Theorem ltanqg 7349

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

Assertion

Ref	Expression
ltanqg	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))

Proof of Theorem ltanqg

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 7297	. 2 ⊢ Q = ((N × N) / ~Q )
2		breq1 3990	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ))
3		oveq2 5858	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑣, 𝑢⟩] ~Q +Q [⟨𝑥, 𝑦⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q +Q 𝐴))
4	3	breq1d 3997	. . 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 3991	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q 𝐵))
7		oveq2 5858	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → ([⟨𝑣, 𝑢⟩] ~Q +Q [⟨𝑧, 𝑤⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q +Q 𝐵))
8	7	breq2d 3999	. . 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 5857	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q +Q 𝐴) = (𝐶 +Q 𝐴))
11		oveq1 5857	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q +Q 𝐵) = (𝐶 +Q 𝐵))
12	10, 11	breq12d 4000	. . 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		addclpi 7276	. . . . . 6 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 +N 𝑔) ∈ N)
15	14	adantl 275	. . . . 5 ⊢ ((((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) → (𝑓 +N 𝑔) ∈ N)
16		simp3l 1020	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑣 ∈ N)
17		simp1r 1017	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑦 ∈ N)
18		mulclpi 7277	. . . . . 6 ⊢ ((𝑣 ∈ N ∧ 𝑦 ∈ N) → (𝑣 ·N 𝑦) ∈ N)
19	16, 17, 18	syl2anc 409	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑦) ∈ N)
20		simp3r 1021	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑢 ∈ N)
21		simp1l 1016	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑥 ∈ N)
22		mulclpi 7277	. . . . . 6 ⊢ ((𝑢 ∈ N ∧ 𝑥 ∈ N) → (𝑢 ·N 𝑥) ∈ N)
23	20, 21, 22	syl2anc 409	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑥) ∈ N)
24	15, 19, 23	caovcld 6003	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ∈ N)
25		mulclpi 7277	. . . . 5 ⊢ ((𝑢 ∈ N ∧ 𝑦 ∈ N) → (𝑢 ·N 𝑦) ∈ N)
26	20, 17, 25	syl2anc 409	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑦) ∈ N)
27		mulclpi 7277	. . . . . . 7 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) ∈ N)
28	27	adantl 275	. . . . . 6 ⊢ ((((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) → (𝑓 ·N 𝑔) ∈ N)
29		simp2r 1019	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑤 ∈ N)
30	28, 16, 29	caovcld 6003	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑤) ∈ N)
31		simp2l 1018	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑧 ∈ N)
32		mulclpi 7277	. . . . . 6 ⊢ ((𝑢 ∈ N ∧ 𝑧 ∈ N) → (𝑢 ·N 𝑧) ∈ N)
33	20, 31, 32	syl2anc 409	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑧) ∈ N)
34	15, 30, 33	caovcld 6003	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)) ∈ N)
35		mulclpi 7277	. . . . 5 ⊢ ((𝑢 ∈ N ∧ 𝑤 ∈ N) → (𝑢 ·N 𝑤) ∈ N)
36	20, 29, 35	syl2anc 409	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑤) ∈ N)
37		ordpipqqs 7323	. . . 4 ⊢ (((((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ∈ N ∧ (𝑢 ·N 𝑦) ∈ N) ∧ (((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)) ∈ N ∧ (𝑢 ·N 𝑤) ∈ N)) → ([⟨((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)), (𝑢 ·N 𝑦)⟩] ~Q <Q [⟨((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)), (𝑢 ·N 𝑤)⟩] ~Q ↔ (((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)))))
38	24, 26, 34, 36, 37	syl22anc 1234	. . 3 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)), (𝑢 ·N 𝑦)⟩] ~Q <Q [⟨((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)), (𝑢 ·N 𝑤)⟩] ~Q ↔ (((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)))))
39		simp3 994	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ∈ N ∧ 𝑢 ∈ N))
