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Theorem prarloclemarch2 6574
 Description: Like prarloclemarch 6573 but the integer must be at least two, and there is also 𝐵 added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 6658. (Contributed by Jim Kingdon, 25-Nov-2019.)
Assertion
Ref Expression
prarloclemarch2 ((𝐴Q𝐵Q𝐶Q) → ∃𝑥N (1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem prarloclemarch2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 prarloclemarch 6573 . . 3 ((𝐴Q𝐶Q) → ∃𝑧N 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))
213adant2 934 . 2 ((𝐴Q𝐵Q𝐶Q) → ∃𝑧N 𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))
3 pinn 6464 . . . . . . . 8 (𝑧N𝑧 ∈ ω)
4 1pi 6470 . . . . . . . . . . . 12 1𝑜N
54elexi 2584 . . . . . . . . . . 11 1𝑜 ∈ V
65sucid 4181 . . . . . . . . . 10 1𝑜 ∈ suc 1𝑜
7 df-2o 6032 . . . . . . . . . 10 2𝑜 = suc 1𝑜
86, 7eleqtrri 2129 . . . . . . . . 9 1𝑜 ∈ 2𝑜
9 2onn 6124 . . . . . . . . . . 11 2𝑜 ∈ ω
10 nnaword2 6117 . . . . . . . . . . 11 ((2𝑜 ∈ ω ∧ 𝑧 ∈ ω) → 2𝑜 ⊆ (𝑧 +𝑜 2𝑜))
119, 10mpan 408 . . . . . . . . . 10 (𝑧 ∈ ω → 2𝑜 ⊆ (𝑧 +𝑜 2𝑜))
1211sseld 2971 . . . . . . . . 9 (𝑧 ∈ ω → (1𝑜 ∈ 2𝑜 → 1𝑜 ∈ (𝑧 +𝑜 2𝑜)))
138, 12mpi 15 . . . . . . . 8 (𝑧 ∈ ω → 1𝑜 ∈ (𝑧 +𝑜 2𝑜))
143, 13syl 14 . . . . . . 7 (𝑧N → 1𝑜 ∈ (𝑧 +𝑜 2𝑜))
15 o1p1e2 6078 . . . . . . . . 9 (1𝑜 +𝑜 1𝑜) = 2𝑜
16 addpiord 6471 . . . . . . . . . . 11 ((1𝑜N ∧ 1𝑜N) → (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜))
174, 4, 16mp2an 410 . . . . . . . . . 10 (1𝑜 +N 1𝑜) = (1𝑜 +𝑜 1𝑜)
18 addclpi 6482 . . . . . . . . . . 11 ((1𝑜N ∧ 1𝑜N) → (1𝑜 +N 1𝑜) ∈ N)
194, 4, 18mp2an 410 . . . . . . . . . 10 (1𝑜 +N 1𝑜) ∈ N
2017, 19eqeltrri 2127 . . . . . . . . 9 (1𝑜 +𝑜 1𝑜) ∈ N
2115, 20eqeltrri 2127 . . . . . . . 8 2𝑜N
22 addpiord 6471 . . . . . . . 8 ((𝑧N ∧ 2𝑜N) → (𝑧 +N 2𝑜) = (𝑧 +𝑜 2𝑜))
2321, 22mpan2 409 . . . . . . 7 (𝑧N → (𝑧 +N 2𝑜) = (𝑧 +𝑜 2𝑜))
2414, 23eleqtrrd 2133 . . . . . 6 (𝑧N → 1𝑜 ∈ (𝑧 +N 2𝑜))
25 addclpi 6482 . . . . . . . 8 ((𝑧N ∧ 2𝑜N) → (𝑧 +N 2𝑜) ∈ N)
2621, 25mpan2 409 . . . . . . 7 (𝑧N → (𝑧 +N 2𝑜) ∈ N)
27 ltpiord 6474 . . . . . . . 8 ((1𝑜N ∧ (𝑧 +N 2𝑜) ∈ N) → (1𝑜 <N (𝑧 +N 2𝑜) ↔ 1𝑜 ∈ (𝑧 +N 2𝑜)))
284, 27mpan 408 . . . . . . 7 ((𝑧 +N 2𝑜) ∈ N → (1𝑜 <N (𝑧 +N 2𝑜) ↔ 1𝑜 ∈ (𝑧 +N 2𝑜)))
2926, 28syl 14 . . . . . 6 (𝑧N → (1𝑜 <N (𝑧 +N 2𝑜) ↔ 1𝑜 ∈ (𝑧 +N 2𝑜)))
3024, 29mpbird 160 . . . . 5 (𝑧N → 1𝑜 <N (𝑧 +N 2𝑜))
3130adantl 266 . . . 4 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → 1𝑜 <N (𝑧 +N 2𝑜))
3231adantrr 456 . . 3 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → 1𝑜 <N (𝑧 +N 2𝑜))
33 nna0 6083 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ω → (𝑧 +𝑜 ∅) = 𝑧)
34 0lt1o 6053 . . . . . . . . . . . . . . . . . . . 20 ∅ ∈ 1𝑜
35 1on 6038 . . . . . . . . . . . . . . . . . . . . . 22 1𝑜 ∈ On
3635onsuci 4269 . . . . . . . . . . . . . . . . . . . . 21 suc 1𝑜 ∈ On
37 ontr1 4153 . . . . . . . . . . . . . . . . . . . . 21 (suc 1𝑜 ∈ On → ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ suc 1𝑜) → ∅ ∈ suc 1𝑜))
3836, 37ax-mp 7 . . . . . . . . . . . . . . . . . . . 20 ((∅ ∈ 1𝑜 ∧ 1𝑜 ∈ suc 1𝑜) → ∅ ∈ suc 1𝑜)
3934, 6, 38mp2an 410 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ suc 1𝑜
4039, 7eleqtrri 2129 . . . . . . . . . . . . . . . . . 18 ∅ ∈ 2𝑜
41 nnaordi 6111 . . . . . . . . . . . . . . . . . . 19 ((2𝑜 ∈ ω ∧ 𝑧 ∈ ω) → (∅ ∈ 2𝑜 → (𝑧 +𝑜 ∅) ∈ (𝑧 +𝑜 2𝑜)))
429, 41mpan 408 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ω → (∅ ∈ 2𝑜 → (𝑧 +𝑜 ∅) ∈ (𝑧 +𝑜 2𝑜)))
