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Theorem caucvgprprlemloc 7665
Description: Lemma for caucvgprpr 7674. 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 7371 . . . . 5 (𝑠 <Q 𝑡 → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑡)
21adantl 275 . . . 4 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑡)
3 subhalfnqq 7376 . . . . . 6 (𝑦Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
43ad2antrl 487 . . . . 5 ((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
5 archrecnq 7625 . . . . . . 7 (𝑥Q → ∃𝑐N (*Q‘[⟨𝑐, 1o⟩] ~Q ) <Q 𝑥)
65ad2antrl 487 . . . . . 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 529 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → 𝑠 <Q 𝑡)
87adantr 274 . . . . . . . . 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 530 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → 𝑦Q)
109adantr 274 . . . . . . . . 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 531 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 +Q 𝑦) = 𝑡)
1211adantr 274 . . . . . . . . 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 530 . . . . . . . . 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 531 . . . . . . . . 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 526 . . . . . . . . 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 527 . . . . . . . . 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 7646 . . . . . . . 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 530 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → 𝑠Q)
1918ad3antrrr 489 . . . . . . . . . . . 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 7384 . . . . . . . . . . . . . 14 (𝑐N → [⟨𝑐, 1o⟩] ~QQ)
2120ad2antrl 487 . . . . . . . . . . . . 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 7354 . . . . . . . . . . . . 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 7337 . . . . . . . . . . . 12 ((𝑠Q ∧ (*Q‘[⟨𝑐, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∈ Q)
2519, 23, 24syl2anc 409 . . . . . . . . . . 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 7509 . . . . . . . . . . 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 7509 . . . . . . . . . . 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 7499 . . . . . . . . . 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 409 . . . . . . . . 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 531 . . . . . . . . . . 11 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → 𝑡Q)
3332ad3antrrr 489 . . . . . . . . . 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 7509 . . . . . . . . . 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 493 . . . . . . . . . . 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 5632 . . . . . . . . . 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 7327 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4039brel 4663 . . . . . . . . . . . . 13 ((*Q‘[⟨𝑐, 1o⟩] ~Q ) <Q 𝑥 → ((*Q‘[⟨𝑐, 1o⟩] ~Q ) ∈ Q𝑥Q))
4140simpld 111 . . . . . . . . . . . 12 ((*Q‘[⟨𝑐, 1o⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑐, 1o⟩] ~Q ) ∈ Q)
4241ad2antll 488 . . . . . . . . . . 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 7499 . . . . . . . . . 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 409 . . . . . . . . 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 7558 . . . . . . . . . 10 <P Or P
47 sowlin 4305 . . . . . . . . . 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 422 . . . . . . . . 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 1233 . . . . . . . 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 274 . . . . . . . . . 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 530 . . . . . . . . . . 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 109 . . . . . . . . . . . 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 7581 . . . . . . . . . . . . . 14 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
5554adantl 275 . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . 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 7540 . . . . . . . . . . . . . 14 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
6261adantl 275 . . . . . . . . . . . . 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 6022 . . . . . . . . . . . 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 166 . . . . . . . . . . 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 3765 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ⟨𝑎, 1o⟩ = ⟨𝑐, 1o⟩)
6665eceq1d 6549 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑐 → [⟨𝑎, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
6766fveq2d 5500 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
6867oveq2d 5869 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → (𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
6968breq2d 4001 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → (𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
7069abbidv 2288 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))})
7168breq1d 3999 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → ((𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) <Q 𝑞))
7271abbidv 2288 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) <Q 𝑞})
7370, 72opeq12d 3773 . . . . . . . . . . . . 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 5496 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
7573, 74breq12d 4002 . . . . . . . . . . . 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 2834 . . . . . . . . . . 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 409 . . . . . . . . . 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 7647 . . . . . . . . . 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 415 . . . . . . . . 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 114 . . . . . . . 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 274 . . . . . . . . . 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 5496 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
84 opeq1 3765 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑐 → ⟨𝑏, 1o⟩ = ⟨𝑐, 1o⟩)
8584eceq1d 6549 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → [⟨𝑏, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
8685fveq2d 5500 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑐 → (*Q‘[⟨𝑏, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
8786breq2d 4001 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → (𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
8887abbidv 2288 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → {𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑐, 1o⟩] ~Q )})
8986breq1d 3999 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑐, 1o⟩] ~Q ) <Q 𝑞))
9089abbidv 2288 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑐, 1o⟩] ~Q ) <Q 𝑞})
9188, 90opeq12d 3773 . . . . . . . . . . . . . 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 5871 . . . . . . . . . . . . 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 3999 . . . . . . . . . . . 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 2834 . . . . . . . . . . 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 281 . . . . . . . . . 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 7648 . . . . . . . . . 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 415 . . . . . . . . 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 114 . . . . . . . 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 781 . . . . . . 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 2592 . . . . 5 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1024, 101rexlimddv 2592 . . . 4 ((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1032, 102rexlimddv 2592 . . 3 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
104103ex 114 . 2 ((𝜑 ∧ (𝑠Q𝑡Q)) → (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
105104ralrimivva 2552 1 (𝜑 → ∀𝑠Q𝑡Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 703  w3a 973   = wceq 1348  wcel 2141  {cab 2156  wral 2448  wrex 2449  {crab 2452  cop 3586   class class class wbr 3989   Or wor 4280  wf 5194  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  1oc1o 6388  [cec 6511  Ncnpi 7234   <N clti 7237   ~Q ceq 7241  Qcnq 7242   +Q cplq 7244  *Qcrq 7246   <Q cltq 7247  Pcnp 7253   +P cpp 7255  <P cltp 7257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432
This theorem is referenced by:  caucvgprprlemcl  7666
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