Theorem prarloclemlt 6591

Description: Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (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 6094	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ ω
2		nnacl 6059	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 2𝑜 ∈ ω) → (𝑦 +𝑜 2𝑜) ∈ ω)
3	1, 2	mpan2 401	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ ω)
4		nnaword1 6086	. . . . . . . . . . 11 ⊢ (((𝑦 +𝑜 2𝑜) ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 2𝑜) ⊆ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
5	3, 4	sylan 267	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 2𝑜) ⊆ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
6		1onn 6093	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 ∈ ω
7	6	elexi 2567	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ V
8	7	sucid 4154	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ suc 1𝑜
9		df-2o 6002	. . . . . . . . . . . . 13 ⊢ 2𝑜 = suc 1𝑜
10	8, 9	eleqtrri 2113	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ 2𝑜
11		nnaordi 6081	. . . . . . . . . . . . 13 ⊢ ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
12	1, 11	mpan 400	. . . . . . . . . . . 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 261	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
15	5, 14	sseldd 2946	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
16	15	ancoms 255	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
17		1pi 6413	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ N
18		nnppipi 6441	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 1𝑜 ∈ N) → (𝑦 +𝑜 1𝑜) ∈ N)
19	17, 18	mpan2 401	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ N)
20	19	adantl 262	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ N)
21		o1p1e2 6048	. . . . . . . . . . . . . 14 ⊢ (1𝑜 +𝑜 1𝑜) = 2𝑜
22		nnppipi 6441	. . . . . . . . . . . . . . 15 ⊢ ((1𝑜 ∈ ω ∧ 1𝑜 ∈ N) → (1𝑜 +𝑜 1𝑜) ∈ N)
23	6, 17, 22	mp2an 402	. . . . . . . . . . . . . 14 ⊢ (1𝑜 +𝑜 1𝑜) ∈ N
24	21, 23	eqeltrri 2111	. . . . . . . . . . . . 13 ⊢ 2𝑜 ∈ N
25		nnppipi 6441	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 2𝑜 ∈ N) → (𝑦 +𝑜 2𝑜) ∈ N)
26	24, 25	mpan2 401	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ N)
27		pinn 6407	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 2𝑜) ∈ N → (𝑦 +𝑜 2𝑜) ∈ ω)
28	26, 27	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ ω)
29		nnacom 6063	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ω ∧ (𝑦 +𝑜 2𝑜) ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
30	28, 29	sylan2 270	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
31		nnppipi 6441	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ω ∧ (𝑦 +𝑜 2𝑜) ∈ N) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) ∈ N)
32	26, 31	sylan2 270	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) ∈ N)
33	30, 32	eqeltrrd 2115	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N)
34		ltpiord 6417	. . . . . . . . 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 391	. . . . . . . 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 156	. . . . . . 7 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
37		mulidpi 6416	. . . . . . . . 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 6429	. . . . . . . . . 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 392	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
41		mulidpi 6416	. . . . . . . . . 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 2074	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
44	38, 43	breq12d 3777	. . . . . . 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 156	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
46		simpr 103	. . . . . . 7 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
47		ordpipqqs 6472	. . . . . . . . . 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 411	. . . . . . . . 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 414	. . . . . . . 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 267	. . . . . . 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 391	. . . . . 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 156	. . . . 5 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )
53	52	adantlr 446	. . . 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 4376	. . . . . . . . 9 ⊢ (((𝑦 +𝑜 1𝑜) ∈ N ∧ 1𝑜 ∈ N) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
55	20, 17, 54	sylancl 392	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
56		enqex 6458	. . . . . . . . 9 ⊢ ~Q ∈ V
57	56	ecelqsi 6160	. . . . . . . 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 6446	. . . . . . 7 ⊢ Q = ((N × N) / ~Q )
60	58, 59	syl6eleqr 2131	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q)
61	60	adantlr 446	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q)
62		opelxpi 4376	. . . . . . . . 9 ⊢ ((((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜 ∈ N) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
63	33, 17, 62	sylancl 392	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
64	56	ecelqsi 6160	. . . . . . . 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 2131	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q)
67	66	adantlr 446	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q)
68		simplr3 948	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → 𝑃 ∈ Q)
69		ltmnqg 6499	. . . . 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 1135	. . . 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 135	. . 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 6481	. . . . 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 391	. . . 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 6481	. . . . 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 391	. . . 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 3777	. . 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 135	. 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 6474	. . . 4 ⊢ (([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q ∧ 𝑃 ∈ Q) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
79	61, 68, 78	syl2anc 391	. . 3 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
80		mulclnq 6474	. . . 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 391	. . 3 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
82		simplr1 946	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ⟨𝐿, 𝑈⟩ ∈ P)
83		simplr2 947	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝐿)
84		elprnql 6579	. . . 4 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿) → 𝐴 ∈ Q)
85	82, 83, 84	syl2anc 391	. . 3 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → 𝐴 ∈ Q)
86		ltanqg 6498	. . 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 1135	. 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 135	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 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  suc csuc 4102  ωcom 4313   × cxp 4343  (class class class)co 5512  1_𝑜c1o 5994  2_𝑜c2o 5995   +_𝑜 coa 5998  [cec 6104   / cqs 6105  Ncnpi 6370   ·_N cmi 6372   <_N clti 6373   ~_Q ceq 6377  Qcnq 6378   +_Q cplq 6380   ·_Q cmq 6381   <_Q cltq 6383  Pcnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-ltnqqs 6451  df-inp 6564

This theorem is referenced by:  prarloclem3step  6594