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Theorem caucvgprprlemloc 7241
Description: Lemma for caucvgprpr 7250. 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 6947 . . . . 5 (𝑠 <Q 𝑡 → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑡)
21adantl 271 . . . 4 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → ∃𝑦Q (𝑠 +Q 𝑦) = 𝑡)
3 subhalfnqq 6952 . . . . . 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 7201 . . . . . . 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 7222 . . . . . . . 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 6960 . . . . . . . . . . . . . 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 6930 . . . . . . . . . . . . 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 6913 . . . . . . . . . . . 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 7085 . . . . . . . . . . 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 7085 . . . . . . . . . . 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 7075 . . . . . . . . . 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 7085 . . . . . . . . . 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 5419 . . . . . . . . . 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 6903 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
4039brel 4478 . . . . . . . . . . . . 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 7075 . . . . . . . . . 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 7134 . . . . . . . . . 10 <P Or P
47 sowlin 4138 . . . . . . . . . 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 1174 . . . . . . . 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 7157 . . . . . . . . . . . . . 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 7116 . . . . . . . . . . . . . 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 5796 . . . . . . . . . . . 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 3617 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ⟨𝑎, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
6665eceq1d 6308 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑐 → [⟨𝑎, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
6766fveq2d 5293 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
6867oveq2d 5650 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑐 → (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
6968breq2d 3849 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → (𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
7069abbidv 2205 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))})
7168breq1d 3847 . . . . . . . . . . . . . . 15 (𝑎 = 𝑐 → ((𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞))
7271abbidv 2205 . . . . . . . . . . . . . 14 (𝑎 = 𝑐 → {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) <Q 𝑞})
7370, 72opeq12d 3625 . . . . . . . . . . . . 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 5289 . . . . . . . . . . . . 13 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
7573, 74breq12d 3850 . . . . . . . . . . . 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 2722 . . . . . . . . . . 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 7223 . . . . . . . . . 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 5289 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → (𝐹𝑏) = (𝐹𝑐))
84 opeq1 3617 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑐 → ⟨𝑏, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
8584eceq1d 6308 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → [⟨𝑏, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
8685fveq2d 5293 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑐 → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
8786breq2d 3849 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → (𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
8887abbidv 2205 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → {𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )})
8986breq1d 3847 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞))
9089abbidv 2205 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ) <Q 𝑞})
9188, 90opeq12d 3625 . . . . . . . . . . . . . 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 5652 . . . . . . . . . . . . 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 3847 . . . . . . . . . . . 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 2722 . . . . . . . . . . 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 7224 . . . . . . . . . 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 735 . . . . . . 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 2493 . . . . 5 (((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1024, 101rexlimddv 2493 . . . 4 ((((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) ∧ (𝑦Q ∧ (𝑠 +Q 𝑦) = 𝑡)) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
1032, 102rexlimddv 2493 . . 3 (((𝜑 ∧ (𝑠Q𝑡Q)) ∧ 𝑠 <Q 𝑡) → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))
104103ex 113 . 2 ((𝜑 ∧ (𝑠Q𝑡Q)) → (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
105104ralrimivva 2455 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 664  w3a 924   = wceq 1289  wcel 1438  {cab 2074  wral 2359  wrex 2360  {crab 2363  cop 3444   class class class wbr 3837   Or wor 4113  wf 4998  cfv 5002  (class class class)co 5634  1st c1st 5891  2nd c2nd 5892  1𝑜c1o 6156  [cec 6270  Ncnpi 6810   <N clti 6813   ~Q ceq 6817  Qcnq 6818   +Q cplq 6820  *Qcrq 6822   <Q cltq 6823  Pcnp 6829   +P cpp 6831  <P cltp 6833
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-po 4114  df-iso 4115  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-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  caucvgprprlemcl  7242
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