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Theorem caucvgprprlemloc 6955
 Description: Lemma for caucvgprpr 6964. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
caucvgprpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
Assertion
Ref Expression
caucvgprprlemloc (𝜑 → ∀𝑠Q𝑡Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐹,𝑙,𝑟   𝑢,𝐹,𝑟   𝑞,𝑝,𝑠,𝑡   𝜑,𝑠,𝑡   𝑝,𝑙,𝑞,𝑠,𝑡,𝑟   𝑢,𝑝,𝑞,𝑠,𝑡
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑡,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑡,𝑘,𝑛,𝑠,𝑞,𝑝)   𝐿(𝑢,𝑡,𝑘,𝑚,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemloc
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 6661 . . . . 5 (𝑠 <Q 𝑡 → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑡)
21adantl 271 . . . 4 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑡)
3 subhalfnqq 6666 . . . . . 6 (𝑦Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
43ad2antrl 474 . . . . 5 ((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
5 archrecnq 6915 . . . . . . 7 (𝑥Q → ∃𝑐N (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)
65ad2antrl 474 . . . . . 6 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → ∃𝑐N (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)
7 simpllr 501 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → 𝑠 <Q 𝑡)
87adantr 270 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → 𝑠 <Q 𝑡)
9 simplrl 502 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → 𝑦Q)
109adantr 270 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → 𝑦Q)
11 simplrr 503 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 +Q 𝑦) = 𝑡)
1211adantr 270 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q 𝑦) = 𝑡)
13 simplrl 502 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → 𝑥Q)
14 simplrr 503 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (𝑥 +Q 𝑥) <Q 𝑦)
15 simprl 498 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → 𝑐N)
16 simprr 499 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)
178, 10, 12, 13, 14, 15, 16caucvgprprlemloccalc 6936 . . . . . . . 8 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
18 simplrl 502 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → 𝑠Q)
1918ad3antrrr 476 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → 𝑠Q)
20 nnnq 6674 . . . . . . . . . . . . . 14 (𝑐N → [⟨𝑐, 1𝑜⟩] ~QQ)
2120ad2antrl 474 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → [⟨𝑐, 1𝑜⟩] ~QQ)
22 recclnq 6644 . . . . . . . . . . . . 13 ([⟨𝑐, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) ∈ Q)
2321, 22syl 14 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) ∈ Q)
24 addclnq 6627 . . . . . . . . . . . 12 ((𝑠Q ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∈ Q)
2519, 23, 24syl2anc 403 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∈ Q)
26 nqprlu 6799 . . . . . . . . . . 11 ((𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∈ Q → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ ∈ P)
2725, 26syl 14 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ ∈ P)
28 nqprlu 6799 . . . . . . . . . . 11 ((*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) ∈ Q → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
2923, 28syl 14 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
30 addclpr 6789 . . . . . . . . . 10 ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
3127, 29, 30syl2anc 403 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
32 simplrr 503 . . . . . . . . . . 11 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → 𝑡Q)
3332ad3antrrr 476 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → 𝑡Q)
34 nqprlu 6799 . . . . . . . . . 10 (𝑡Q → ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ∈ P)
3533, 34syl 14 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ∈ P)
36 caucvgprpr.f . . . . . . . . . . . 12 (𝜑𝐹:NP)
3736ad5antr 480 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → 𝐹:NP)
3837, 15ffvelrnd 5335 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (𝐹𝑐) ∈ P)
39 ltrelnq 6617 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4039brel 4418 . . . . . . . . . . . . 13 ((*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥 → ((*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) ∈ Q𝑥Q))
4140simpld 110 . . . . . . . . . . . 12 ((*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) ∈ Q)
4241ad2antll 475 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) ∈ Q)
4342, 28syl 14 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
44 addclpr 6789 . . . . . . . . . 10 (((𝐹𝑐) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
4538, 43, 44syl2anc 403 . . . . . . . . 9 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
46 ltsopr 6848 . . . . . . . . . 10 <P Or P
47 sowlin 4083 . . . . . . . . . 10 ((<P Or P ∧ ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ∈ P ∧ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)) → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∨ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)))
4846, 47mpan 415 . . . . . . . . 9 (((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ∈ P ∧ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P) → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∨ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)))
4931, 35, 45, 48syl3anc 1170 . . . . . . . 8 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∨ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)))
5017, 49mpd 13 . . . . . . 7 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∨ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))
5119adantr 270 . . . . . . . . . 10 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → 𝑠Q)
52 simplrl 502 . . . . . . . . . . 11 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → 𝑐N)
53 simpr 108 . . . . . . . . . . . 12 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
54 ltaprg 6871 . . . . . . . . . . . . . 14 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
