Theorem ltmnqg 6499

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 6446	. 2 ⊢ Q = ((N × N) / ~Q )
2		breq1 3767	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ))
3		oveq2 5520	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴))
4	3	breq1d 3774	. . 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 224	. 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 3768	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q 𝐵))
7		oveq2 5520	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵))
8	7	breq2d 3776	. . 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 224	. 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 5519	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) = (𝐶 ·Q 𝐴))
11		oveq1 5519	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵) = (𝐶 ·Q 𝐵))
12	10, 11	breq12d 3777	. . 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 221	. 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 6426	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) ∈ N)
15	14	adantl 262	. . . . . . 7 ⊢ ((((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) → (𝑓 ·N 𝑔) ∈ N)
16		simp1l 928	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑥 ∈ N)
17		simp2r 931	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑤 ∈ N)
18	15, 16, 17	caovcld 5654	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑥 ·N 𝑤) ∈ N)
19		simp1r 929	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑦 ∈ N)
20		simp2l 930	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑧 ∈ N)
21	15, 19, 20	caovcld 5654	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑦 ·N 𝑧) ∈ N)
22		mulclpi 6426	. . . . . . 7 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → (𝑣 ·N 𝑢) ∈ N)
23	22	3ad2ant3 927	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑢) ∈ N)
24		ltmpig 6437	. . . . . 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 1135	. . . . 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 932	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑣 ∈ N)
27		simp3r 933	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑢 ∈ N)
28		mulcompig 6429	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
29	28	adantl 262	. . . . . . 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 6430	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
31	30	adantl 262	. . . . . . 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 5685	. . . . . 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 5685	. . . . . . 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 6429	. . . . . . . . . 10 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
35	34	oveq1d 5527	. . . . . . . . 9 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → ((𝑢 ·N 𝑣) ·N (𝑦 ·N 𝑧)) = ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧)))
36	35	ancoms 255	. . . . . . . 8 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → ((𝑢 ·N 𝑣) ·N (𝑦 ·N 𝑧)) = ((𝑣 ·N 𝑢) ·N (𝑦 ·N 𝑧)))
37	36	3ad2ant3 927	. . . . . . 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 2072	. . . . . 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 3777	. . . . 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 180	. . . 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 6472	. . . . 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 924	. . . 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 5654	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑥) ∈ N)
44	15, 27, 19	caovcld 5654	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑦) ∈ N)
45	15, 26, 20	caovcld 5654	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑧) ∈ N)
46	15, 27, 17	caovcld 5654	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑤) ∈ N)
47		ordpipqqs 6472	. . . . 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 1136	. . . 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 209	. . 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 6471	. . . . . 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 255	. . . . 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 923	. . . 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 6471	. . . . . 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 255	. . . . 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 922	. . . 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 3777	. . 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 180	. 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 6195	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  [cec 6104  Ncnpi 6370   ·_N cmi 6372   <_N clti 6373   ~_Q ceq 6377  Qcnq 6378   ·_Q cmq 6381   <_Q cltq 6383

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448  df-ltnqqs 6451

This theorem is referenced by:  ltmnqi  6501  lt2mulnq  6503  ltaddnq  6505  prarloclemarch  6516  prarloclemarch2  6517  ltrnqg  6518  prarloclemlt  6591  addnqprllem  6625  addnqprulem  6626  appdivnq  6661  mulnqprl  6666  mulnqpru  6667  mullocprlem  6668  mulclpr  6670  distrlem4prl  6682  distrlem4pru  6683  1idprl  6688  1idpru  6689  recexprlem1ssl  6731  recexprlem1ssu  6732