Theorem ltmnqg 7233

Description: Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)

Assertion

Ref	Expression
ltmnqg	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Proof of Theorem ltmnqg

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

Step	Hyp	Ref	Expression
1		df-nqqs 7180	. 2 ⊢ Q = ((N × N) / ~Q )
2		breq1 3940	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ))
3		oveq2 5790	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴))
4	3	breq1d 3947	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → (([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) ↔ ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q )))
5	2, 4	bibi12d 234	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q )) ↔ (𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ↔ ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ))))
6		breq2 3941	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q 𝐵))
7		oveq2 5790	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵))
8	7	breq2d 3949	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → (([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) ↔ ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵)))
9	6, 8	bibi12d 234	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → ((𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ↔ ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q )) ↔ (𝐴 <Q 𝐵 ↔ ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵))))
10		oveq1 5789	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) = (𝐶 ·Q 𝐴))
11		oveq1 5789	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵) = (𝐶 ·Q 𝐵))
12	10, 11	breq12d 3950	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → (([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵) ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
13	12	bibi2d 231	. 2 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ((𝐴 <Q 𝐵 ↔ ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵)) ↔ (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))))
14		mulclpi 7160	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) ∈ N)
15	14	adantl 275	. . . . . . 7 ⊢ ((((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) → (𝑓 ·N 𝑔) ∈ N)
16		simp1l 1006	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑥 ∈ N)
17		simp2r 1009	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑤 ∈ N)
18	15, 16, 17	caovcld 5932	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑥 ·N 𝑤) ∈ N)
19		simp1r 1007	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑦 ∈ N)
20		simp2l 1008	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑧 ∈ N)
21	15, 19, 20	caovcld 5932	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑦 ·N 𝑧) ∈ N)
22		mulclpi 7160	. . . . . . 7 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → (𝑣 ·N 𝑢) ∈ N)
23	22	3ad2ant3 1005	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑢) ∈ N)
24		ltmpig 7171	. . . . . 6 ⊢ (((𝑥 ·N 𝑤) ∈ N ∧ (𝑦 ·N 𝑧) ∈ N ∧ (𝑣 ·N 𝑢) ∈ N) → ((𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧) ↔ ((𝑣 ·N 𝑢) ·N (𝑥 ·N 𝑤)) <N ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧))))
25	18, 21, 23, 24	syl3anc 1217	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧) ↔ ((𝑣 ·N 𝑢) ·N (𝑥 ·N 𝑤)) <N ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧))))
26		simp3l 1010	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑣 ∈ N)
27		simp3r 1011	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑢 ∈ N)
28		mulcompig 7163	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
29	28	adantl 275	. . . . . . 7 ⊢ ((((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
30		mulasspig 7164	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
31	30	adantl 275	. . . . . . 7 ⊢ ((((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N)) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
32	26, 16, 27, 29, 31, 17, 15	caov4d 5963	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑣 ·N 𝑥) ·N (𝑢 ·N 𝑤)) = ((𝑣 ·N 𝑢) ·N (𝑥 ·N 𝑤)))
33	27, 19, 26, 29, 31, 20, 15	caov4d 5963	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑧)) = ((𝑢 ·N 𝑣) ·N (𝑦 ·N 𝑧)))
