Theorem ltexnqq 6598

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 6538	. . 3 ⊢ Q = ((N × N) / ~Q )
2		breq1 3788	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q ↔ 𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ))
3		oveq1 5539	. . . . . 6 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = (𝐴 +Q 𝑥))
4	3	eqeq1d 2089	. . . . 5 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
5	4	rexbidv 2369	. . . 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 232	. . 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 3789	. . . 4 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ↔ 𝐴 <Q 𝐵))
8		eqeq2 2090	. . . . 5 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → ((𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = 𝐵))
9	8	rexbidv 2369	. . . 4 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))
10	7, 9	imbi12d 232	. . 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 6564	. . . 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 6518	. . . . . . . . 9 ⊢ ((𝑦 ∈ N ∧ 𝑣 ∈ N) → (𝑦 ·N 𝑣) ∈ N)
13		mulclpi 6518	. . . . . . . . 9 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N) → (𝑧 ·N 𝑤) ∈ N)
14	12, 13	anim12i 331	. . . . . . . 8 ⊢ (((𝑦 ∈ N ∧ 𝑣 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
15	14	an42s 553	. . . . . . 7 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
16		ltexpi 6527	. . . . . . 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 2354	. . . . . 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	syl6bb 194	. . . . 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 495	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → (𝑦 ∈ N ∧ 𝑧 ∈ N))
21		simpr 108	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → 𝑢 ∈ N)
22		simpr 108	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ N ∧ 𝑧 ∈ N) → 𝑧 ∈ N)
23		simpr 108	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ N) → 𝑣 ∈ N)
24	22, 23	anim12i 331	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑣 ∈ N))
25	24	adantr 270	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → (𝑧 ∈ N ∧ 𝑣 ∈ N))
26		mulclpi 6518	. . . . . . . . . . . . 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 303	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → ((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N)))
29	28	adantrr 462	. . . . . . . . . 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 6560	. . . . . . . . . 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 499	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑦 ∈ N)
33		simpllr 500	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑧 ∈ N)
34		simplrr 502	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑣 ∈ N)
35		mulcompig 6521	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
36	35	adantl 271	. . . . . . . . . . . . . . 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 6522	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
38	37	adantl 271	. . . . . . . . . . . . . . 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 5702	. . . . . . . . . . . . . 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 5547	. . . . . . . . . . . . 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 403	. . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑢 ∈ N)
43		distrpig 6523	. . . . . . . . . . . . . 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 1169	. . . . . . . . . . . . 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 2116	. . . . . . . . . . . 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 3576	. . . . . . . . . . 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 6165	. . . . . . . . . 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 500	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → 𝑧 ∈ N)
49	12	ad2ant2rl 494	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑦 ·N 𝑣) ∈ N)
50		addclpi 6517	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ·N 𝑣) ∈ N ∧ 𝑢 ∈ N) → ((𝑦 ·N 𝑣) +N 𝑢) ∈ N)
51	49, 50	sylan 277	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → ((𝑦 ·N 𝑣) +N 𝑢) ∈ N)
52	48, 51, 27	3jca 1118	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 6576	. . . . . . . . . . 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 2113	. . . . . . . . 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 923	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) ↔ (𝑧 ∈ N ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)))
58	57	biimpri 131	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ N ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N))
59	58	adantll 459	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N))
60	59	anim1i 333	. . . . . . . . . . 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 461	. . . . . . . . . 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 3570	. . . . . . . . . . . 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 6165	. . . . . . . . . . 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 6576	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) → [⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
65	63, 64	sylan9eqr 2135	. . . . . . . . . 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 2117	. . . . . . . 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 403	. . . . . . . . . . 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 4394	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N) → ⟨𝑢, (𝑧 ·N 𝑣)⟩ ∈ (N × N))
70		enqex 6550	. . . . . . . . . . . . 13 ⊢ ~Q ∈ V
71	70	ecelqsi 6183	. . . . . . . . . . . 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 403	. . . . . . . . . 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	syl6eleqr 2172	. . . . . . . . 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 5540	. . . . . . . . . . 11 ⊢ (𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ))
76	75	eqeq1d 2089	. . . . . . . . . 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 271	. . . . . . . . 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 2705	. . . . . . . 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 113	. . . . . 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 1740	. . . . 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 148	. . . 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 148	. . 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 6217	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))
85		ltaddnq 6597	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
86		breq2 3789	. . . . 5 ⊢ ((𝐴 +Q 𝑥) = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑥) ↔ 𝐴 <Q 𝐵))
87	85, 86	syl5ibcom 153	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → ((𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
88	87	rexlimdva 2477	. . 3 ⊢ (𝐴 ∈ Q → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
89	88	adantr 270	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
90	84, 89	impbid 127	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349  ⟨cop 3401   class class class wbr 3785   × cxp 4361  (class class class)co 5532  [cec 6127   / cqs 6128  Ncnpi 6462   +_N cpli 6463   ·_N cmi 6464   <_N clti 6465   ~_Q ceq 6469  Qcnq 6470   +_Q cplq 6472   <_Q cltq 6475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-ltnqqs 6543

This theorem is referenced by:  ltexnqi  6599  addlocpr  6726  ltexprlemopl  6791  ltexprlemopu  6793  ltexprlemrl  6800  ltexprlemru  6802  cauappcvgprlemopl  6836  caucvgprlemopl  6859  caucvgprprlemopl  6887