Theorem prarloclemarch2 6609

Description: Like prarloclemarch 6608 but the integer must be at least two, and there is also 𝐵 added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 6693. (Contributed by Jim Kingdon, 25-Nov-2019.)

Assertion

Ref	Expression
prarloclemarch2	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑥 ∈ N (1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem prarloclemarch2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prarloclemarch 6608	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑧 ∈ N 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))
2	1	3adant2 957	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑧 ∈ N 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))
3		pinn 6499	. . . . . . . 8 ⊢ (𝑧 ∈ N → 𝑧 ∈ ω)
4		1pi 6505	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ N
5	4	elexi 2611	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ V
6	5	sucid 4172	. . . . . . . . . 10 ⊢ 1𝑜 ∈ suc 1𝑜
7		df-2o 6025	. . . . . . . . . 10 ⊢ 2𝑜 = suc 1𝑜
8	6, 7	eleqtrri 2154	. . . . . . . . 9 ⊢ 1𝑜 ∈ 2𝑜
9		2onn 6117	. . . . . . . . . . 11 ⊢ 2𝑜 ∈ ω
10		nnaword2 6110	. . . . . . . . . . 11 ⊢ ((2𝑜 ∈ ω ∧ 𝑧 ∈ ω) → 2𝑜 ⊆ (𝑧 +𝑜 2𝑜))
11	9, 10	mpan 414	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → 2𝑜 ⊆ (𝑧 +𝑜 2𝑜))
12	11	sseld 2998	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → (1𝑜 ∈ 2𝑜 → 1𝑜 ∈ (𝑧 +𝑜 2𝑜)))
13	8, 12	mpi 15	. . . . . . . 8 ⊢ (𝑧 ∈ ω → 1𝑜 ∈ (𝑧 +𝑜 2𝑜))
14	3, 13	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ N → 1𝑜 ∈ (𝑧 +𝑜 2𝑜))
15		o1p1e2 6071	. . . . . . . . 9 ⊢ (1𝑜 +𝑜 1𝑜) = 2𝑜
16		addpiord 6506	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜))
17	4, 4, 16	mp2an 416	. . . . . . . . . 10 ⊢ (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜)
18		addclpi 6517	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) ∈ N)
19	4, 4, 18	mp2an 416	. . . . . . . . . 10 ⊢ (1𝑜 +N 1𝑜) ∈ N
20	17, 19	eqeltrri 2152	. . . . . . . . 9 ⊢ (1𝑜 +𝑜 1𝑜) ∈ N
21	15, 20	eqeltrri 2152	. . . . . . . 8 ⊢ 2𝑜 ∈ N
22		addpiord 6506	. . . . . . . 8 ⊢ ((𝑧 ∈ N ∧ 2𝑜 ∈ N) → (𝑧 +N 2𝑜) = (𝑧 +𝑜 2𝑜))
23	21, 22	mpan2 415	. . . . . . 7 ⊢ (𝑧 ∈ N → (𝑧 +N 2𝑜) = (𝑧 +𝑜 2𝑜))
24	14, 23	eleqtrrd 2158	. . . . . 6 ⊢ (𝑧 ∈ N → 1𝑜 ∈ (𝑧 +N 2𝑜))
25		addclpi 6517	. . . . . . . 8 ⊢ ((𝑧 ∈ N ∧ 2𝑜 ∈ N) → (𝑧 +N 2𝑜) ∈ N)
26	21, 25	mpan2 415	. . . . . . 7 ⊢ (𝑧 ∈ N → (𝑧 +N 2𝑜) ∈ N)
27		ltpiord 6509	. . . . . . . 8 ⊢ ((1𝑜 ∈ N ∧ (𝑧 +N 2𝑜) ∈ N) → (1𝑜 <N (𝑧 +N 2𝑜) ↔ 1𝑜 ∈ (𝑧 +N 2𝑜)))
28	4, 27	mpan 414	. . . . . . 7 ⊢ ((𝑧 +N 2𝑜) ∈ N → (1𝑜 <N (𝑧 +N 2𝑜) ↔ 1𝑜 ∈ (𝑧 +N 2𝑜)))
29	26, 28	syl 14	. . . . . 6 ⊢ (𝑧 ∈ N → (1𝑜 <N (𝑧 +N 2𝑜) ↔ 1𝑜 ∈ (𝑧 +N 2𝑜)))
30	24, 29	mpbird 165	. . . . 5 ⊢ (𝑧 ∈ N → 1𝑜 <N (𝑧 +N 2𝑜))
31	30	adantl 271	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → 1𝑜 <N (𝑧 +N 2𝑜))
32	31	adantrr 462	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → 1𝑜 <N (𝑧 +N 2𝑜))
33		nna0 6076	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ω → (𝑧 +𝑜 ∅) = 𝑧)
34		0lt1o 6046	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ 1𝑜
35		1on 6031	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1𝑜 ∈ On
36	35	onsuci 4260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ suc 1𝑜 ∈ On
37		ontr1 4144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (suc 1𝑜 ∈ On → ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ suc 1𝑜) → ∅ ∈ suc 1𝑜))
38	36, 37	ax-mp 7	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ suc 1𝑜) → ∅ ∈ suc 1𝑜)
39	34, 6, 38	mp2an 416	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ suc 1𝑜
40	39, 7	eleqtrri 2154	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 2𝑜
41		nnaordi 6104	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2𝑜 ∈ ω ∧ 𝑧 ∈ ω) → (∅ ∈ 2𝑜 → (𝑧 +𝑜 ∅) ∈ (𝑧 +𝑜 2𝑜)))
42	9, 41	mpan 414	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ω → (∅ ∈ 2𝑜 → (𝑧 +𝑜 ∅) ∈ (𝑧 +𝑜 2𝑜)))
