Theorem ltexnqq 7349

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 7289	. . 3 ⊢ Q = ((N × N) / ~Q )
2		breq1 3985	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q ↔ 𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ))
3		oveq1 5849	. . . . . 6 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = (𝐴 +Q 𝑥))
4	3	eqeq1d 2174	. . . . 5 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
5	4	rexbidv 2467	. . . 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 233	. . 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 3986	. . . 4 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ↔ 𝐴 <Q 𝐵))
8		eqeq2 2175	. . . . 5 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → ((𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = 𝐵))
9	8	rexbidv 2467	. . . 4 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))
10	7, 9	imbi12d 233	. . 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 7315	. . . 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 7269	. . . . . . . . 9 ⊢ ((𝑦 ∈ N ∧ 𝑣 ∈ N) → (𝑦 ·N 𝑣) ∈ N)
13		mulclpi 7269	. . . . . . . . 9 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N) → (𝑧 ·N 𝑤) ∈ N)
14	12, 13	anim12i 336	. . . . . . . 8 ⊢ (((𝑦 ∈ N ∧ 𝑣 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
15	14	an42s 579	. . . . . . 7 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
16		ltexpi 7278	. . . . . . 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 2450	. . . . . 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 195	. . . . 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 519	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → (𝑦 ∈ N ∧ 𝑧 ∈ N))
21		simpr 109	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → 𝑢 ∈ N)
22		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ N ∧ 𝑧 ∈ N) → 𝑧 ∈ N)
23		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ N) → 𝑣 ∈ N)
24	22, 23	anim12i 336	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑣 ∈ N))
25	24	adantr 274	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → (𝑧 ∈ N ∧ 𝑣 ∈ N))
26		mulclpi 7269	. . . . . . . . . . . . 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 308	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → ((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N)))
29	28	adantrr 471	. . . . . . . . . 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 7311	. . . . . . . . . 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 523	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑦 ∈ N)
33		simpllr 524	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑧 ∈ N)
34		simplrr 526	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑣 ∈ N)
35		mulcompig 7272	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
36	35	adantl 275	. . . . . . . . . . . . . . 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 7273	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
38	37	adantl 275	. . . . . . . . . . . . . . 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 6023	. . . . . . . . . . . . . 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 5857	. . . . . . . . . . . . 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 409	. . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑢 ∈ N)
43		distrpig 7274	. . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . 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 2201	. . . . . . . . . . . 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 3764	. . . . . . . . . . 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 6537	. . . . . . . . . 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 524	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → 𝑧 ∈ N)
49	12	ad2ant2rl 503	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑦 ·N 𝑣) ∈ N)
50		addclpi 7268	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ·N 𝑣) ∈ N ∧ 𝑢 ∈ N) → ((𝑦 ·N 𝑣) +N 𝑢) ∈ N)
51	49, 50	sylan 281	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → ((𝑦 ·N 𝑣) +N 𝑢) ∈ N)
52	48, 51, 27	3jca 1167	. . . . . . . . . . . 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 471	. . . . . . . . . . 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 7327	. . . . . . . . . . 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 2198	. . . . . . . . 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 972	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) ↔ (𝑧 ∈ N ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
58	57	biimpri 132	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ N ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N))
59	58	adantll 468	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N))
60	59	anim1i 338	. . . . . . . . . . 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 470	. . . . . . . . . 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 3758	. . . . . . . . . . . 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 6537	. . . . . . . . . . 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 7327	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) → [⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
65	63, 64	sylan9eqr 2221	. . . . . . . . . 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 2202	. . . . . . . 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 409	. . . . . . . . . . 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 4636	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N) → ⟨𝑢, (𝑧 ·N 𝑣)⟩ ∈ (N × N))
70		enqex 7301	. . . . . . . . . . . . 13 ⊢ ~Q ∈ V
71	70	ecelqsi 6555	. . . . . . . . . . . 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 409	. . . . . . . . . 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 2260	. . . . . . . . 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 5850	. . . . . . . . . . 11 ⊢ (𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ))
76	75	eqeq1d 2174	. . . . . . . . . 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 275	. . . . . . . . 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 2834	. . . . . . . 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 114	. . . . . 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 1807	. . . . 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 149	. . . 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 149	. . 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 6589	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))
85		ltaddnq 7348	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
86		breq2 3986	. . . . 5 ⊢ ((𝐴 +Q 𝑥) = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑥) ↔ 𝐴 <Q 𝐵))
87	85, 86	syl5ibcom 154	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → ((𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
88	87	rexlimdva 2583	. . 3 ⊢ (𝐴 ∈ Q → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
89	88	adantr 274	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
90	84, 89	impbid 128	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  ⟨cop 3579   class class class wbr 3982   × cxp 4602  (class class class)co 5842  [cec 6499   / cqs 6500  Ncnpi 7213   +_N cpli 7214   ·_N cmi 7215   <_N clti 7216   ~_Q ceq 7220  Qcnq 7221   +_Q cplq 7223   <_Q cltq 7226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-ltnqqs 7294

This theorem is referenced by:  ltexnqi  7350  addlocpr  7477  ltexprlemopl  7542  ltexprlemopu  7544  ltexprlemrl  7551  ltexprlemru  7553  cauappcvgprlemopl  7587  caucvgprlemopl  7610  caucvgprprlemopl  7638