40		simp1 992	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑥 ∈ N ∧ 𝑦 ∈ N))
41		addpipqqs 7319	. . . . 5 ⊢ (((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ (𝑥 ∈ N ∧ 𝑦 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q +Q [⟨𝑥, 𝑦⟩] ~Q ) = [⟨((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)), (𝑢 ·N 𝑦)⟩] ~Q )
42	39, 40, 41	syl2anc 409	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q +Q [⟨𝑥, 𝑦⟩] ~Q ) = [⟨((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)), (𝑢 ·N 𝑦)⟩] ~Q )
43		simp2 993	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑧 ∈ N ∧ 𝑤 ∈ N))
44		addpipqqs 7319	. . . . 5 ⊢ (((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q +Q [⟨𝑧, 𝑤⟩] ~Q ) = [⟨((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)), (𝑢 ·N 𝑤)⟩] ~Q )
45	39, 43, 44	syl2anc 409	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑣, 𝑢⟩] ~Q +Q [⟨𝑧, 𝑤⟩] ~Q ) = [⟨((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)), (𝑢 ·N 𝑤)⟩] ~Q )
46	42, 45	breq12d 4000	. . 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 𝑦) +N (𝑢 ·N 𝑥)), (𝑢 ·N 𝑦)⟩] ~Q <Q [⟨((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)), (𝑢 ·N 𝑤)⟩] ~Q ))
47		ordpipqqs 7323	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ (𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧)))
48	47	3adant3 1012	. . . 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)))
49		mulclpi 7277	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑤 ∈ N) → (𝑥 ·N 𝑤) ∈ N)
50	21, 29, 49	syl2anc 409	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑥 ·N 𝑤) ∈ N)
51		mulclpi 7277	. . . . . 6 ⊢ ((𝑦 ∈ N ∧ 𝑧 ∈ N) → (𝑦 ·N 𝑧) ∈ N)
52	17, 31, 51	syl2anc 409	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑦 ·N 𝑧) ∈ N)
53		mulclpi 7277	. . . . . 6 ⊢ ((𝑢 ∈ N ∧ 𝑢 ∈ N) → (𝑢 ·N 𝑢) ∈ N)
54	20, 20, 53	syl2anc 409	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑢) ∈ N)
55		ltmpig 7288	. . . . 5 ⊢ (((𝑥 ·N 𝑤) ∈ N ∧ (𝑦 ·N 𝑧) ∈ N ∧ (𝑢 ·N 𝑢) ∈ N) → ((𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧) ↔ ((𝑢 ·N 𝑢) ·N (𝑥 ·N 𝑤)) <N ((𝑢 ·N 𝑢) ·N (𝑦 ·N 𝑧))))
56	50, 52, 54, 55	syl3anc 1233	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑥 ·N 𝑤) <N (𝑦 ·N 𝑧) ↔ ((𝑢 ·N 𝑢) ·N (𝑥 ·N 𝑤)) <N ((𝑢 ·N 𝑢) ·N (𝑦 ·N 𝑧))))
57		mulclpi 7277	. . . . . . 7 ⊢ (((𝑢 ·N 𝑥) ∈ N ∧ (𝑢 ·N 𝑤) ∈ N) → ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)) ∈ N)
58	23, 36, 57	syl2anc 409	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)) ∈ N)
59		mulclpi 7277	. . . . . . 7 ⊢ (((𝑢 ·N 𝑦) ∈ N ∧ (𝑢 ·N 𝑧) ∈ N) → ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)) ∈ N)
60	26, 33, 59	syl2anc 409	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)) ∈ N)
61		mulclpi 7277	. . . . . . 7 ⊢ (((𝑣 ·N 𝑦) ∈ N ∧ (𝑢 ·N 𝑤) ∈ N) → ((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) ∈ N)
62	19, 36, 61	syl2anc 409	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) ∈ N)
63		ltapig 7287	. . . . . 6 ⊢ ((((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)) ∈ N ∧ ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)) ∈ N ∧ ((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) ∈ N) → (((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)) ↔ (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤))) <N (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)))))
64	58, 60, 62, 63	syl3anc 1233	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)) ↔ (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤))) <N (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)))))
65		mulcompig 7280	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
66	65	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))
67		mulasspig 7281	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
68	67	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)))
