Theorem prarloclemlt 7031

Description: Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7041. (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 6260	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ ω
2		nnacl 6223	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 2𝑜 ∈ ω) → (𝑦 +𝑜 2𝑜) ∈ ω)
3	1, 2	mpan2 416	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ ω)
4		nnaword1 6252	. . . . . . . . . . 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		1oex 6171	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ V
7	6	sucid 4235	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ suc 1𝑜
8		df-2o 6164	. . . . . . . . . . . . 13 ⊢ 2𝑜 = suc 1𝑜
9	7, 8	eleqtrri 2163	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ 2𝑜
10		nnaordi 6247	. . . . . . . . . . . . 13 ⊢ ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
11	1, 10	mpan 415	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
12	9, 11	mpi 15	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
13	12	adantr 270	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
14	5, 13	sseldd 3024	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
15	14	ancoms 264	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
16		1pi 6853	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ N
17		nnppipi 6881	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 1𝑜 ∈ N) → (𝑦 +𝑜 1𝑜) ∈ N)
18	16, 17	mpan2 416	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ N)
19	18	adantl 271	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ N)
20		o1p1e2 6211	. . . . . . . . . . . . . 14 ⊢ (1𝑜 +𝑜 1𝑜) = 2𝑜
21		1onn 6259	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 ∈ ω
22		nnppipi 6881	. . . . . . . . . . . . . . 15 ⊢ ((1𝑜 ∈ ω ∧ 1𝑜 ∈ N) → (1𝑜 +𝑜 1𝑜) ∈ N)
23	21, 16, 22	mp2an 417	. . . . . . . . . . . . . 14 ⊢ (1𝑜 +𝑜 1𝑜) ∈ N
24	20, 23	eqeltrri 2161	. . . . . . . . . . . . 13 ⊢ 2𝑜 ∈ N
25		nnppipi 6881	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ω ∧ 2𝑜 ∈ N) → (𝑦 +𝑜 2𝑜) ∈ N)
26	24, 25	mpan2 416	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ N)
27		pinn 6847	. . . . . . . . . . . 12 ⊢ ((𝑦 +𝑜 2𝑜) ∈ N → (𝑦 +𝑜 2𝑜) ∈ ω)
28	26, 27	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ ω)
29		nnacom 6227	. . . . . . . . . . 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 6881	. . . . . . . . . . 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 2165	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N)
34		ltpiord 6857	. . . . . . . . 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	19, 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	15, 35	mpbird 165	. . . . . . 7 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
37		mulidpi 6856	. . . . . . . . 9 ⊢ ((𝑦 +𝑜 1𝑜) ∈ N → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) = (𝑦 +𝑜 1𝑜))
38	19, 37	syl 14	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) = (𝑦 +𝑜 1𝑜))
39		mulcompig 6869	. . . . . . . . . 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, 16, 39	sylancl 404	. . . . . . . . 9 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
41		mulidpi 6856	. . . . . . . . . 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 2122	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
44	38, 43	breq12d 3850	. . . . . . 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 6912	. . . . . . . . . 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	16, 47	mpanl2 426	. . . . . . . . 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	16, 48	mpanr2 429	. . . . . . . 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	18, 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 461	. . . 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 4459	. . . . . . . . 9 ⊢ (((𝑦 +𝑜 1𝑜) ∈ N ∧ 1𝑜 ∈ N) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
55	19, 16, 54	sylancl 404	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
56		enqex 6898	. . . . . . . . 9 ⊢ ~Q ∈ V
57	56	ecelqsi 6326	. . . . . . . 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 6886	. . . . . . 7 ⊢ Q = ((N × N) / ~Q )
60	58, 59	syl6eleqr 2181	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q)
61	60	adantlr 461	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ Q)
62		opelxpi 4459	. . . . . . . . 9 ⊢ ((((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜 ∈ N) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
63	33, 16, 62	sylancl 404	. . . . . . . 8 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
64	56	ecelqsi 6326	. . . . . . . 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 2181	. . . . . 6 ⊢ ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q)
67	66	adantlr 461	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ Q)
68		simplr3 987	. . . . 5 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → 𝑃 ∈ Q)
69		ltmnqg 6939	. . . . 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 1174	. . . 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 6921	. . . . 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 6921	. . . . 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 3850	. . 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 6914	. . . 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 6914	. . . 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 985	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → ⟨𝐿, 𝑈⟩ ∈ P)
83		simplr2 986	. . . 4 ⊢ (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ 𝐿 ∧ 𝑃 ∈ Q)) ∧ 𝑦 ∈ ω) → 𝐴 ∈ 𝐿)
84		elprnql 7019	. . . 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 6938	. . 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 1174	. 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 924   = wceq 1289   ∈ wcel 1438   ⊆ wss 2997  ⟨cop 3444   class class class wbr 3837  suc csuc 4183  ωcom 4395   × cxp 4426  (class class class)co 5634  1_𝑜c1o 6156  2_𝑜c2o 6157   +_𝑜 coa 6160  [cec 6270   / cqs 6271  Ncnpi 6810   ·_N cmi 6812   <_N clti 6813   ~_Q ceq 6817  Qcnq 6818   +_Q cplq 6820   ·_Q cmq 6821   <_Q cltq 6823  Pcnp 6829

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-ltnqqs 6891  df-inp 7004

This theorem is referenced by:  prarloclem3step  7034