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Theorem caucvgprprlemmu 6947
Description: Lemma for caucvgprpr 6964. 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 6567 . . . . 5 1𝑜N
32a1i 9 . . . 4 (𝜑 → 1𝑜N)
41, 3ffvelrnd 5335 . . 3 (𝜑 → (𝐹‘1𝑜) ∈ P)
5 prop 6727 . . 3 ((𝐹‘1𝑜) ∈ P → ⟨(1st ‘(𝐹‘1𝑜)), (2nd ‘(𝐹‘1𝑜))⟩ ∈ P)
6 prmu 6730 . . 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 498 . . . 4 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → 𝑥Q)
9 1nq 6618 . . . 4 1QQ
10 addclnq 6627 . . . 4 ((𝑥Q ∧ 1QQ) → (𝑥 +Q 1Q) ∈ Q)
118, 9, 10sylancl 404 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝑥 +Q 1Q) ∈ Q)
122a1i 9 . . . . 5 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → 1𝑜N)
13 simprr 499 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → 𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))
144adantr 270 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝐹‘1𝑜) ∈ P)
15 nqpru 6804 . . . . . . . . 9 ((𝑥Q ∧ (𝐹‘1𝑜) ∈ P) → (𝑥 ∈ (2nd ‘(𝐹‘1𝑜)) ↔ (𝐹‘1𝑜)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
168, 14, 15syl2anc 403 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝑥 ∈ (2nd ‘(𝐹‘1𝑜)) ↔ (𝐹‘1𝑜)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
1713, 16mpbid 145 . . . . . . 7 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝐹‘1𝑜)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
18 ltaprg 6871 . . . . . . . . 9 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
1918adantl 271 . . . . . . . 8 (((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
20 nqprlu 6799 . . . . . . . . 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 6799 . . . . . . . . 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 6830 . . . . . . . . 9 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2524adantl 271 . . . . . . . 8 (((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2619, 14, 21, 23, 25caovord2d 5701 . . . . . . 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 145 . . . . . 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 6603 . . . . . . . . . . . . 13 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
2928fveq2i 5212 . . . . . . . . . . . 12 (*Q‘1Q) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )
30 rec1nq 6647 . . . . . . . . . . . 12 (*Q‘1Q) = 1Q
3129, 30eqtr3i 2104 . . . . . . . . . . 11 (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) = 1Q
3231breq2i 3801 . . . . . . . . . 10 (𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q 1Q)
3332abbii 2195 . . . . . . . . 9 {𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q 1Q}
3431breq1i 3800 . . . . . . . . . 10 ((*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ 1Q <Q 𝑞)
3534abbii 2195 . . . . . . . . 9 {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ 1Q <Q 𝑞}
3633, 35opeq12i 3583 . . . . . . . 8 ⟨{𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩
3736oveq2i 5554 . . . . . . 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 6813 . . . . . . 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 404 . . . . . 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 3823 . . . . 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 5209 . . . . . . . 8 (𝑟 = 1𝑜 → (𝐹𝑟) = (𝐹‘1𝑜))
43 opeq1 3578 . . . . . . . . . . . . 13 (𝑟 = 1𝑜 → ⟨𝑟, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
4443eceq1d 6208 . . . . . . . . . . . 12 (𝑟 = 1𝑜 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
4544fveq2d 5213 . . . . . . . . . . 11 (𝑟 = 1𝑜 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ))
4645breq2d 3805 . . . . . . . . . 10 (𝑟 = 1𝑜 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )))
4746abbidv 2197 . . . . . . . . 9 (𝑟 = 1𝑜 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q )})
4845breq1d 3803 . . . . . . . . . 10 (𝑟 = 1𝑜 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞))
4948abbidv 2197 . . . . . . . . 9 (𝑟 = 1𝑜 → {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨1𝑜, 1𝑜⟩] ~Q ) <Q 𝑞})
5047, 49opeq12d 3586 . . . . . . . 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 5561 . . . . . . 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 3803 . . . . . 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 2702 . . . . 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 403 . . . 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 3797 . . . . . . . . 9 (𝑢 = (𝑥 +Q 1Q) → (𝑝 <Q 𝑢𝑝 <Q (𝑥 +Q 1Q)))
5655abbidv 2197 . . . . . . . 8 (𝑢 = (𝑥 +Q 1Q) → {𝑝𝑝 <Q 𝑢} = {𝑝𝑝 <Q (𝑥 +Q 1Q)})
57 breq1 3796 . . . . . . . . 9 (𝑢 = (𝑥 +Q 1Q) → (𝑢 <Q 𝑞 ↔ (𝑥 +Q 1Q) <Q 𝑞))
5857abbidv 2197 . . . . . . . 8 (𝑢 = (𝑥 +Q 1Q) → {𝑞𝑢 <Q 𝑞} = {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞})
5956, 58opeq12d 3586 . . . . . . 7 (𝑢 = (𝑥 +Q 1Q) → ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
6059breq2d 3805 . . . . . 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 2370 . . . . 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 5212 . . . . . 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 6615 . . . . . . . 8 Q ∈ V
6564rabex 3930 . . . . . . 7 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
6664rabex 3930 . . . . . . 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 5805 . . . . . 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 2102 . . . . 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 2752 . . . 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 408 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → (𝑥 +Q 1Q) ∈ (2nd𝐿))
71 eleq1 2142 . . . 4 (𝑡 = (𝑥 +Q 1Q) → (𝑡 ∈ (2nd𝐿) ↔ (𝑥 +Q 1Q) ∈ (2nd𝐿)))
7271rspcev 2702 . . 3 (((𝑥 +Q 1Q) ∈ Q ∧ (𝑥 +Q 1Q) ∈ (2nd𝐿)) → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
7311, 70, 72syl2anc 403 . 2 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1𝑜)))) → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
747, 73rexlimddv 2482 1 (𝜑 → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  {cab 2068  wral 2349  wrex 2350  {crab 2353  cop 3409   class class class wbr 3793  wf 4928  cfv 4932  (class class class)co 5543  1st c1st 5796  2nd c2nd 5797  1𝑜c1o 6058  [cec 6170  Ncnpi 6524   <N clti 6527   ~Q ceq 6531  Qcnq 6532  1Qc1q 6533   +Q cplq 6534  *Qcrq 6536   <Q cltq 6537  Pcnp 6543   +P cpp 6545  <P cltp 6547
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  caucvgprprlemm  6948
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