43	40, 42	mpi 15	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ω → (𝑧 +𝑜 ∅) ∈ (𝑧 +𝑜 2𝑜))
44	33, 43	eqeltrrd 2156	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ (𝑧 +𝑜 2𝑜))
45	3, 44	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ N → 𝑧 ∈ (𝑧 +𝑜 2𝑜))
46	45, 23	eleqtrrd 2158	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ N → 𝑧 ∈ (𝑧 +N 2𝑜))
47		ltpiord 6509	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ N ∧ (𝑧 +N 2𝑜) ∈ N) → (𝑧 <N (𝑧 +N 2𝑜) ↔ 𝑧 ∈ (𝑧 +N 2𝑜)))
48	26, 47	mpdan 412	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ N → (𝑧 <N (𝑧 +N 2𝑜) ↔ 𝑧 ∈ (𝑧 +N 2𝑜)))
49	46, 48	mpbird 165	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ N → 𝑧 <N (𝑧 +N 2𝑜))
50		mulidpi 6508	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ N → (𝑧 ·N 1𝑜) = 𝑧)
51		mulcompig 6521	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 +N 2𝑜) ∈ N ∧ 1𝑜 ∈ N) → ((𝑧 +N 2𝑜) ·N 1𝑜) = (1𝑜 ·N (𝑧 +N 2𝑜)))
52	4, 51	mpan2 415	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 +N 2𝑜) ∈ N → ((𝑧 +N 2𝑜) ·N 1𝑜) = (1𝑜 ·N (𝑧 +N 2𝑜)))
53	26, 52	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ N → ((𝑧 +N 2𝑜) ·N 1𝑜) = (1𝑜 ·N (𝑧 +N 2𝑜)))
54		mulidpi 6508	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 +N 2𝑜) ∈ N → ((𝑧 +N 2𝑜) ·N 1𝑜) = (𝑧 +N 2𝑜))
55	26, 54	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ N → ((𝑧 +N 2𝑜) ·N 1𝑜) = (𝑧 +N 2𝑜))
56	53, 55	eqtr3d 2115	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ N → (1𝑜 ·N (𝑧 +N 2𝑜)) = (𝑧 +N 2𝑜))
57	49, 50, 56	3brtr4d 3815	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ N → (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜)))
58		ordpipqqs 6564	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ N ∧ 1𝑜 ∈ N) ∧ ((𝑧 +N 2𝑜) ∈ N ∧ 1𝑜 ∈ N)) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜))))
59	4, 58	mpanl2 425	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ N ∧ ((𝑧 +N 2𝑜) ∈ N ∧ 1𝑜 ∈ N)) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜))))
60	4, 59	mpanr2 428	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ N ∧ (𝑧 +N 2𝑜) ∈ N) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜))))
61	26, 60	mpdan 412	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ N → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜))))
62	57, 61	mpbird 165	. . . . . . . . . . 11 ⊢ (𝑧 ∈ N → [⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )
63	62	adantl 271	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Q ∧ 𝑧 ∈ N) → [⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )
64		opelxpi 4394	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 +N 2𝑜) ∈ N ∧ 1𝑜 ∈ N) → ⟨(𝑧 +N 2𝑜), 1𝑜⟩ ∈ (N × N))
65	4, 64	mpan2 415	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 +N 2𝑜) ∈ N → ⟨(𝑧 +N 2𝑜), 1𝑜⟩ ∈ (N × N))
66		enqex 6550	. . . . . . . . . . . . . . . 16 ⊢ ~Q ∈ V
67	66	ecelqsi 6183	. . . . . . . . . . . . . . 15 ⊢ (⟨(𝑧 +N 2𝑜), 1𝑜⟩ ∈ (N × N) → [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
68	26, 65, 67	3syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ N → [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
69		df-nqqs 6538	. . . . . . . . . . . . . 14 ⊢ Q = ((N × N) / ~Q )
70	68, 69	syl6eleqr 2172	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ N → [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ Q)
71		opelxpi 4394	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ N ∧ 1𝑜 ∈ N) → ⟨𝑧, 1𝑜⟩ ∈ (N × N))
72	4, 71	mpan2 415	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ N → ⟨𝑧, 1𝑜⟩ ∈ (N × N))
73	66	ecelqsi 6183	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑧, 1𝑜⟩ ∈ (N × N) → [⟨𝑧, 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
74	72, 73	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ N → [⟨𝑧, 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
