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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⟩ ∈ 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 6260 . . . . . . . . . . . 12 2𝑜 ∈ ω
2 nnacl 6223 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 2𝑜 ∈ ω) → (𝑦 +𝑜 2𝑜) ∈ ω)
31, 2mpan2 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))
53, 4sylan 277 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 2𝑜) ⊆ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
6 1oex 6171 . . . . . . . . . . . . . 14 1𝑜 ∈ V
76sucid 4235 . . . . . . . . . . . . 13 1𝑜 ∈ suc 1𝑜
8 df-2o 6164 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
97, 8eleqtrri 2163 . . . . . . . . . . . 12 1𝑜 ∈ 2𝑜
10 nnaordi 6247 . . . . . . . . . . . . 13 ((2𝑜 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
111, 10mpan 415 . . . . . . . . . . . 12 (𝑦 ∈ ω → (1𝑜 ∈ 2𝑜 → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜)))
129, 11mpi 15 . . . . . . . . . . 11 (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
1312adantr 270 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ (𝑦 +𝑜 2𝑜))
145, 13sseldd 3024 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝑋 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
1514ancoms 264 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
16 1pi 6853 . . . . . . . . . . 11 1𝑜N
17 nnppipi 6881 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 1𝑜N) → (𝑦 +𝑜 1𝑜) ∈ N)
1816, 17mpan2 416 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑦 +𝑜 1𝑜) ∈ N)
1918adantl 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 ∧ 1𝑜N) → (1𝑜 +𝑜 1𝑜) ∈ N)
2321, 16, 22mp2an 417 . . . . . . . . . . . . . 14 (1𝑜 +𝑜 1𝑜) ∈ N
2420, 23eqeltrri 2161 . . . . . . . . . . . . 13 2𝑜N
25 nnppipi 6881 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ 2𝑜N) → (𝑦 +𝑜 2𝑜) ∈ N)
2624, 25mpan2 416 . . . . . . . . . . . 12 (𝑦 ∈ ω → (𝑦 +𝑜 2𝑜) ∈ N)
27 pinn 6847 . . . . . . . . . . . 12 ((𝑦 +𝑜 2𝑜) ∈ N → (𝑦 +𝑜 2𝑜) ∈ ω)
2826, 27syl 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))
3028, 29sylan2 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)
3226, 31sylan2 280 . . . . . . . . . 10 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (𝑋 +𝑜 (𝑦 +𝑜 2𝑜)) ∈ N)
3330, 32eqeltrrd 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)))
3519, 33, 34syl2anc 403 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) <N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ↔ (𝑦 +𝑜 1𝑜) ∈ ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
3615, 35mpbird 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))
3819, 37syl 14 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ((𝑦 +𝑜 1𝑜) ·N 1𝑜) = (𝑦 +𝑜 1𝑜))
39 mulcompig 6869 . . . . . . . . . 10 ((((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜N) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)))
4033, 16, 39sylancl 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))
4233, 41syl 14 . . . . . . . . 9 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ·N 1𝑜) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
4340, 42eqtr3d 2122 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → (1𝑜 ·N ((𝑦 +𝑜 2𝑜) +𝑜 𝑋)) = ((𝑦 +𝑜 2𝑜) +𝑜 𝑋))
4438, 43breq12d 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)))
4536, 44mpbird 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 ∧ 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))))
4816, 47mpanl2 426 . . . . . . . . 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))))
4916, 48mpanr2 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))))
5018, 49sylan 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))))
5146, 33, 50syl2anc 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))))
5245, 51mpbird 165 . . . . 5 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )
5352adantlr 461 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q <Q [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q )
54 opelxpi 4459 . . . . . . . . 9 (((𝑦 +𝑜 1𝑜) ∈ N ∧ 1𝑜N) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
5519, 16, 54sylancl 404 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩ ∈ (N × N))
56 enqex 6898 . . . . . . . . 9 ~Q ∈ V
5756ecelqsi 6326 . . . . . . . 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 6886 . . . . . . 7 Q = ((N × N) / ~Q )
6058, 59syl6eleqr 2181 . . . . . 6 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ)
6160adantlr 461 . . . . 5 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ)
62 opelxpi 4459 . . . . . . . . 9 ((((𝑦 +𝑜 2𝑜) +𝑜 𝑋) ∈ N ∧ 1𝑜N) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
6333, 16, 62sylancl 404 . . . . . . . 8 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → ⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩ ∈ (N × N))
6456ecelqsi 6326 . . . . . . . 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 2181 . . . . . 6 ((𝑋 ∈ ω ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~QQ)
6766adantlr 461 . . . . 5 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → [⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~QQ)
68 simplr3 987 . . . . 5 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → 𝑃Q)
69 ltmnqg 6939 . . . . 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 1174 . . . 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 145 . . 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 6921 . . . . 5 ((𝑃Q ∧ [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ) → (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃))
7368, 61, 72syl2anc 403 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → (𝑃 ·Q [⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃))
74 mulcomnqg 6921 . . . . 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 403 . . . 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 3850 . . 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 145 . 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 6914 . . . 4 (([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~QQ𝑃Q) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
7961, 68, 78syl2anc 403 . . 3 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ([⟨(𝑦 +𝑜 1𝑜), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
80 mulclnq 6914 . . . 4 (([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~QQ𝑃Q) → ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
8167, 68, 80syl2anc 403 . . 3 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃) ∈ Q)
82 simplr1 985 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ⟨𝐿, 𝑈⟩ ∈ P)
83 simplr2 986 . . . 4 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → 𝐴𝐿)
84 elprnql 7019 . . . 4 ((⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿) → 𝐴Q)
8582, 83, 84syl2anc 403 . . 3 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → 𝐴Q)
86 ltanqg 6938 . . 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 1174 . 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 145 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 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  1𝑜c1o 6156  2𝑜c2o 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
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