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Theorem caucvgprprlemml 7197
Description: Lemma for caucvgprpr 7215. 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 5268 . . . . . 6 (𝑚 = 1𝑜 → (𝐹𝑚) = (𝐹‘1𝑜))
21breq2d 3832 . . . . 5 (𝑚 = 1𝑜 → (𝐴<P (𝐹𝑚) ↔ 𝐴<P (𝐹‘1𝑜)))
3 caucvgprpr.bnd . . . . 5 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
4 1pi 6818 . . . . . 6 1𝑜N
54a1i 9 . . . . 5 (𝜑 → 1𝑜N)
62, 3, 5rspcdva 2720 . . . 4 (𝜑𝐴<P (𝐹‘1𝑜))
7 ltrelpr 7008 . . . . . 6 <P ⊆ (P × P)
87brel 4458 . . . . 5 (𝐴<P (𝐹‘1𝑜) → (𝐴P ∧ (𝐹‘1𝑜) ∈ P))
98simpld 110 . . . 4 (𝐴<P (𝐹‘1𝑜) → 𝐴P)
106, 9syl 14 . . 3 (𝜑𝐴P)
11 prop 6978 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 prml 6980 . . . 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 6917 . . . 4 (𝑥Q → ∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥)
1615ad2antrl 474 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → ∃𝑠Q (𝑠 +Q 𝑠) <Q 𝑥)
17 simplr 497 . . . . . 6 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠Q)
18 archrecnq 7166 . . . . . . . 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 108 . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑟N)
22 nnnq 6925 . . . . . . . . . . . . . . . 16 (𝑟N → [⟨𝑟, 1𝑜⟩] ~QQ)
23 recclnq 6895 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑠Q)
26 ltanqg 6903 . . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . . 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 145 . . . . . . . . . . . . 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 501 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) <Q 𝑥)
30 ltsonq 6901 . . . . . . . . . . . . . 14 <Q Or Q
31 ltrelnq 6868 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
3230, 31sotri 4794 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝐴P)
35 simprr 499 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) → 𝑥 ∈ (1st𝐴))
3635ad4antr 478 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝑥 ∈ (1st𝐴))
37 prcdnql 6987 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ∈ (1st𝐴)))
3811, 37sylan 277 . . . . . . . . . . . . 13 ((𝐴P𝑥 ∈ (1st𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ∈ (1st𝐴)))
3934, 36, 38syl2anc 403 . . . . . . . . . . . 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 6878 . . . . . . . . . . . . 13 ((𝑠Q ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ∈ Q)
4225, 24, 41syl2anc 403 . . . . . . . . . . . 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 7054 . . . . . . . . . . . 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 403 . . . . . . . . . . 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 145 . . . . . . . . . 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 5268 . . . . . . . . . . . 12 (𝑚 = 𝑟 → (𝐹𝑚) = (𝐹𝑟))
4746breq2d 3832 . . . . . . . . . . 11 (𝑚 = 𝑟 → (𝐴<P (𝐹𝑚) ↔ 𝐴<P (𝐹𝑟)))
483ad5antr 480 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → ∀𝑚N 𝐴<P (𝐹𝑚))
4947, 48, 21rspcdva 2720 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟N) ∧ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑠) → 𝐴<P (𝐹𝑟))
50 ltsopr 7099 . . . . . . . . . . 11 <P Or P
5150, 7sotri 4794 . . . . . . . . . 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 403 . . . . . . . . 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 113 . . . . . . . 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 2471 . . . . . . 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 5620 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
5756breq2d 3832 . . . . . . . . . . 11 (𝑙 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))))
5857abbidv 2202 . . . . . . . . . 10 (𝑙 = 𝑠 → {𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))})
5956breq1d 3830 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞))
6059abbidv 2202 . . . . . . . . . 10 (𝑙 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞})
6158, 60opeq12d 3613 . . . . . . . . 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 3830 . . . . . . . 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 2377 . . . . . . 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 5271 . . . . . . . 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 6866 . . . . . . . . . 10 Q ∈ V
6766rabex 3958 . . . . . . . . 9 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
6866rabex 3958 . . . . . . . . 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 5874 . . . . . . . 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 2105 . . . . . . 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 2765 . . . . . 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 408 . . . . 5 ((((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠 ∈ (1st𝐿))
7372ex 113 . . . 4 (((𝜑 ∧ (𝑥Q𝑥 ∈ (1st𝐴))) ∧ 𝑠Q) → ((𝑠 +Q 𝑠) <Q 𝑥𝑠 ∈ (1st𝐿)))
7473reximdva 2471 . . 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 2489 1 (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  {cab 2071  wral 2355  wrex 2356  {crab 2359  cop 3434   class class class wbr 3820  wf 4977  cfv 4981  (class class class)co 5613  1st c1st 5866  2nd c2nd 5867  1𝑜c1o 6128  [cec 6242  Ncnpi 6775   <N clti 6778   ~Q ceq 6782  Qcnq 6783   +Q cplq 6785  *Qcrq 6787   <Q cltq 6788  Pcnp 6794   +P cpp 6796  <P cltp 6798
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-inp 6969  df-iltp 6973
This theorem is referenced by:  caucvgprprlemm  7199
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