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Theorem caucvgprprlemml 7555
 Description: Lemma for caucvgprpr 7573. 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 5433 . . . . . 6 (𝑚 = 1o → (𝐹𝑚) = (𝐹‘1o))
21breq2d 3951 . . . . 5 (𝑚 = 1o → (𝐴<P (𝐹𝑚) ↔ 𝐴<P (𝐹‘1o)))
3 caucvgprpr.bnd . . . . 5 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
4 1pi 7176 . . . . . 6 1oN
54a1i 9 . . . . 5 (𝜑 → 1oN)
62, 3, 5rspcdva 2800 . . . 4 (𝜑𝐴<P (𝐹‘1o))
7 ltrelpr 7366 . . . . . 6 <P ⊆ (P × P)
87brel 4603 . . . . 5 (𝐴<P (𝐹‘1o) → (𝐴P ∧ (𝐹‘1o) ∈ P))
98simpld 111 . . . 4 (𝐴<P (𝐹‘1o) → 𝐴P)
106, 9syl 14 . . 3 (𝜑𝐴P)
11 prop 7336 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 prml 7338 . . . 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 7275 . . . 4 (𝑥Q → ∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥)
1615ad2antrl 482 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥)
17 simplr 520 . . . . . 6 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠Q)
18 archrecnq 7524 . . . . . . . 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 109 . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑟N)
22 nnnq 7283 . . . . . . . . . . . . . . . 16 (𝑟N → [⟨𝑟, 1o⟩] ~QQ)
23 recclnq 7253 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑠Q)
26 ltanqg 7261 . . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . . 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 146 . . . . . . . . . . . . 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 524 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) <Q 𝑥)
30 ltsonq 7259 . . . . . . . . . . . . . 14 <Q Or Q
31 ltrelnq 7226 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
3230, 31sotri 4946 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 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 488 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝐴P)
35 simprr 522 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐴))
3635ad4antr 486 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑥 ∈ (1st𝐴))
37 prcdnql 7345 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st𝐴)))
3811, 37sylan 281 . . . . . . . . . . . . 13 ((𝐴P𝑥 ∈ (1st𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st𝐴)))
3934, 36, 38syl2anc 409 . . . . . . . . . . . 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 7236 . . . . . . . . . . . . 13 ((𝑠Q ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ Q)
4225, 24, 41syl2anc 409 . . . . . . . . . . . 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 7412 . . . . . . . . . . . 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 409 . . . . . . . . . . 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 146 . . . . . . . . . 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 5433 . . . . . . . . . . . 12 (𝑚 = 𝑟 → (𝐹𝑚) = (𝐹𝑟))
4746breq2d 3951 . . . . . . . . . . 11 (𝑚 = 𝑟 → (𝐴<P (𝐹𝑚) ↔ 𝐴<P (𝐹𝑟)))
483ad5antr 488 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ∀𝑚N 𝐴<P (𝐹𝑚))
4947, 48, 21rspcdva 2800 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝐴<P (𝐹𝑟))
50 ltsopr 7457 . . . . . . . . . . 11 <P Or P
5150, 7sotri 4946 . . . . . . . . . 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 409 . . . . . . . . 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 114 . . . . . . . 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 2539 . . . . . . 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 5793 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
5756breq2d 3951 . . . . . . . . . . 11 (𝑙 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
5857abbidv 2259 . . . . . . . . . 10 (𝑙 = 𝑠 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
5956breq1d 3949 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
6059abbidv 2259 . . . . . . . . . 10 (𝑙 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞})
6158, 60opeq12d 3723 . . . . . . . . 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 3949 . . . . . . . 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 2441 . . . . . . 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 5436 . . . . . . . 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 7224 . . . . . . . . . 10 Q ∈ V
6766rabex 4082 . . . . . . . . 9 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
6866rabex 4082 . . . . . . . . 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 6056 . . . . . . . 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 2162 . . . . . . 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 2849 . . . . . 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 414 . . . . 5 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠 ∈ (1st𝐿))
7372ex 114 . . . 4 (((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) → ((𝑠 +Q 𝑠) <Q 𝑥𝑠 ∈ (1st𝐿)))
7473reximdva 2539 . . 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 2559 1 (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2112  {cab 2127  ∀wral 2418  ∃wrex 2419  {crab 2422  ⟨cop 3537   class class class wbr 3939  ⟶wf 5131  ‘cfv 5135  (class class class)co 5786  1st c1st 6048  2nd c2nd 6049  1oc1o 6318  [cec 6439  Ncnpi 7133
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