Theorem ltanqg 7398

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 7346	. 2 ⊢ Q = ((N × N) / ~Q )
2		breq1 4006	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑦⟩] ~Q <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ))
3		oveq2 5882	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~Q = 𝐴 → ([⟨𝑣, 𝑢⟩] ~Q +Q [⟨𝑥, 𝑦⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q +Q 𝐴))
4	3	breq1d 4013	. . 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 235	. 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 4007	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → (𝐴 <Q [⟨𝑧, 𝑤⟩] ~Q ↔ 𝐴 <Q 𝐵))
7		oveq2 5882	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~Q = 𝐵 → ([⟨𝑣, 𝑢⟩] ~Q +Q [⟨𝑧, 𝑤⟩] ~Q ) = ([⟨𝑣, 𝑢⟩] ~Q +Q 𝐵))
8	7	breq2d 4015	. . 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 235	. 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 5881	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q +Q 𝐴) = (𝐶 +Q 𝐴))
11		oveq1 5881	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~Q = 𝐶 → ([⟨𝑣, 𝑢⟩] ~Q +Q 𝐵) = (𝐶 +Q 𝐵))
12	10, 11	breq12d 4016	. . 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 232	. 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 7325	. . . . . 6 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 +N 𝑔) ∈ N)
15	14	adantl 277	. . . . 5 ⊢ ((((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) → (𝑓 +N 𝑔) ∈ N)
16		simp3l 1025	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑣 ∈ N)
17		simp1r 1022	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑦 ∈ N)
18		mulclpi 7326	. . . . . 6 ⊢ ((𝑣 ∈ N ∧ 𝑦 ∈ N) → (𝑣 ·N 𝑦) ∈ N)
19	16, 17, 18	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑦) ∈ N)
20		simp3r 1026	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑢 ∈ N)
21		simp1l 1021	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑥 ∈ N)
22		mulclpi 7326	. . . . . 6 ⊢ ((𝑢 ∈ N ∧ 𝑥 ∈ N) → (𝑢 ·N 𝑥) ∈ N)
23	20, 21, 22	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑥) ∈ N)
24	15, 19, 23	caovcld 6027	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑣 ·N 𝑦) +N (𝑢 ·N 𝑥)) ∈ N)
25		mulclpi 7326	. . . . 5 ⊢ ((𝑢 ∈ N ∧ 𝑦 ∈ N) → (𝑢 ·N 𝑦) ∈ N)
26	20, 17, 25	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑦) ∈ N)
27		mulclpi 7326	. . . . . . 7 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) ∈ N)
28	27	adantl 277	. . . . . 6 ⊢ ((((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝑓 ∈ N ∧ 𝑔 ∈ N)) → (𝑓 ·N 𝑔) ∈ N)
29		simp2r 1024	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑤 ∈ N)
30	28, 16, 29	caovcld 6027	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ·N 𝑤) ∈ N)
31		simp2l 1023	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → 𝑧 ∈ N)
32		mulclpi 7326	. . . . . 6 ⊢ ((𝑢 ∈ N ∧ 𝑧 ∈ N) → (𝑢 ·N 𝑧) ∈ N)
33	20, 31, 32	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑧) ∈ N)
34	15, 30, 33	caovcld 6027	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑣 ·N 𝑤) +N (𝑢 ·N 𝑧)) ∈ N)
35		mulclpi 7326	. . . . 5 ⊢ ((𝑢 ∈ N ∧ 𝑤 ∈ N) → (𝑢 ·N 𝑤) ∈ N)
36	20, 29, 35	syl2anc 411	. . . 4 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑤) ∈ N)
37		ordpipqqs 7372	. . . 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 1239	. . 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 999	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑣 ∈ N ∧ 𝑢 ∈ N))
40		simp1 997	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑥 ∈ N ∧ 𝑦 ∈ N))
41		addpipqqs 7368	. . . . 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 411	. . . 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 998	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑧 ∈ N ∧ 𝑤 ∈ N))
44		addpipqqs 7368	. . . . 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 411	. . . 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 4016	. . 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 7372	. . . . 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 1017	. . . 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 7326	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑤 ∈ N) → (𝑥 ·N 𝑤) ∈ N)