75	74, 69	syl6eleqr 2172	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ N → [⟨𝑧, 1𝑜⟩] ~Q ∈ Q)
76		ltmnqg 6591	. . . . . . . . . . . . . 14 ⊢ (([⟨𝑧, 1𝑜⟩] ~Q ∈ Q ∧ [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ Q ∧ 𝐶 ∈ Q) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
77	75, 76	syl3an1 1202	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ N ∧ [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ Q ∧ 𝐶 ∈ Q) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
78	70, 77	syl3an2 1203	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ N ∧ 𝑧 ∈ N ∧ 𝐶 ∈ Q) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
79	78	3anidm12 1226	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝐶 ∈ Q) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
80	79	ancoms 264	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Q ∧ 𝑧 ∈ N) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
81	63, 80	mpbid 145	. . . . . . . . 9 ⊢ ((𝐶 ∈ Q ∧ 𝑧 ∈ N) → (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ))
82		mulcomnqg 6573	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Q ∧ [⟨𝑧, 1𝑜⟩] ~Q ∈ Q) → (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) = ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))
83	75, 82	sylan2 280	. . . . . . . . 9 ⊢ ((𝐶 ∈ Q ∧ 𝑧 ∈ N) → (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) = ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))
84		mulcomnqg 6573	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Q ∧ [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ Q) → (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
85	70, 84	sylan2 280	. . . . . . . . 9 ⊢ ((𝐶 ∈ Q ∧ 𝑧 ∈ N) → (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
86	81, 83, 85	3brtr3d 3814	. . . . . . . 8 ⊢ ((𝐶 ∈ Q ∧ 𝑧 ∈ N) → ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
87	86	3ad2antl3 1102	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
88	87	adantrr 462	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
89		ltsonq 6588	. . . . . . . . . 10 ⊢ <Q Or Q
90		ltrelnq 6555	. . . . . . . . . 10 ⊢ <Q ⊆ (Q × Q)
91	89, 90	sotri 4740	. . . . . . . . 9 ⊢ ((𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) ∧ ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
92	91	ex 113	. . . . . . . 8 ⊢ (𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) → (([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
93	92	adantl 271	. . . . . . 7 ⊢ ((𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶)) → (([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
94	93	adantl 271	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → (([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
95	88, 94	mpd 13	. . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
96		mulclnq 6566	. . . . . . . . . 10 ⊢ (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ Q ∧ 𝐶 ∈ Q) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q)
97	70, 96	sylan 277	. . . . . . . . 9 ⊢ ((𝑧 ∈ N ∧ 𝐶 ∈ Q) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q)
98	97	ancoms 264	. . . . . . . 8 ⊢ ((𝐶 ∈ Q ∧ 𝑧 ∈ N) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q)
99	98	3ad2antl3 1102	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q)
100		simpl2 942	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → 𝐵 ∈ Q)
101		ltaddnq 6597	. . . . . . 7 ⊢ ((([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q ∧ 𝐵 ∈ Q) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
102	99, 100, 101	syl2anc 403	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
103	102	adantrr 462	. . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
104	89, 90	sotri 4740	. . . . 5 ⊢ ((𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∧ ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵)) → 𝐴 <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
