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Theorem caucvgprprlemmu 6791
Description: Lemma for caucvgprpr 6808. The upper 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
caucvgprprlemmu (𝜑 → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟,𝑚   𝐹,𝑟,𝑢   𝑡,𝐿   𝑞,𝑝,𝑟,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑡,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑡,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑡,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐿(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemmu
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . . . 4 (𝜑𝐹:NP)
2 1pi 6411 . . . . 5 1𝑜N
32a1i 9 . . . 4 (𝜑 → 1𝑜N)
41, 3ffvelrnd 5303 . . 3 (𝜑 → (𝐹‘1𝑜) ∈ P)
5 prop 6571 . . 3 ((𝐹‘1𝑜) ∈ P → ⟨(1st ‘(𝐹‘1𝑜)), (2nd ‘(𝐹‘1𝑜))⟩ ∈ P)
6 prmu 6574 . . 3 (⟨(1st ‘(𝐹‘1𝑜)), (2nd ‘(𝐹‘1𝑜))⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))
74, 5, 63syl 17 . 2 (𝜑 → ∃𝑥Q 𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))
8 simprl 483 . . . 4 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → 𝑥Q)
9 1nq 6462 . . . 4 1QQ
10 addclnq 6471 . . . 4 ((𝑥Q ∧ 1QQ) → (𝑥 +Q 1Q) ∈ Q)
118, 9, 10sylancl 392 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝑥 +Q 1Q) ∈ Q)
122a1i 9 . . . . 5 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → 1𝑜N)
13 simprr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → 𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))
144adantr 261 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝐹‘1𝑜) ∈ P)
15 nqpru 6648 . . . . . . . . 9 ((𝑥Q ∧ (𝐹‘1𝑜) ∈ P) → (𝑥 ∈ (2nd ‘(𝐹‘1𝑜)) ↔ (𝐹‘1𝑜)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
168, 14, 15syl2anc 391 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝑥 ∈ (2nd ‘(𝐹‘1𝑜)) ↔ (𝐹‘1𝑜)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
1713, 16mpbid 135 . . . . . . 7 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝐹‘1𝑜)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
18 ltaprg 6715 . . . . . . . . 9 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
1918adantl 262 . . . . . . . 8 (((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
20 nqprlu 6643 . . . . . . . . 9 (𝑥Q → ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩ ∈ P)
218, 20syl 14 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩ ∈ P)
22 nqprlu 6643 . . . . . . . . 9 (1QQ → ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩ ∈ P)
239, 22mp1i 10 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩ ∈ P)
24 addcomprg 6674 . . . . . . . . 9 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2524adantl 262 . . . . . . . 8 (((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2619, 14, 21, 23, 25caovord2d 5670 . . . . . . 7 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ((𝐹‘1𝑜)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩ ↔ ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩)<P (⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩)))
2717, 26mpbid 135 . . . . . 6 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩)<P (⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩))
28 df-1nqqs 6447 . . . . . . . . . . . . 13 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
2928fveq2i 5181 . . . . . . . . . . . 12 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
30 rec1nq 6491 . . . . . . . . . . . 12 (*Q‘1Q) = 1Q
3129, 30eqtr3i 2062 . . . . . . . . . . 11 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
3231breq2i 3772 . . . . . . . . . 10 (𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q 1Q)
3332abbii 2153 . . . . . . . . 9 {𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q 1Q}
3431breq1i 3771 . . . . . . . . . 10 ((*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ 1Q <Q 𝑞)
3534abbii 2153 . . . . . . . . 9 {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ 1Q <Q 𝑞}
3633, 35opeq12i 3554 . . . . . . . 8 ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩
3736oveq2i 5523 . . . . . . 7 ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩)
3837a1i 9 . . . . . 6 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩))