50	21, 29, 49	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑥 ·N 𝑤) ∈ N)
51		mulclpi 7326	. . . . . 6 ⊢ ((𝑦 ∈ N ∧ 𝑧 ∈ N) → (𝑦 ·N 𝑧) ∈ N)
52	17, 31, 51	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑦 ·N 𝑧) ∈ N)
53		mulclpi 7326	. . . . . 6 ⊢ ((𝑢 ∈ N ∧ 𝑢 ∈ N) → (𝑢 ·N 𝑢) ∈ N)
54	20, 20, 53	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑢 ·N 𝑢) ∈ N)
55		ltmpig 7337	. . . . 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 1238	. . . 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 7326	. . . . . . 7 ⊢ (((𝑢 ·N 𝑥) ∈ N ∧ (𝑢 ·N 𝑤) ∈ N) → ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)) ∈ N)
58	23, 36, 57	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑥) ·N (𝑢 ·N 𝑤)) ∈ N)
59		mulclpi 7326	. . . . . . 7 ⊢ (((𝑢 ·N 𝑦) ∈ N ∧ (𝑢 ·N 𝑧) ∈ N) → ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)) ∈ N)
60	26, 33, 59	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑢 ·N 𝑦) ·N (𝑢 ·N 𝑧)) ∈ N)
61		mulclpi 7326	. . . . . . 7 ⊢ (((𝑣 ·N 𝑦) ∈ N ∧ (𝑢 ·N 𝑤) ∈ N) → ((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) ∈ N)
62	19, 36, 61	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑣 ·N 𝑦) ·N (𝑢 ·N 𝑤)) ∈ N)
63		ltapig 7336	. . . . . 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 1238	. . . . 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 7329	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N) → (𝑓 ·N 𝑔) = (𝑔 ·N 𝑓))
66	65	adantl 277	. . . . . . 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 7330	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → ((𝑓 ·N 𝑔) ·N ℎ) = (𝑓 ·N (𝑔 ·N ℎ)))
68	67	adantl 277	. . . . . . 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 6058	. . . . . 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 6058	. . . . . 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 4016	. . . . 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 7331	. . . . . . . 8 ⊢ ((𝑓 ∈ N ∧ 𝑔 ∈ N ∧ ℎ ∈ N) → (𝑓 ·N (𝑔 +N ℎ)) = ((𝑓 ·N 𝑔) +N (𝑓 ·N ℎ)))
73	72	adantl 277	. . . . . . 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 6050	. . . . . 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 6049	. . . . . . 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 6059	. . . . . . . 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 5889	. . . . . . 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 2210	. . . . . 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 4016	. . . . 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 220	. . . 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 214	. . 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 221	. 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 6623	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ⟨cop 3595   class class class wbr 4003  (class class class)co 5874  [cec 6532  Ncnpi 7270   +_N cpli 7271   ·_N cmi 7272   <_N clti 7273   ~_Q ceq 7277  Qcnq 7278   +_Q cplq 7280   <_Q cltq 7283

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-ltnqqs 7351

This theorem is referenced by:  ltanqi  7400  lt2addnq  7402  ltaddnq  7405  prarloclemlt  7491  prarloclemcalc  7500  addlocprlemgt  7532  addclpr  7535  prmuloclemcalc  7563  distrlem4prl  7582  distrlem4pru  7583  ltexprlemopl  7599  ltexprlemopu  7601  ltexprlemdisj  7604  ltexprlemloc  7605  ltexprlemfl  7607  ltexprlemfu  7609  aptiprleml  7637  aptiprlemu  7638  cauappcvgprlemopl  7644  cauappcvgprlemlol  7645  cauappcvgprlemdisj  7649  cauappcvgprlemloc  7650  cauappcvgprlemladdfu  7652  cauappcvgprlemladdru  7654  cauappcvgprlemladdrl  7655  cauappcvgprlem1  7657  caucvgprlemnkj  7664  caucvgprlemnbj  7665  caucvgprlemm  7666  caucvgprlemopl  7667  caucvgprlemlol  7668  caucvgprlemloc  7673  caucvgprlemladdfu  7675  caucvgprlemladdrl  7676  caucvgprprlemml  7692  caucvgprprlemopl  7695  caucvgprprlemlol  7696