34		mulcompig 7163	. . . . . . . . . 10 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
35	34	oveq1d 5797	. . . . . . . . 9 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → ((𝑢 ·N 𝑣) ·N (𝑦 ·N 𝑧)) = ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧)))
36	35	ancoms 266	. . . . . . . 8 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → ((𝑢 ·N 𝑣) ·N (𝑦 ·N 𝑧)) = ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧)))
37	36	3ad2ant3 1005	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑣) ·N (𝑦 ·N 𝑧)) = ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧)))
38	33, 37	eqtrd 2173	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑧)) = ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧)))
39	32, 38	breq12d 3950	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (((𝑣 ·N 𝑥) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑧)) ↔ ((𝑣 ·N 𝑢) ·N (𝑥 ·N 𝑤)) <N ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧))))
40	25, 39	bitr4d 190	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧) ↔ ((𝑣 ·N 𝑥) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑧))))
41		ordpipqqs 7206	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ (𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧)))
42	41	3adant3 1002	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ (𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧)))
43	15, 26, 16	caovcld 5932	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑥) ∈ N)
44	15, 27, 19	caovcld 5932	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑦) ∈ N)
45	15, 26, 20	caovcld 5932	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑧) ∈ N)
46	15, 27, 17	caovcld 5932	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑤) ∈ N)
47		ordpipqqs 7206	. . . . 5 ⊢ ((((𝑣 ·N 𝑥) ∈ N ∧ (𝑢 ·N 𝑦) ∈ N) ∧ ((𝑣 ·N 𝑧) ∈ N ∧ (𝑢 ·N 𝑤) ∈ N)) → ([⟨(𝑣 ·N 𝑥), (𝑢 ·N 𝑦)⟩] ~Q <Q [⟨(𝑣 ·N 𝑧), (𝑢 ·N 𝑤)⟩] ~Q ↔ ((𝑣 ·N 𝑥) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑧))))
48	43, 44, 45, 46, 47	syl22anc 1218	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨(𝑣 ·N 𝑥), (𝑢 ·N 𝑦)⟩] ~Q <Q [⟨(𝑣 ·N 𝑧), (𝑢 ·N 𝑤)⟩] ~Q ↔ ((𝑣 ·N 𝑥) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑧))))
49	40, 42, 48	3bitr4d 219	. . 3 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ [⟨(𝑣 ·N 𝑥), (𝑢 ·N 𝑦)⟩] ~Q <Q [⟨(𝑣 ·N 𝑧), (𝑢 ·N 𝑤)⟩] ~Q ))
50		mulpipqqs 7205	. . . . . 6 ⊢ (((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ (𝑥 ∈ N ∧ 𝑦 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) = [⟨(𝑣 ·N 𝑥), (𝑢 ·N 𝑦)⟩] ~Q )
51	50	ancoms 266	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) = [⟨(𝑣 ·N 𝑥), (𝑢 ·N 𝑦)⟩] ~Q )
52	51	3adant2 1001	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) = [⟨(𝑣 ·N 𝑥), (𝑢 ·N 𝑦)⟩] ~Q )
53		mulpipqqs 7205	. . . . . 6 ⊢ (((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) = [⟨(𝑣 ·N 𝑧), (𝑢 ·N 𝑤)⟩] ~Q )
54	53	ancoms 266	. . . . 5 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) = [⟨(𝑣 ·N 𝑧), (𝑢 ·N 𝑤)⟩] ~Q )
55	54	3adant1 1000	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) = [⟨(𝑣 ·N 𝑧), (𝑢 ·N 𝑤)⟩] ~Q )
56	52, 55	breq12d 3950	. . 3 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) ↔ [⟨(𝑣 ·N 𝑥), (𝑢 ·N 𝑦)⟩] ~Q <Q [⟨(𝑣 ·N 𝑧), (𝑢 ·N 𝑤)⟩] ~Q ))
57	49, 56	bitr4d 190	. 2 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) <Q ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q )))
58	1, 5, 9, 13, 57	3ecoptocl 6526	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ⟨cop 3535   class class class wbr 3937  (class class class)co 5782  [cec 6435  Ncnpi 7104   ·_N cmi 7106   <_N clti 7107   ~_Q ceq 7111  Qcnq 7112   ·_Q cmq 7115   <_Q cltq 7117

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-mi 7138  df-lti 7139  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-mqqs 7182  df-ltnqqs 7185

This theorem is referenced by:  ltmnqi  7235  lt2mulnq  7237  ltaddnq  7239  prarloclemarch  7250  prarloclemarch2  7251  ltrnqg  7252  prarloclemlt  7325  addnqprllem  7359  addnqprulem  7360  appdivnq  7395  mulnqprl  7400  mulnqpru  7401  mullocprlem  7402  mulclpr  7404  distrlem4prl  7416  distrlem4pru  7417  1idprl  7422  1idpru  7423  recexprlem1ssl  7465  recexprlem1ssu  7466