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Theorem prarloclemlt 6648
Description: Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6658. (Contributed by Jim Kingdon, 10-Nov-2019.)
Assertion
Ref Expression
prarloclemlt (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 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
StepHypRef Expression
1 2onn 6124 . . . . . . . . . . . 12 2𝑜 ∈ ω
2 nnacl 6089 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 2𝑜 ∈ ω) → (𝑦 +𝑜 2𝑜) ∈ ω)
31, 2mpan2 409 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ ω)
4 nnaword1 6116 . . . . . . . . . . 11 (((𝑦 +𝑜 2𝑜) ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 2𝑜) ⊆ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
53, 4sylan 271 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 2𝑜) ⊆ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
6 1onn 6123 . . . . . . . . . . . . . . 15 1𝑜 ∈ ω
76elexi 2584 . . . . . . . . . . . . . 14 1𝑜 ∈ V
87sucid 4181 . . . . . . . . . . . . 13 1𝑜 ∈ suc 1𝑜
9 df-2o 6032 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
108, 9eleqtrri 2129 . . . . . . . . . . . 12 1𝑜 ∈ 2𝑜
11 nnaordi 6111 . . . . . . . . . . . . 13 ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
121, 11mpan 408 . . . . . . . . . . . 12 (𝑦 ∈ ω → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
1310, 12mpi 15 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
1413adantr 265 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
155, 14sseldd 2973 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
1615ancoms 259 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
17 1pi 6470 . . . . . . . . . . 11 1𝑜N
18 nnppipi 6498 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 1𝑜N) → (𝑦 +𝑜 1𝑜) ∈ N)
1917, 18mpan2 409 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ N)
2019adantl 266 . . . . . . . . 9 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ N)
21 o1p1e2 6078 . . . . . . . . . . . . . 14 (1𝑜 +𝑜 1𝑜) = 2𝑜
22 nnppipi 6498 . . . . . . . . . . . . . . 15 ((1𝑜 ∈ ω ∧ 1𝑜N) → (1𝑜 +𝑜 1𝑜) ∈ N)
236, 17, 22mp2an 410 . . . . . . . . . . . . . 14 (1𝑜 +𝑜 1𝑜) ∈ N
2421, 23eqeltrri 2127 . . . . . . . . . . . . 13 2𝑜N
25 nnppipi 6498 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 2𝑜N) → (𝑦 +𝑜 2𝑜) ∈ N)
2624, 25mpan2 409 . . . . . . . . . . . 12 (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ N)
27 pinn 6464 . . . . . . . . . . . 12 ((𝑦 +𝑜 2𝑜) ∈ N → (𝑦 +𝑜 2𝑜) ∈ ω)
2826, 27syl 14 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ ω)
29 nnacom 6093 . . . . . . . . . . 11 ((𝑋 ∈ ω ∧ (𝑦 +𝑜 2𝑜) ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
3028, 29sylan2 274 . . . . . . . . . 10 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
31 nnppipi 6498 . . . . . . . . . . 11 ((𝑋 ∈ ω ∧ (𝑦 +𝑜 2𝑜) ∈ N) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) ∈ N)
3226, 31sylan2 274 . . . . . . . . . 10 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) ∈ N)
3330, 32eqeltrrd 2131 . . . . . . . . 9 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N)
34 ltpiord 6474 . . . . . . . . 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)))
3520, 33, 34syl2anc 397 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ↔ (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
3616, 35mpbird 160 . . . . . . 7 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
37 mulidpi 6473 . . . . . . . . 9 ((𝑦 +𝑜 1𝑜) ∈ N → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) = (𝑦 +𝑜 1𝑜))
3820, 37syl 14 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) = (𝑦 +𝑜 1𝑜))
39 mulcompig 6486 . . . . . . . . . 10 ((((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜N) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
4033, 17, 39sylancl 398 . . . . . . . . 9 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
41 mulidpi 6473 . . . . . . . . . 10 (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
4233, 41syl 14 . . . . . . . . 9 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
4340, 42eqtr3d 2090 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
4438, 43breq12d 3804 . . . . . . 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)))
4536, 44mpbird 160 . . . . . 6 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) <N (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
46 simpr 107 . . . . . . 7 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ∈ ω)
47 ordpipqqs 6529 . . . . . . . . . 10 ((((𝑦 +𝑜 1𝑜) ∈ N ∧ 1𝑜N) ∧ (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜N)) → ([⟨(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))))
4817, 47mpanl2 419 . . . . . . . . 9 (((𝑦 +𝑜 1𝑜) ∈ N ∧ (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜N)) → ([⟨(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))))
4917, 48mpanr2 422 . . . . . . . 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))))
