Theorem prarloclemlt 6683

Description: Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6693. (Contributed by Jim Kingdon, 10-Nov-2019.)

Assertion

Ref	Expression
prarloclemlt	⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → (𝐴 +Q ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)))

Proof of Theorem prarloclemlt

Step	Hyp	Ref	Expression
1		2onn 6117	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ ω
2		nnacl 6082	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 2𝑜 ∈ ω) → (𝑦 +𝑜 2𝑜) ∈ ω)
3	1, 2	mpan2 415	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ ω)
4		nnaword1 6109	. . . . . . . . . . 11 ⊢ (((𝑦 +𝑜 2𝑜) ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 2𝑜) ⊆ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
5	3, 4	sylan 277	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 2𝑜) ⊆ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
6		1onn 6116	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 ∈ ω
7	6	elexi 2611	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ V
8	7	sucid 4172	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ suc 1𝑜
9		df-2o 6025	. . . . . . . . . . . . 13 ⊢ 2𝑜 = suc 1𝑜
10	8, 9	eleqtrri 2154	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ 2𝑜
11		nnaordi 6104	. . . . . . . . . . . . 13 ⊢ ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
12	1, 11	mpan 414	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
13	10, 12	mpi 15	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
14	13	adantr 270	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
15	5, 14	sseldd 3000	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
16	15	ancoms 264	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
17		1pi 6505	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ N
18		nnppipi 6533	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 1𝑜 ∈ N) → (𝑦 +𝑜 1𝑜) ∈ N)
19	17, 18	mpan2 415	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ N)
20	19	adantl 271	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ N)
21		o1p1e2 6071	. . . . . . . . . . . . . 14 ⊢ (1𝑜 +𝑜 1𝑜) = 2𝑜
22		nnppipi 6533	. . . . . . . . . . . . . . 15 ⊢ ((1𝑜 ∈ ω ∧ 1𝑜 ∈ N) → (1𝑜 +𝑜 1𝑜) ∈ N)
23	6, 17, 22	mp2an 416	. . . . . . . . . . . . . 14 ⊢ (1𝑜 +𝑜 1𝑜) ∈ N
24	21, 23	eqeltrri 2152	. . . . . . . . . . . . 13 ⊢ 2𝑜 ∈ N
25		nnppipi 6533	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 2𝑜 ∈ N) → (𝑦 +𝑜 2𝑜) ∈ N)
26	24, 25	mpan2 415	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ N)
27		pinn 6499	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 2𝑜) ∈ N → (𝑦 +𝑜 2𝑜) ∈ ω)
28	26, 27	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ ω)
29		nnacom 6086	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ω ∧ (𝑦 +𝑜 2𝑜) ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
30	28, 29	sylan2 280	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
31		nnppipi 6533	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ω ∧ (𝑦 +𝑜 2𝑜) ∈ N) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) ∈ N)
32	26, 31	sylan2 280	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) ∈ N)
33	30, 32	eqeltrrd 2156	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N)
34		ltpiord 6509	. . . . . . . . 9 ⊢ (((𝑦 +𝑜 1𝑜) ∈ N ∧ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N) → ((𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ↔ (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
35	20, 33, 34	syl2anc 403	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ↔ (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
36	16, 35	mpbird 165	. . . . . . 7 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
37		mulidpi 6508	. . . . . . . . 9 ⊢ ((𝑦 +𝑜 1𝑜) ∈ N → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) = (𝑦 +𝑜 1𝑜))
38	20, 37	syl 14	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) = (𝑦 +𝑜 1𝑜))
39		mulcompig 6521	. . . . . . . . . 10 ⊢ ((((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜 ∈ N) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
40	33, 17, 39	sylancl 404	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
41		mulidpi 6508	. . . . . . . . . 10 ⊢ (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
42	33, 41	syl 14	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
43	40, 42	eqtr3d 2115	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
44	38, 43	breq12d 3798	. . . . . . 7 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)) ↔ (𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
45	36, 44	mpbird 165	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
46		simpr 108	. . . . . . 7 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
47		ordpipqqs 6564	. . . . . . . . . 10 ⊢ ((((𝑦 +𝑜 1𝑜) ∈ N ∧ 1𝑜 ∈ N) ∧ (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜 ∈ N)) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ↔ ((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))))
48	17, 47	mpanl2 425	. . . . . . . . 9 ⊢ (((𝑦 +𝑜 1𝑜) ∈ N ∧ (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜 ∈ N)) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ↔ ((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))))
49	17, 48	mpanr2 428	. . . . . . . 8 ⊢ (((𝑦 +𝑜 1𝑜) ∈ N ∧ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ↔ ((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))))