4340, 42mpi 15 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ω → (𝑧 +𝑜 ∅) ∈ (𝑧 +𝑜 2𝑜))
4433, 43eqeltrrd 2131 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ω → 𝑧 ∈ (𝑧 +𝑜 2𝑜))
453, 44syl 14 . . . . . . . . . . . . . . 15 (𝑧N𝑧 ∈ (𝑧 +𝑜 2𝑜))
4645, 23eleqtrrd 2133 . . . . . . . . . . . . . 14 (𝑧N𝑧 ∈ (𝑧 +N 2𝑜))
47 ltpiord 6474 . . . . . . . . . . . . . . 15 ((𝑧N ∧ (𝑧 +N 2𝑜) ∈ N) → (𝑧 <N (𝑧 +N 2𝑜) ↔ 𝑧 ∈ (𝑧 +N 2𝑜)))
4826, 47mpdan 406 . . . . . . . . . . . . . 14 (𝑧N → (𝑧 <N (𝑧 +N 2𝑜) ↔ 𝑧 ∈ (𝑧 +N 2𝑜)))
4946, 48mpbird 160 . . . . . . . . . . . . 13 (𝑧N𝑧 <N (𝑧 +N 2𝑜))
50 mulidpi 6473 . . . . . . . . . . . . 13 (𝑧N → (𝑧 ·N 1𝑜) = 𝑧)
51 mulcompig 6486 . . . . . . . . . . . . . . . 16 (((𝑧 +N 2𝑜) ∈ N ∧ 1𝑜N) → ((𝑧 +N 2𝑜) ·N 1𝑜) = (1𝑜 ·N (𝑧 +N 2𝑜)))
524, 51mpan2 409 . . . . . . . . . . . . . . 15 ((𝑧 +N 2𝑜) ∈ N → ((𝑧 +N 2𝑜) ·N 1𝑜) = (1𝑜 ·N (𝑧 +N 2𝑜)))
5326, 52syl 14 . . . . . . . . . . . . . 14 (𝑧N → ((𝑧 +N 2𝑜) ·N 1𝑜) = (1𝑜 ·N (𝑧 +N 2𝑜)))
54 mulidpi 6473 . . . . . . . . . . . . . . 15 ((𝑧 +N 2𝑜) ∈ N → ((𝑧 +N 2𝑜) ·N 1𝑜) = (𝑧 +N 2𝑜))
5526, 54syl 14 . . . . . . . . . . . . . 14 (𝑧N → ((𝑧 +N 2𝑜) ·N 1𝑜) = (𝑧 +N 2𝑜))
5653, 55eqtr3d 2090 . . . . . . . . . . . . 13 (𝑧N → (1𝑜 ·N (𝑧 +N 2𝑜)) = (𝑧 +N 2𝑜))
5749, 50, 563brtr4d 3821 . . . . . . . . . . . 12 (𝑧N → (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜)))
58 ordpipqqs 6529 . . . . . . . . . . . . . . 15 (((𝑧N ∧ 1𝑜N) ∧ ((𝑧 +N 2𝑜) ∈ N ∧ 1𝑜N)) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜))))
594, 58mpanl2 419 . . . . . . . . . . . . . 14 ((𝑧N ∧ ((𝑧 +N 2𝑜) ∈ N ∧ 1𝑜N)) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜))))
604, 59mpanr2 422 . . . . . . . . . . . . 13 ((𝑧N ∧ (𝑧 +N 2𝑜) ∈ N) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜))))
6126, 60mpdan 406 . . . . . . . . . . . 12 (𝑧N → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝑧 ·N 1𝑜) <N (1𝑜 ·N (𝑧 +N 2𝑜))))
6257, 61mpbird 160 . . . . . . . . . . 11 (𝑧N → [⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )
6362adantl 266 . . . . . . . . . 10 ((𝐶Q𝑧N) → [⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )
64 opelxpi 4403 . . . . . . . . . . . . . . . 16 (((𝑧 +N 2𝑜) ∈ N ∧ 1𝑜N) → ⟨(𝑧 +N 2𝑜), 1𝑜⟩ ∈ (N × N))
654, 64mpan2 409 . . . . . . . . . . . . . . 15 ((𝑧 +N 2𝑜) ∈ N → ⟨(𝑧 +N 2𝑜), 1𝑜⟩ ∈ (N × N))
66 enqex 6515 . . . . . . . . . . . . . . . 16 ~Q ∈ V
6766ecelqsi 6190 . . . . . . . . . . . . . . 15 (⟨(𝑧 +N 2𝑜), 1𝑜⟩ ∈ (N × N) → [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
6826, 65, 673syl 17 . . . . . . . . . . . . . 14 (𝑧N → [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
69 df-nqqs 6503 . . . . . . . . . . . . . 14 Q = ((N × N) / ~Q )
7068, 69syl6eleqr 2147 . . . . . . . . . . . . 13 (𝑧N → [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~QQ)
71 opelxpi 4403 . . . . . . . . . . . . . . . . 17 ((𝑧N ∧ 1𝑜N) → ⟨𝑧, 1𝑜⟩ ∈ (N × N))
724, 71mpan2 409 . . . . . . . . . . . . . . . 16 (𝑧N → ⟨𝑧, 1𝑜⟩ ∈ (N × N))
7366ecelqsi 6190 . . . . . . . . . . . . . . . 16 (⟨𝑧, 1𝑜⟩ ∈ (N × N) → [⟨𝑧, 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
7472, 73syl 14 . . . . . . . . . . . . . . 15 (𝑧N → [⟨𝑧, 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
7574, 69syl6eleqr 2147 . . . . . . . . . . . . . 14 (𝑧N → [⟨𝑧, 1𝑜⟩] ~QQ)
76 ltmnqg 6556 . . . . . . . . . . . . . 14 (([⟨𝑧, 1𝑜⟩] ~QQ ∧ [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~QQ𝐶Q) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