5019, 49sylan 271 . . . . . . 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))))
5146, 33, 50syl2anc 397 . . . . . 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))))
5245, 51mpbird 160 . . . . 5 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )
5352adantlr 454 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )
54 opelxpi 4403 . . . . . . . . 9 (((𝑦 +𝑜 1𝑜) ∈ N ∧ 1𝑜N) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
5520, 17, 54sylancl 398 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
56 enqex 6515 . . . . . . . . 9 ~Q ∈ V
5756ecelqsi 6190 . . . . . . . 8 (⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
5855, 57syl 14 . . . . . . 7 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
59 df-nqqs 6503 . . . . . . 7 Q = ((N × N) / ~Q )
6058, 59syl6eleqr 2147 . . . . . 6 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ)
6160adantlr 454 . . . . 5 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ)
62 opelxpi 4403 . . . . . . . . 9 ((((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜N) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
6333, 17, 62sylancl 398 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
6456ecelqsi 6190 . . . . . . . 8 (⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
6563, 64syl 14 . . . . . . 7 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
6665, 59syl6eleqr 2147 . . . . . 6 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~QQ)
6766adantlr 454 . . . . 5 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~QQ)
68 simplr3 959 . . . . 5 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → 𝑃Q)
69 ltmnqg 6556 . . . . 5 (([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ ∧ [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~QQ𝑃Q) → ([⟨(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 )))
7061, 67, 68, 69syl3anc 1146 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 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 )))
7153, 70mpbid 139 . . 3 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) <Q (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ))
72 mulcomnqg 6538 . . . . 5 ((𝑃Q ∧ [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ) → (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃))
7368, 61, 72syl2anc 397 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃))
74 mulcomnqg 6538 . . . . 5 ((𝑃Q ∧ [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~QQ) → (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ) = ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃))
7568, 67, 74syl2anc 397 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → (𝑃 ·Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ) = ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃))
7673, 75breq12d 3804 . . 3 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 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)))
7771, 76mpbid 139 . 2 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) <Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃))
78 mulclnq 6531 . . . 4 (([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ𝑃Q) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
7961, 68, 78syl2anc 397 . . 3 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
80 mulclnq 6531 . . . 4 (([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~QQ𝑃Q) → ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
8167, 68, 80syl2anc 397 . . 3 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
82 simplr1 957 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ⟨𝐿, 𝑈⟩ ∈ P)
83 simplr2 958 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → 𝐴𝐿)
84 elprnql 6636 . . . 4 ((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) → 𝐴Q)
8582, 83, 84syl2anc 397 . . 3 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → 𝐴Q)
86 ltanqg 6555 . . 3 ((([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q ∧ ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q𝐴Q) → (([⟨(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))))
8779, 81, 85, 86syl3anc 1146 . 2 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 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))))
8877, 87mpbid 139 1 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → (𝐴 +Q ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃)) <Q (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wss 2944  cop 3405   class class class wbr 3791  suc csuc 4129  ωcom 4340   × cxp 4370  (class class class)co 5539  1𝑜c1o 6024  2𝑜c2o 6025   +𝑜 coa 6028  [cec 6134   / cqs 6135  Ncnpi 6427   ·N cmi 6429   <N clti 6430   ~Q ceq 6434  Qcnq 6435   +Q cplq 6437   ·Q cmq 6438   <Q cltq 6440  Pcnp 6446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-ltnqqs 6508  df-inp 6621
This theorem is referenced by:  prarloclem3step  6651
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