 ∈ Q → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
89	88	adantr 276	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
90	84, 89	impbid 129	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  ⟨cop 3708   class class class wbr 4125   × cxp 4767  (class class class)co 6075  [cec 6795   / cqs 6796  Ncnpi 7629   +_N cpli 7630   ·_N cmi 7631   <_N clti 7632   ~_Q ceq 7636  Qcnq 7637   +_Q cplq 7639   <_Q cltq 7642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-ltnqqs 7710

This theorem is referenced by:  ltexnqi  7766  addlocpr  7893  ltexprlemopl  7958  ltexprlemopu  7960  ltexprlemrl  7967  ltexprlemru  7969  cauappcvgprlemopl  8003  caucvgprlemopl  8026  caucvgprprlemopl  8054