Theorem ltexnqq 7503

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 7443	. . 3 ⊢ Q = ((N × N) / ~Q )
2		breq1 4046	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q <Q [⟨𝑤, 𝑣⟩] ~Q ↔ 𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ))
3		oveq1 5941	. . . . . 6 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = (𝐴 +Q 𝑥))
4	3	eqeq1d 2213	. . . . 5 ⊢ ([⟨𝑦, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ))
5	4	rexbidv 2506	. . . 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 4047	. . . 4 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑤, 𝑣⟩] ~Q ↔ 𝐴 <Q 𝐵))
8		eqeq2 2214	. . . . 5 ⊢ ([⟨𝑤, 𝑣⟩] ~Q = 𝐵 → ((𝐴 +Q 𝑥) = [⟨𝑤, 𝑣⟩] ~Q ↔ (𝐴 +Q 𝑥) = 𝐵))
9	8	rexbidv 2506	. . . 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 7469	. . . 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 7423	. . . . . . . . 9 ⊢ ((𝑦 ∈ N ∧ 𝑣 ∈ N) → (𝑦 ·N 𝑣) ∈ N)
13		mulclpi 7423	. . . . . . . . 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 589	. . . . . . 7 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → ((𝑦 ·N 𝑣) ∈ N ∧ (𝑧 ·N 𝑤) ∈ N))
16		ltexpi 7432	. . . . . . 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 2489	. . . . . 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 527	. . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 479	. . . . . . . . . 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 7465	. . . . . . . . . 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 533	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑦 ∈ N)
33		simpllr 534	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑧 ∈ N)
34		simplrr 536	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑣 ∈ N)
35		mulcompig 7426	. . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . 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 6118	. . . . . . . . . . . . . 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 5949	. . . . . . . . . . . . 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 411	. . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ (𝑢 ∈ N ∧ ((𝑦 ·N 𝑣) +N 𝑢) = (𝑧 ·N 𝑤))) → 𝑢 ∈ N)
43		distrpig 7428	. . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . 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 2240	. . . . . . . . . . . 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 3824	. . . . . . . . . . 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 6646	. . . . . . . . . 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 534	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) ∧ 𝑢 ∈ N) → 𝑧 ∈ N)
49	12	ad2ant2rl 511	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑤 ∈ N ∧ 𝑣 ∈ N)) → (𝑦 ·N 𝑣) ∈ N)
50		addclpi 7422	. . . . . . . . . . . . . 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 1179	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 7481	. . . . . . . . . . 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 2237	. . . . . . . . 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 984	. . . . . . . . . . . . . 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 476	. . . . . . . . . . . 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 478	. . . . . . . . . 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 3818	. . . . . . . . . . . 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 6646	. . . . . . . . . . 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 7481	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N ∧ 𝑣 ∈ N) → [⟨(𝑧 ·N 𝑤), (𝑧 ·N 𝑣)⟩] ~Q = [⟨𝑤, 𝑣⟩] ~Q )
65	63, 64	sylan9eqr 2259	. . . . . . . . . 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 2241	. . . . . . . 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 411	. . . . . . . . . . 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 4705	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ N ∧ (𝑧 ·N 𝑣) ∈ N) → ⟨𝑢, (𝑧 ·N 𝑣)⟩ ∈ (N × N))
70		enqex 7455	. . . . . . . . . . . . 13 ⊢ ~Q ∈ V
71	70	ecelqsi 6666	. . . . . . . . . . . 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 411	. . . . . . . . . 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 2298	. . . . . . . . 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 5942	. . . . . . . . . . 11 ⊢ (𝑥 = [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q → ([⟨𝑦, 𝑧⟩] ~Q +Q 𝑥) = ([⟨𝑦, 𝑧⟩] ~Q +Q [⟨𝑢, (𝑧 ·N 𝑣)⟩] ~Q ))
76	75	eqeq1d 2213	. . . . . . . . . 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 2880	. . . . . . . 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 1841	. . . . 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 6700	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵))
85		ltaddnq 7502	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
86		breq2 4047	. . . . 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 2622	. . 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 980   = wceq 1372  ∃wex 1514   ∈ wcel 2175  ∃wrex 2484  ⟨cop 3635   class class class wbr 4043   × cxp 4671  (class class class)co 5934  [cec 6608   / cqs 6609  Ncnpi 7367   +_N cpli 7368   ·_N cmi 7369   <_N clti 7370   ~_Q ceq 7374  Qcnq 7375   +_Q cplq 7377   <_Q cltq 7380

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4334  df-id 4338  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-1o 6492  df-oadd 6496  df-omul 6497  df-er 6610  df-ec 6612  df-qs 6616  df-ni 7399  df-pli 7400  df-mi 7401  df-lti 7402  df-plpq 7439  df-mpq 7440  df-enq 7442  df-nqqs 7443  df-plqqs 7444  df-mqqs 7445  df-1nqqs 7446  df-ltnqqs 7448

This theorem is referenced by:  ltexnqi  7504  addlocpr  7631  ltexprlemopl  7696  ltexprlemopu  7698  ltexprlemrl  7705  ltexprlemru  7707  cauappcvgprlemopl  7741  caucvgprlemopl  7764  caucvgprprlemopl  7792