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Theorem caucvgprprlemloc 7475
Description: Lemma for caucvgprpr 7484. 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 7181 . . . . 5 (𝑠 <Q 𝑡 → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑡)
21adantl 273 . . . 4 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑡)
3 subhalfnqq 7186 . . . . . 6 (𝑦Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
43ad2antrl 479 . . . . 5 ((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝑦)
5 archrecnq 7435 . . . . . . 7 (𝑥Q → ∃𝑐N (*Q‘[⟨𝑐, 1o⟩] ~Q ) <Q 𝑥)
65ad2antrl 479 . . . . . 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 506 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → 𝑠 <Q 𝑡)
87adantr 272 . . . . . . . . 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 507 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → 𝑦Q)
109adantr 272 . . . . . . . . 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 508 . . . . . . . . . 10 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 +Q 𝑦) = 𝑡)
1211adantr 272 . . . . . . . . 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 507 . . . . . . . . 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 508 . . . . . . . . 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 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)) → 𝑐N)
16 simprr 504 . . . . . . . . 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 7456 . . . . . . . 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 507 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → 𝑠Q)
1918ad3antrrr 481 . . . . . . . . . . . 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 7194 . . . . . . . . . . . . . 14 (𝑐N → [⟨𝑐, 1o⟩] ~QQ)
2120ad2antrl 479 . . . . . . . . . . . . 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 7164 . . . . . . . . . . . . 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 7147 . . . . . . . . . . . 12 ((𝑠Q ∧ (*Q‘[⟨𝑐, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) ∈ Q)
2519, 23, 24syl2anc 406 . . . . . . . . . . 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 7319 . . . . . . . . . . 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 7319 . . . . . . . . . . 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 7309 . . . . . . . . . 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 406 . . . . . . . . 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 508 . . . . . . . . . . 11 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → 𝑡Q)
3332ad3antrrr 481 . . . . . . . . . 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 7319 . . . . . . . . . 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 485 . . . . . . . . . . 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 5522 . . . . . . . . . 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 7137 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4039brel 4559 . . . . . . . . . . . . 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 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)) → (*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 7309 . . . . . . . . . 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 406 . . . . . . . . 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 7368 . . . . . . . . . 10 <P Or P
47 sowlin 4210 . . . . . . . . . 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 418 . . . . . . . . 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 1199 . . . . . . . 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 272 . . . . . . . . . 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 507 . . . . . . . . . . 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 7391 . . . . . . . . . . . . . 14 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
5554adantl 273 . . . . . . . . . . . . 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 272 . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 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 272 . . . . . . . . . . . . 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 7350 . . . . . . . . . . . . . 14 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
6261adantl 273 . . . . . . . . . . . . 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 5906 . . . . . . . . . . . 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 3673 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ⟨𝑎, 1o⟩ = ⟨𝑐, 1o⟩)
6665eceq1d 6431 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑐 → [⟨𝑎, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
6766fveq2d 5391 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
6867oveq2d 5756 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → (𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
6968breq2d 3909 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → (𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))))
7069abbidv 2233 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q ))})
7168breq1d 3907 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → ((𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) <Q 𝑞))
7271abbidv 2233 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1o⟩] ~Q )) <Q 𝑞})
7370, 72opeq12d 3681 . . . . . . . . . . . . 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 5387 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
7573, 74breq12d 3910 . . . . . . . . . . . 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 2761 . . . . . . . . . . 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 406 . . . . . . . . . 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 7457 . . . . . . . . . 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 411 . . . . . . . . 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 272 . . . . . . . . . 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 5387 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
84 opeq1 3673 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑐 → ⟨𝑏, 1o⟩ = ⟨𝑐, 1o⟩)
8584eceq1d 6431 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → [⟨𝑏, 1o⟩] ~Q = [⟨𝑐, 1o⟩] ~Q )
8685fveq2d 5391 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑐 → (*Q‘[⟨𝑏, 1o⟩] ~Q ) = (*Q‘[⟨𝑐, 1o⟩] ~Q ))
8786breq2d 3909 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → (𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑐, 1o⟩] ~Q )))
8887abbidv 2233 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → {𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑐, 1o⟩] ~Q )})
8986breq1d 3907 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑐, 1o⟩] ~Q ) <Q 𝑞))
9089abbidv 2233 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑐, 1o⟩] ~Q ) <Q 𝑞})
9188, 90opeq12d 3681 . . . . . . . . . . . . . 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 5758 . . . . . . . . . . . . 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 3907 . . . . . . . . . . . 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 2761 . . . . . . . . . . 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 279 . . . . . . . . . 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 7458 . . . . . . . . . 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 411 . . . . . . . . 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 758 . . . . . . 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 2529 . . . . 5 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1024, 101rexlimddv 2529 . . . 4 ((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1032, 102rexlimddv 2529 . . 3 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
104103ex 114 . 2 ((𝜑 ∧ (𝑠Q𝑡Q)) → (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
105104ralrimivva 2489 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 680  w3a 945   = wceq 1314  wcel 1463  {cab 2101  wral 2391  wrex 2392  {crab 2395  cop 3498   class class class wbr 3897   Or wor 4185  wf 5087  cfv 5091  (class class class)co 5740  1st c1st 6002  2nd c2nd 6003  1oc1o 6272  [cec 6393  Ncnpi 7044   <N clti 7047   ~Q ceq 7051  Qcnq 7052   +Q cplq 7054  *Qcrq 7056   <Q cltq 7057  Pcnp 7063   +P cpp 7065  <P cltp 7067
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242
This theorem is referenced by:  caucvgprprlemcl  7476
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