7775, 76syl3an1 1179 . . . . . . . . . . . . 13 ((𝑧N ∧ [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~QQ𝐶Q) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
7870, 77syl3an2 1180 . . . . . . . . . . . 12 ((𝑧N𝑧N𝐶Q) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
79783anidm12 1203 . . . . . . . . . . 11 ((𝑧N𝐶Q) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
8079ancoms 259 . . . . . . . . . 10 ((𝐶Q𝑧N) → ([⟨𝑧, 1𝑜⟩] ~Q <Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ↔ (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )))
8163, 80mpbid 139 . . . . . . . . 9 ((𝐶Q𝑧N) → (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) <Q (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ))
82 mulcomnqg 6538 . . . . . . . . . 10 ((𝐶Q ∧ [⟨𝑧, 1𝑜⟩] ~QQ) → (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) = ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))
8375, 82sylan2 274 . . . . . . . . 9 ((𝐶Q𝑧N) → (𝐶 ·Q [⟨𝑧, 1𝑜⟩] ~Q ) = ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))
84 mulcomnqg 6538 . . . . . . . . . 10 ((𝐶Q ∧ [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~QQ) → (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
8570, 84sylan2 274 . . . . . . . . 9 ((𝐶Q𝑧N) → (𝐶 ·Q [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ) = ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
8681, 83, 853brtr3d 3820 . . . . . . . 8 ((𝐶Q𝑧N) → ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
87863ad2antl3 1079 . . . . . . 7 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
8887adantrr 456 . . . . . 6 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
89 ltsonq 6553 . . . . . . . . . 10 <Q Or Q
90 ltrelnq 6520 . . . . . . . . . 10 <Q ⊆ (Q × Q)
9189, 90sotri 4747 . . . . . . . . 9 ((𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) ∧ ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
9291ex 112 . . . . . . . 8 (𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) → (([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
9392adantl 266 . . . . . . 7 ((𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶)) → (([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
9493adantl 266 . . . . . 6 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → (([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶) <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
9588, 94mpd 13 . . . . 5 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → 𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
96 mulclnq 6531 . . . . . . . . . 10 (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~QQ𝐶Q) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q)
9770, 96sylan 271 . . . . . . . . 9 ((𝑧N𝐶Q) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q)
9897ancoms 259 . . . . . . . 8 ((𝐶Q𝑧N) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q)
99983ad2antl3 1079 . . . . . . 7 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q)
100 simpl2 919 . . . . . . 7 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → 𝐵Q)
101 ltaddnq 6562 . . . . . . 7 ((([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q𝐵Q) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
10299, 100, 101syl2anc 397 . . . . . 6 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
103102adantrr 456 . . . . 5 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