69	20, 20, 21, 66, 68, 29, 28	caov4d 6034	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑢) ·N (𝑥 ·N 𝑤)) = ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)))
70	20, 20, 17, 66, 68, 31, 28	caov4d 6034	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑢) ·N (𝑦 ·N 𝑧)) = ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)))
71	69, 70	breq12d 4000	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (((𝑢 ·N 𝑢) ·N (𝑥 ·N 𝑤)) <N ((𝑢 ·N 𝑢) ·N (𝑦 ·N 𝑧)) ↔ ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧))))
72		distrpig 7282	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → (𝑓 ·N (𝑔 +N ℎ)) = ((𝑓 ·N 𝑔) +N (𝑓 ·N ℎ)))
73	72	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 (𝑓 ·N ℎ)))
74	73, 19, 23, 36, 15, 66	caovdir2d 6026	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ·N (𝑢 ·N 𝑤)) = (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤))))
75	73, 26, 30, 33	caovdid 6025	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑦) ·N ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧))) = (((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑤)) +N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧))))
76	20, 17, 16, 66, 68, 29, 28	caov411d 6035	. . . . . . . 8 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑤)) = ((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)))
77	76	oveq1d 5865	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (((𝑢 ·N 𝑦) ·N (𝑣 ·N 𝑤)) +N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧))) = (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧))))
78	75, 77	eqtrd 2203	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑦) ·N ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧))) = (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧))))
79	74, 78	breq12d 4000	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧))) ↔ (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤))) <N (((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) +N ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)))))
80	64, 71, 79	3bitr4d 219	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (((𝑢 ·N 𝑢) ·N (𝑥 ·N 𝑤)) <N ((𝑢 ·N 𝑢) ·N (𝑦 ·N 𝑧)) ↔ (((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)))))
81	48, 56, 80	3bitrd 213	. . 3 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ (((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ·N (𝑢 ·N 𝑤)) <N ((𝑢 ·N 𝑦) ·N ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)))))
82	38, 46, 81	3bitr4rd 220	. 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 )))
83	1, 5, 9, 13, 82	3ecoptocl 6598	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 973   = wceq 1348   ∈ wcel 2141  ⟨cop 3584   class class class wbr 3987  (class class class)co 5850  [cec 6507  Ncnpi 7221   +_N cpli 7222   ·_N cmi 7223   <_N clti 7224   ~_Q ceq 7228  Qcnq 7229   +_Q cplq 7231   <_Q cltq 7234

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-ltnqqs 7302

This theorem is referenced by:  ltanqi  7351  lt2addnq  7353  ltaddnq  7356  prarloclemlt  7442  prarloclemcalc  7451  addlocprlemgt  7483  addclpr  7486  prmuloclemcalc  7514  distrlem4prl  7533  distrlem4pru  7534  ltexprlemopl  7550  ltexprlemopu  7552  ltexprlemdisj  7555  ltexprlemloc  7556  ltexprlemfl  7558  ltexprlemfu  7560  aptiprleml  7588  aptiprlemu  7589  cauappcvgprlemopl  7595  cauappcvgprlemlol  7596  cauappcvgprlemdisj  7600  cauappcvgprlemloc  7601  cauappcvgprlemladdfu  7603  cauappcvgprlemladdru  7605  cauappcvgprlemladdrl  7606  cauappcvgprlem1  7608  caucvgprlemnkj  7615  caucvgprlemnbj  7616  caucvgprlemm  7617  caucvgprlemopl  7618  caucvgprlemlol  7619  caucvgprlemloc  7624  caucvgprlemladdfu  7626  caucvgprlemladdrl  7627  caucvgprprlemml  7643  caucvgprprlemopl  7646  caucvgprprlemlol  7647