50	19, 49	sylan 277	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ↔ ((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))))
51	46, 33, 50	syl2anc 403	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ↔ ((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))))
52	45, 51	mpbird 165	. . . . 5 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )
53	52	adantlr 460	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )
54		opelxpi 4394	. . . . . . . . 9 ⊢ (((𝑦 +𝑜 1𝑜) ∈ N ∧ 1𝑜 ∈ N) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
55	20, 17, 54	sylancl 404	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
56		enqex 6550	. . . . . . . . 9 ⊢ ~Q ∈ V
57	56	ecelqsi 6183	. . . . . . . 8 ⊢ (⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
58	55, 57	syl 14	. . . . . . 7 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
59		df-nqqs 6538	. . . . . . 7 ⊢ Q = ((N × N) / ~Q )
60	58, 59	syl6eleqr 2172	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q)
61	60	adantlr 460	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q)
62		opelxpi 4394	. . . . . . . . 9 ⊢ ((((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜 ∈ N) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
63	33, 17, 62	sylancl 404	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
64	56	ecelqsi 6183	. . . . . . . 8 ⊢ (⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
65	63, 64	syl 14	. . . . . . 7 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
66	65, 59	syl6eleqr 2172	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q)
67	66	adantlr 460	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q)
68		simplr3 982	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → 𝑃 ∈ Q)
69		ltmnqg 6591	. . . . 5 ⊢ (([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q ∧ [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q ∧ 𝑃 ∈ Q) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ↔ (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) <Q (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )))
70	61, 67, 68, 69	syl3anc 1169	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ↔ (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) <Q (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )))
71	53, 70	mpbid 145	. . 3 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) <Q (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ))
72		mulcomnqg 6573	. . . . 5 ⊢ ((𝑃 ∈ Q ∧ [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q) → (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃))
73	68, 61, 72	syl2anc 403	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃))
74		mulcomnqg 6573	. . . . 5 ⊢ ((𝑃 ∈ Q ∧ [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q) → (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ) = ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃))
75	68, 67, 74	syl2anc 403	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ) = ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃))
76	73, 75	breq12d 3798	. . 3 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ((𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) <Q (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ) ↔ ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) <Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)))
77	71, 76	mpbid 145	. 2 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) <Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃))
78		mulclnq 6566	. . . 4 ⊢ (([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q ∧ 𝑃 ∈ Q) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
79	61, 68, 78	syl2anc 403	. . 3 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
80		mulclnq 6566	. . . 4 ⊢ (([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q ∧ 𝑃 ∈ Q) → ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
81	67, 68, 80	syl2anc 403	. . 3 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
82		simplr1 980	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ⟨𝐿, 𝑈⟩ ∈ P)
83		simplr2 981	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝐿)
84		elprnql 6671	. . . 4 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿) → 𝐴 ∈ Q)
85	82, 83, 84	syl2anc 403	. . 3 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → 𝐴 ∈ Q)
86		ltanqg 6590	. . 3 ⊢ ((([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q ∧ ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q ∧ 𝐴 ∈ Q) → (([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) <Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ↔ (𝐴 +Q ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃))))
87	79, 81, 85, 86	syl3anc 1169	. 2 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → (([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) <Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ↔ (𝐴 +Q ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃))))
88	77, 87	mpbid 145	1 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → (𝐴 +Q ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∈ wcel 1433   ⊆ wss 2973  ⟨cop 3401   class class class wbr 3785  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 cmi 6464   <_N clti 6465   ~_Q ceq 6469  Qcnq 6470   +_Q cplq 6472   ·_Q cmq 6473   <_Q cltq 6475  Pcnp 6481

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-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-ltnqqs 6543  df-inp 6656

This theorem is referenced by:  prarloclem3step  6686