5554adantl 271 . . . . . . . . . . . . 13 ((((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
5642adantr 270 . . . . . . . . . . . . . . 15 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) ∈ Q)
5751, 56, 24syl2anc 403 . . . . . . . . . . . . . 14 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∈ Q)
5857, 26syl 14 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ ∈ P)
5938adantr 270 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (𝐹𝑐) ∈ P)
6056, 28syl 14 . . . . . . . . . . . . 13 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
61 addcomprg 6830 . . . . . . . . . . . . . 14 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
6261adantl 271 . . . . . . . . . . . . 13 ((((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
6355, 58, 59, 60, 62caovord2d 5701 . . . . . . . . . . . 12 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑐) ↔ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
6453, 63mpbird 165 . . . . . . . . . . 11 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑐))
65 opeq1 3578 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ⟨𝑎, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
6665eceq1d 6208 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑐 → [⟨𝑎, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
6766fveq2d 5213 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
6867oveq2d 5559 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
6968breq2d 3805 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → (𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
7069abbidv 2197 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))})
7168breq1d 3803 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → ((𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞))
7271abbidv 2197 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞})
7370, 72opeq12d 3586 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
74 fveq2 5209 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
7573, 74breq12d 3806 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ↔ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑐)))
7675rspcev 2702 . . . . . . . . . . 11 ((𝑐N ∧ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑐)) → ∃𝑎N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎))
7752, 64, 76syl2anc 403 . . . . . . . . . 10 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → ∃𝑎N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎))
78 caucvgprpr.lim . . . . . . . . . . 11 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
7978caucvgprprlemell 6937 . . . . . . . . . 10 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑎N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎)))
8051, 77, 79sylanbrc 408 . . . . . . . . 9 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → 𝑠 ∈ (1st𝐿))
8180ex 113 . . . . . . . 8 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) → 𝑠 ∈ (1st𝐿)))
8233adantr 270 . . . . . . . . . 10 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) → 𝑡Q)
83 fveq2 5209 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
84 opeq1 3578 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑐 → ⟨𝑏, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
8584eceq1d 6208 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → [⟨𝑏, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
8685fveq2d 5213 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑐 → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
8786breq2d 3805 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → (𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
8887abbidv 2197 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → {𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )})
8986breq1d 3803 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞))
9089abbidv 2197 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞})
9188, 90opeq12d 3586 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)
9283, 91oveq12d 5561 . . . . . . . . . . . . 13 (𝑏 = 𝑐 → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
9392breq1d 3803 . . . . . . . . . . . 12 (𝑏 = 𝑐 → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ↔ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))
9493rspcev 2702 . . . . . . . . . . 11 ((𝑐N ∧ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
9515, 94sylan 277 . . . . . . . . . 10 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
9678caucvgprprlemelu 6938 . . . . . . . . . 10 (𝑡 ∈ (2nd𝐿) ↔ (𝑡Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))
9782, 95, 96sylanbrc 408 . . . . . . . . 9 (((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) → 𝑡 ∈ (2nd𝐿))
9897ex 113 . . . . . . . 8 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ → 𝑡 ∈ (2nd𝐿)))
9981, 98orim12d 733 . . . . . . 7 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∨ ((𝐹𝑐) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
10050, 99mpd 13 . . . . . 6 ((((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑐N ∧ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑥)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1016, 100rexlimddv 2482 . . . . 5 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1024, 101rexlimddv 2482 . . . 4 ((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1032, 102rexlimddv 2482 . . 3 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
104103ex 113 . 2 ((𝜑 ∧ (𝑠Q𝑡Q)) → (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
105104ralrimivva 2444 1 (𝜑 → ∀𝑠Q𝑡Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 662   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  {cab 2068  ∀wral 2349  ∃wrex 2350  {crab 2353  ⟨cop 3409   class class class wbr 3793   Or wor 4058  ⟶wf 4928  ‘cfv 4932  (class class class)co 5543  1st c1st 5796  2nd c2nd 5797  1𝑜c1o 6058  [cec 6170  Ncnpi 6524
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