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Theorem caucvgprprlemml 6792
 Description: Lemma for caucvgprpr 6810. 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 1pi 6413 . . . . 5 1𝑜N
2 caucvgprpr.bnd . . . . 5 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
3 fveq2 5178 . . . . . . 7 (𝑚 = 1𝑜 → (𝐹𝑚) = (𝐹‘1𝑜))
43breq2d 3776 . . . . . 6 (𝑚 = 1𝑜 → (𝐴<P (𝐹𝑚) ↔ 𝐴<P (𝐹‘1𝑜)))
54rspcv 2652 . . . . 5 (1𝑜N → (∀𝑚N 𝐴<P (𝐹𝑚) → 𝐴<P (𝐹‘1𝑜)))
61, 2, 5mpsyl 59 . . . 4 (𝜑𝐴<P (𝐹‘1𝑜))
7 ltrelpr 6603 . . . . . 6 <P ⊆ (P × P)
87brel 4392 . . . . 5 (𝐴<P (𝐹‘1𝑜) → (𝐴P ∧ (𝐹‘1𝑜) ∈ P))
98simpld 105 . . . 4 (𝐴<P (𝐹‘1𝑜) → 𝐴P)
106, 9syl 14 . . 3 (𝜑𝐴P)
11 prop 6573 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 prml 6575 . . . 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 6512 . . . 4 (𝑥Q → ∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥)
1615ad2antrl 459 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥)
17 simplr 482 . . . . . 6 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠Q)
18 archrecnq 6761 . . . . . . . 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 103 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑟N)
22 nnnq 6520 . . . . . . . . . . . . . . . 16 (𝑟N → [⟨𝑟, 1𝑜⟩] ~QQ)
23 recclnq 6490 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑠Q)
26 ltanqg 6498 . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . 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 135 . . . . . . . . . . . . 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 486 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) <Q 𝑥)
30 ltsonq 6496 . . . . . . . . . . . . . 14 <Q Or Q
31 ltrelnq 6463 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
3230, 31sotri 4720 . . . . . . . . . . . . 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 391 . . . . . . . . . . . 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 465 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝐴P)
35 simprr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐴))
3635ad4antr 463 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑥 ∈ (1st𝐴))
37 prcdnql 6582 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ∈ (1st𝐴)))
3811, 37sylan 267 . . . . . . . . . . . . 13 ((𝐴P𝑥 ∈ (1st𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ∈ (1st𝐴)))
3934, 36, 38syl2anc 391 . . . . . . . . . . . 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 6473 . . . . . . . . . . . . 13 ((𝑠Q ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ∈ Q)
4225, 24, 41syl2anc 391 . . . . . . . . . . . 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 6649 . . . . . . . . . . . 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 391 . . . . . . . . . . 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 135 . . . . . . . . . 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)
462ad5antr 465 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → ∀𝑚N 𝐴<P (𝐹𝑚))
47 fveq2 5178 . . . . . . . . . . . . 13 (𝑚 = 𝑟 → (𝐹𝑚) = (𝐹𝑟))
4847breq2d 3776 . . . . . . . . . . . 12 (𝑚 = 𝑟 → (𝐴<P (𝐹𝑚) ↔ 𝐴<P (𝐹𝑟)))
4948rspcv 2652 . . . . . . . . . . 11 (𝑟N → (∀𝑚N 𝐴<P (𝐹𝑚) → 𝐴<P (𝐹𝑟)))
5021, 46, 49sylc 56 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝐴<P (𝐹𝑟))
51 ltsopr 6694 . . . . . . . . . . 11 <P Or P
5251, 7sotri 4720 . . . . . . . . . 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))
5345, 50, 52syl2anc 391 . . . . . . . . 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))
5453ex 108 . . . . . . . 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)))
5554reximdva 2421 . . . . . . 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)))
5619, 55mpd 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))
57 oveq1 5519 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
5857breq2d 3776 . . . . . . . . . . 11 (𝑙 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))))
5958abbidv 2155 . . . . . . . . . 10 (𝑙 = 𝑠 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))})
6057breq1d 3774 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞))
6160abbidv 2155 . . . . . . . . . 10 (𝑙 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞})
6259, 61opeq12d 3557 . . . . . . . . 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}⟩)
6362breq1d 3774 . . . . . . . 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)))
6463rexbidv 2327 . . . . . . 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)))
65 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}⟩}⟩
6665fveq2i 5181 . . . . . . . 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}⟩}⟩)
67 nqex 6461 . . . . . . . . . 10 Q ∈ V
6867rabex 3901 . . . . . . . . 9 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
6967rabex 3901 . . . . . . . . 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
7068, 69op1st 5773 . . . . . . . 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)}
7166, 70eqtri 2060 . . . . . . 7 (1st𝐿) = {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}
7264, 71elrab2 2700 . . . . . 6 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
7317, 56, 72sylanbrc 394 . . . . 5 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠 ∈ (1st𝐿))
7473ex 108 . . . 4 (((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) → ((𝑠 +Q 𝑠) <Q 𝑥𝑠 ∈ (1st𝐿)))
7574reximdva 2421 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → (∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥 → ∃𝑠Q 𝑠 ∈ (1st𝐿)))
7616, 75mpd 13 . 2 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑠Q 𝑠 ∈ (1st𝐿))
7714, 76rexlimddv 2437 1 (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  1𝑜c1o 5994  [cec 6104  Ncnpi 6370
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