10489, 90sotri 4747 . . . . 5 ((𝐴 <Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∧ ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵)) → 𝐴 <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
10595, 103, 104syl2anc 397 . . . 4 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → 𝐴 <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵))
106 addcomnqg 6536 . . . . . . 7 ((([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) ∈ Q𝐵Q) → (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵) = (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
10799, 100, 106syl2anc 397 . . . . . 6 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵) = (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
108107breq2d 3803 . . . . 5 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → (𝐴 <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵) ↔ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))))
109108adantrr 456 . . . 4 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → (𝐴 <Q (([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶) +Q 𝐵) ↔ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))))
110105, 109mpbid 139 . . 3 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
111 simpr 107 . . . . 5 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → 𝑧N)
112 breq2 3795 . . . . . . . 8 (𝑥 = (𝑧 +N 2𝑜) → (1𝑜 <N 𝑥 ↔ 1𝑜 <N (𝑧 +N 2𝑜)))
113 opeq1 3576 . . . . . . . . . . . 12 (𝑥 = (𝑧 +N 2𝑜) → ⟨𝑥, 1𝑜⟩ = ⟨(𝑧 +N 2𝑜), 1𝑜⟩)
114113eceq1d 6172 . . . . . . . . . . 11 (𝑥 = (𝑧 +N 2𝑜) → [⟨𝑥, 1𝑜⟩] ~Q = [⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q )
115114oveq1d 5554 . . . . . . . . . 10 (𝑥 = (𝑧 +N 2𝑜) → ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶) = ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))
116115oveq2d 5555 . . . . . . . . 9 (𝑥 = (𝑧 +N 2𝑜) → (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)) = (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))
117116breq2d 3803 . . . . . . . 8 (𝑥 = (𝑧 +N 2𝑜) → (𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)) ↔ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))))
118112, 117anbi12d 450 . . . . . . 7 (𝑥 = (𝑧 +N 2𝑜) → ((1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))) ↔ (1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))))
119118rspcev 2673 . . . . . 6 (((𝑧 +N 2𝑜) ∈ N ∧ (1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶)))) → ∃𝑥N (1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))
120119ex 112 . . . . 5 ((𝑧 +N 2𝑜) ∈ N → ((1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))) → ∃𝑥N (1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)))))
121111, 26, 1203syl 17 . . . 4 (((𝐴Q𝐵Q𝐶Q) ∧ 𝑧N) → ((1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))) → ∃𝑥N (1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)))))
122121adantrr 456 . . 3 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → ((1𝑜 <N (𝑧 +N 2𝑜) ∧ 𝐴 <Q (𝐵 +Q ([⟨(𝑧 +N 2𝑜), 1𝑜⟩] ~Q ·Q 𝐶))) → ∃𝑥N (1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶)))))
12332, 110, 122mp2and 417 . 2 (((𝐴Q𝐵Q𝐶Q) ∧ (𝑧N𝐴 <Q ([⟨𝑧, 1𝑜⟩] ~Q ·Q 𝐶))) → ∃𝑥N (1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))
1242, 123rexlimddv 2454 1 ((𝐴Q𝐵Q𝐶Q) → ∃𝑥N (1𝑜 <N 𝑥𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  ∃wrex 2324   ⊆ wss 2944  ∅c0 3251  ⟨cop 3405   class class class wbr 3791  Oncon0 4127  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 cpli 6428   ·N cmi 6429
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