105	95, 103, 104	syl2anc 403	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → 𝐴 <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
106		addcomnqg 6571	. . . . . . 7 ⊢ ((([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q ∧ 𝐵 ∈ Q) → (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵) = (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
107	99, 100, 106	syl2anc 403	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵) = (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
108	107	breq2d 3797	. . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → (𝐴 <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵) ↔ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))))
109	108	adantrr 462	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → (𝐴 <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵) ↔ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))))
110	105, 109	mpbid 145	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
111		simpr 108	. . . . 5 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → 𝑧 ∈ N)
112		breq2 3789	. . . . . . . 8 ⊢ (𝑥 = (𝑧 +N 2𝑜) → (1𝑜 <N 𝑥 ↔ 1𝑜 <N (𝑧 +N 2𝑜)))
113		opeq1 3570	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑧 +N 2𝑜) → ⟨𝑥, 1𝑜⟩ = ⟨(𝑧 +N 2𝑜), 1𝑜⟩)
114	113	eceq1d 6165	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 +N 2𝑜) → [⟨𝑥, 1𝑜⟩] ~Q = [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )
115	114	oveq1d 5547	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 +N 2𝑜) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶) = ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
116	115	oveq2d 5548	. . . . . . . . 9 ⊢ (𝑥 = (𝑧 +N 2𝑜) → (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)) = (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
117	116	breq2d 3797	. . . . . . . 8 ⊢ (𝑥 = (𝑧 +N 2𝑜) → (𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)) ↔ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))))
118	112, 117	anbi12d 456	. . . . . . 7 ⊢ (𝑥 = (𝑧 +N 2𝑜) → ((1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))) ↔ (1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))))
119	118	rspcev 2701	. . . . . 6 ⊢ (((𝑧 +N 2𝑜) ∈ N ∧ (1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))) → ∃𝑥 ∈ N (1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))
120	119	ex 113	. . . . 5 ⊢ ((𝑧 +N 2𝑜) ∈ N → ((1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))) → ∃𝑥 ∈ N (1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)))))
121	111, 26, 120	3syl 17	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ 𝑧 ∈ N) → ((1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))) → ∃𝑥 ∈ N (1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)))))
122	121	adantrr 462	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → ((1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))) → ∃𝑥 ∈ N (1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)))))
123	32, 110, 122	mp2and 423	. 2 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑧 ∈ N ∧ 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → ∃𝑥 ∈ N (1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))
124	2, 123	rexlimddv 2481	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑥 ∈ N (1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∃wrex 2349   ⊆ wss 2973  ∅c0 3251  ⟨cop 3401   class class class wbr 3785  Oncon0 4118  suc csuc 4120  ωcom 4331   × cxp 4361  (class class class)co 5532  1_𝑜c1o 6017  2_𝑜c2o 6018   +_𝑜 coa 6021  [cec 6127   / cqs 6128  Ncnpi 6462   +_N cpli 6463   ·_N cmi 6464   <_N clti 6465   ~_Q ceq 6469  Qcnq 6470   +_Q cplq 6472   ·_Q cmq 6473   <_Q cltq 6475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543

This theorem is referenced by:  prarloc  6693