39 addnqpr 6657 . . . . . . 7 ((𝑥Q ∧ 1QQ) → ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩ = (⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩))
408, 9, 39sylancl 392 . . . . . 6 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩ = (⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩))
4127, 38, 403brtr4d 3794 . . . . 5 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
42 fveq2 5178 . . . . . . . 8 (𝑟 = 1𝑜 → (𝐹𝑟) = (𝐹‘1𝑜))
43 opeq1 3549 . . . . . . . . . . . . 13 (𝑟 = 1𝑜 → ⟨𝑟, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
4443eceq1d 6142 . . . . . . . . . . . 12 (𝑟 = 1𝑜 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
4544fveq2d 5182 . . . . . . . . . . 11 (𝑟 = 1𝑜 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
4645breq2d 3776 . . . . . . . . . 10 (𝑟 = 1𝑜 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
4746abbidv 2155 . . . . . . . . 9 (𝑟 = 1𝑜 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )})
4845breq1d 3774 . . . . . . . . . 10 (𝑟 = 1𝑜 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞))
4948abbidv 2155 . . . . . . . . 9 (𝑟 = 1𝑜 → {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞})
5047, 49opeq12d 3557 . . . . . . . 8 (𝑟 = 1𝑜 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)
5142, 50oveq12d 5530 . . . . . . 7 (𝑟 = 1𝑜 → ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
5251breq1d 3774 . . . . . 6 (𝑟 = 1𝑜 → (((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩ ↔ ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩))
5352rspcev 2656 . . . . 5 ((1𝑜N ∧ ((𝐹‘1𝑜) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩) → ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
5412, 41, 53syl2anc 391 . . . 4 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
55 breq2 3768 . . . . . . . . 9 (𝑢 = (𝑥 +Q 1Q) → (𝑝 <Q 𝑢𝑝 <Q (𝑥 +Q 1Q)))
5655abbidv 2155 . . . . . . . 8 (𝑢 = (𝑥 +Q 1Q) → {𝑝𝑝 <Q 𝑢} = {𝑝𝑝 <Q (𝑥 +Q 1Q)})
57 breq1 3767 . . . . . . . . 9 (𝑢 = (𝑥 +Q 1Q) → (𝑢 <Q 𝑞 ↔ (𝑥 +Q 1Q) <Q 𝑞))
5857abbidv 2155 . . . . . . . 8 (𝑢 = (𝑥 +Q 1Q) → {𝑞𝑢 <Q 𝑞} = {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞})
5956, 58opeq12d 3557 . . . . . . 7 (𝑢 = (𝑥 +Q 1Q) → ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
6059breq2d 3776 . . . . . 6 (𝑢 = (𝑥 +Q 1Q) → (((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ ↔ ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩))
6160rexbidv 2327 . . . . 5 (𝑢 = (𝑥 +Q 1Q) → (∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ ↔ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩))
62 caucvgprpr.lim . . . . . . 7 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}⟩}⟩
6362fveq2i 5181 . . . . . 6 (2nd𝐿) = (2nd ‘⟨{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}⟩}⟩)
64 nqex 6459 . . . . . . . 8 Q ∈ V
6564rabex 3901 . . . . . . 7 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
6664rabex 3901 . . . . . . 7 {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
6765, 66op2nd 5774 . . . . . 6 (2nd ‘⟨{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}⟩}⟩) = {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}
6863, 67eqtri 2060 . . . . 5 (2nd𝐿) = {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}
6961, 68elrab2 2700 . . . 4 ((𝑥 +Q 1Q) ∈ (2nd𝐿) ↔ ((𝑥 +Q 1Q) ∈ Q ∧ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩))
7011, 54, 69sylanbrc 394 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝑥 +Q 1Q) ∈ (2nd𝐿))
71 eleq1 2100 . . . 4 (𝑡 = (𝑥 +Q 1Q) → (𝑡 ∈ (2nd𝐿) ↔ (𝑥 +Q 1Q) ∈ (2nd𝐿)))
7271rspcev 2656 . . 3 (((𝑥 +Q 1Q) ∈ Q ∧ (𝑥 +Q 1Q) ∈ (2nd𝐿)) → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
7311, 70, 72syl2anc 391 . 2 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
747, 73rexlimddv 2437 1 (𝜑 → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = 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 6368   <N clti 6371   ~Q ceq 6375  Qcnq 6376  1Qc1q 6377   +Q cplq 6378  *Qcrq 6380   <Q cltq 6381  Pcnp 6387   +P cpp 6389  <P cltp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by:  caucvgprprlemm  6792
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