Theorem ltmnqg 7202

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 7149	. 2 ⊢ Q = ((N × N) / ~Q )
2		breq1 3927	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ))
3		oveq2 5775	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑥, 𝑦⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴))
4	3	breq1d 3934	. . 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 3928	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q 𝐵))
7		oveq2 5775	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → ([⟨𝑣, 𝑢⟩] ~Q ·Q [⟨𝑧, 𝑤⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵))
8	7	breq2d 3936	. . 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 5774	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐴) = (𝐶 ·Q 𝐴))
11		oveq1 5774	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q ·Q 𝐵) = (𝐶 ·Q 𝐵))
12	10, 11	breq12d 3937	. . 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 7129	. . . . . . . 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 1005	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑥 ∈ N)
17		simp2r 1008	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑤 ∈ N)
18	15, 16, 17	caovcld 5917	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑥 ·N 𝑤) ∈ N)
19		simp1r 1006	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑦 ∈ N)
20		simp2l 1007	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑧 ∈ N)
21	15, 19, 20	caovcld 5917	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑦 ·N 𝑧) ∈ N)
22		mulclpi 7129	. . . . . . 7 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → (𝑣 ·N 𝑢) ∈ N)
23	22	3ad2ant3 1004	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑢) ∈ N)
24		ltmpig 7140	. . . . . 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 1216	. . . . 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 1009	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑣 ∈ N)
27		simp3r 1010	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑢 ∈ N)
28		mulcompig 7132	. . . . . . . 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 7133	. . . . . . . 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 5948	. . . . . 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 5948	. . . . . . 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 7132	. . . . . . . . . 10 ⊢ ((𝑢 ∈ N ∧ 𝑣 ∈ N) → (𝑢 ·N 𝑣) = (𝑣 ·N 𝑢))
35	34	oveq1d 5782	. . . . . . . . 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 1004	. . . . . . 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 2170	. . . . . 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 3937	. . . . 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 7175	. . . . 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 1001	. . . 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 5917	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑥) ∈ N)
44	15, 27, 19	caovcld 5917	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑦) ∈ N)
45	15, 26, 20	caovcld 5917	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑧) ∈ N)
46	15, 27, 17	caovcld 5917	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑤) ∈ N)
47		ordpipqqs 7175	. . . . 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 1217	. . . 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 7174	. . . . . 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 1000	. . . 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 7174	. . . . . 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 999	. . . 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 3937	. . 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 6511	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 962   = wceq 1331   ∈ wcel 1480  ⟨cop 3525   class class class wbr 3924  (class class class)co 5767  [cec 6420  Ncnpi 7073   ·_N cmi 7075   <_N clti 7076   ~_Q ceq 7080  Qcnq 7081   ·_Q cmq 7084   <_Q cltq 7086

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-lti 7108  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-mqqs 7151  df-ltnqqs 7154

This theorem is referenced by:  ltmnqi  7204  lt2mulnq  7206  ltaddnq  7208  prarloclemarch  7219  prarloclemarch2  7220  ltrnqg  7221  prarloclemlt  7294  addnqprllem  7328  addnqprulem  7329  appdivnq  7364  mulnqprl  7369  mulnqpru  7370  mullocprlem  7371  mulclpr  7373  distrlem4prl  7385  distrlem4pru  7386  1idprl  7391  1idpru  7392  recexprlem1ssl  7434  recexprlem1ssu  7435