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Theorem caucvgprprlemml 7706
Description: Lemma for caucvgprpr 7724. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-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
caucvgprprlemml (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟,𝑚   𝐴,𝑠,𝑟   𝐹,𝑙   𝑝,𝑙,𝑞,𝑟,𝑠   𝑢,𝑙   𝜑,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑢,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝)   𝐿(𝑢,𝑘,𝑚,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemml
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 5527 . . . . . 6 (𝑚 = 1o → (𝐹𝑚) = (𝐹‘1o))
21breq2d 4027 . . . . 5 (𝑚 = 1o → (𝐴<P (𝐹𝑚) ↔ 𝐴<P (𝐹‘1o)))
3 caucvgprpr.bnd . . . . 5 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
4 1pi 7327 . . . . . 6 1oN
54a1i 9 . . . . 5 (𝜑 → 1oN)
62, 3, 5rspcdva 2858 . . . 4 (𝜑𝐴<P (𝐹‘1o))
7 ltrelpr 7517 . . . . . 6 <P ⊆ (P × P)
87brel 4690 . . . . 5 (𝐴<P (𝐹‘1o) → (𝐴P ∧ (𝐹‘1o) ∈ P))
98simpld 112 . . . 4 (𝐴<P (𝐹‘1o) → 𝐴P)
106, 9syl 14 . . 3 (𝜑𝐴P)
11 prop 7487 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 prml 7489 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
1311, 12syl 14 . . 3 (𝐴P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
1410, 13syl 14 . 2 (𝜑 → ∃𝑥Q 𝑥 ∈ (1st𝐴))
15 subhalfnqq 7426 . . . 4 (𝑥Q → ∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥)
1615ad2antrl 490 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥)
17 simplr 528 . . . . . 6 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠Q)
18 archrecnq 7675 . . . . . . . 8 (𝑠Q → ∃𝑟N (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠)
1917, 18syl 14 . . . . . . 7 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → ∃𝑟N (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠)
20 simpr 110 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠)
21 simplr 528 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑟N)
22 nnnq 7434 . . . . . . . . . . . . . . . 16 (𝑟N → [⟨𝑟, 1o⟩] ~QQ)
23 recclnq 7404 . . . . . . . . . . . . . . . 16 ([⟨𝑟, 1o⟩] ~QQ → (*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q)
2421, 22, 233syl 17 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q)
2517ad2antrr 488 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑠Q)
26 ltanqg 7412 . . . . . . . . . . . . . . 15 (((*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q𝑠Q𝑠Q) → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
2724, 25, 25, 26syl3anc 1248 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
2820, 27mpbid 147 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q (𝑠 +Q 𝑠))
29 simpllr 534 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) <Q 𝑥)
30 ltsonq 7410 . . . . . . . . . . . . . 14 <Q Or Q
31 ltrelnq 7377 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
3230, 31sotri 5036 . . . . . . . . . . . . 13 (((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q (𝑠 +Q 𝑠) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥)
3328, 29, 32syl2anc 411 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥)
3410ad5antr 496 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝐴P)
35 simprr 531 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐴))
3635ad4antr 494 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑥 ∈ (1st𝐴))
37 prcdnql 7496 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st𝐴)))
3811, 37sylan 283 . . . . . . . . . . . . 13 ((𝐴P𝑥 ∈ (1st𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st𝐴)))
3934, 36, 38syl2anc 411 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st𝐴)))
4033, 39mpd 13 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st𝐴))
41 addclnq 7387 . . . . . . . . . . . . 13 ((𝑠Q ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ Q)
4225, 24, 41syl2anc 411 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ Q)
43 nqprl 7563 . . . . . . . . . . . 12 (((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ Q𝐴P) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st𝐴) ↔ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P 𝐴))
4442, 34, 43syl2anc 411 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st𝐴) ↔ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P 𝐴))
4540, 44mpbid 147 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P 𝐴)
46 fveq2 5527 . . . . . . . . . . . 12 (𝑚 = 𝑟 → (𝐹𝑚) = (𝐹𝑟))
4746breq2d 4027 . . . . . . . . . . 11 (𝑚 = 𝑟 → (𝐴<P (𝐹𝑚) ↔ 𝐴<P (𝐹𝑟)))
483ad5antr 496 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ∀𝑚N 𝐴<P (𝐹𝑚))
4947, 48, 21rspcdva 2858 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝐴<P (𝐹𝑟))
50 ltsopr 7608 . . . . . . . . . . 11 <P Or P
5150, 7sotri 5036 . . . . . . . . . 10 ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P 𝐴𝐴<P (𝐹𝑟)) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟))
5245, 49, 51syl2anc 411 . . . . . . . . 9 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟))
5352ex 115 . . . . . . . 8 (((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠 → ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
5453reximdva 2589 . . . . . . 7 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → (∃𝑟N (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠 → ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
5519, 54mpd 13 . . . . . 6 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟))
56 oveq1 5895 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
5756breq2d 4027 . . . . . . . . . . 11 (𝑙 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
5857abbidv 2305 . . . . . . . . . 10 (𝑙 = 𝑠 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
5956breq1d 4025 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
6059abbidv 2305 . . . . . . . . . 10 (𝑙 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞})
6158, 60opeq12d 3798 . . . . . . . . 9 (𝑙 = 𝑠 → ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩)
6261breq1d 4025 . . . . . . . 8 (𝑙 = 𝑠 → (⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
6362rexbidv 2488 . . . . . . 7 (𝑙 = 𝑠 → (∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟) ↔ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
64 caucvgprpr.lim . . . . . . . . 9 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}⟩}⟩
6564fveq2i 5530 . . . . . . . 8 (1st𝐿) = (1st ‘⟨{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}⟩}⟩)
66 nqex 7375 . . . . . . . . . 10 Q ∈ V
6766rabex 4159 . . . . . . . . 9 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
6866rabex 4159 . . . . . . . . 9 {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩} ∈ V
6967, 68op1st 6160 . . . . . . . 8 (1st ‘⟨{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}⟩}⟩) = {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}
7065, 69eqtri 2208 . . . . . . 7 (1st𝐿) = {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}
7163, 70elrab2 2908 . . . . . 6 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
7217, 55, 71sylanbrc 417 . . . . 5 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠 ∈ (1st𝐿))
7372ex 115 . . . 4 (((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) → ((𝑠 +Q 𝑠) <Q 𝑥𝑠 ∈ (1st𝐿)))
7473reximdva 2589 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → (∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥 → ∃𝑠Q 𝑠 ∈ (1st𝐿)))
7516, 74mpd 13 . 2 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑠Q 𝑠 ∈ (1st𝐿))
7614, 75rexlimddv 2609 1 (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  {cab 2173  wral 2465  wrex 2466  {crab 2469  cop 3607   class class class wbr 4015  wf 5224  cfv 5228  (class class class)co 5888  1st c1st 6152  2nd c2nd 6153  1oc1o 6423  [cec 6546  Ncnpi 7284   <N clti 7287   ~Q ceq 7291  Qcnq 7292   +Q cplq 7294  *Qcrq 7296   <Q cltq 7297  Pcnp 7303   +P cpp 7305  <P cltp 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-inp 7478  df-iltp 7482
This theorem is referenced by:  caucvgprprlemm  7708
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