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Theorem caucvgprprlemmu 7523
 Description: Lemma for caucvgprpr 7540. 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 7143 . . . . 5 1oN
32a1i 9 . . . 4 (𝜑 → 1oN)
41, 3ffvelrnd 5560 . . 3 (𝜑 → (𝐹‘1o) ∈ P)
5 prop 7303 . . 3 ((𝐹‘1o) ∈ P → ⟨(1st ‘(𝐹‘1o)), (2nd ‘(𝐹‘1o))⟩ ∈ P)
6 prmu 7306 . . 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 521 . . . 4 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → 𝑥Q)
9 1nq 7194 . . . 4 1QQ
10 addclnq 7203 . . . 4 ((𝑥Q ∧ 1QQ) → (𝑥 +Q 1Q) ∈ Q)
118, 9, 10sylancl 410 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝑥 +Q 1Q) ∈ Q)
122a1i 9 . . . . 5 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → 1oN)
13 simprr 522 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → 𝑥 ∈ (2nd ‘(𝐹‘1o)))
144adantr 274 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝐹‘1o) ∈ P)
15 nqpru 7380 . . . . . . . . 9 ((𝑥Q ∧ (𝐹‘1o) ∈ P) → (𝑥 ∈ (2nd ‘(𝐹‘1o)) ↔ (𝐹‘1o)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
168, 14, 15syl2anc 409 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝑥 ∈ (2nd ‘(𝐹‘1o)) ↔ (𝐹‘1o)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩))
1713, 16mpbid 146 . . . . . . 7 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝐹‘1o)<P ⟨{𝑝𝑝 <Q 𝑥}, {𝑞𝑥 <Q 𝑞}⟩)
18 ltaprg 7447 . . . . . . . . 9 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
1918adantl 275 . . . . . . . 8 (((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
20 nqprlu 7375 . . . . . . . . 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 7375 . . . . . . . . 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 7406 . . . . . . . . 9 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2524adantl 275 . . . . . . . 8 (((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2619, 14, 21, 23, 25caovord2d 5944 . . . . . . 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 146 . . . . . 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 7179 . . . . . . . . . . . . 13 1Q = [⟨1o, 1o⟩] ~Q
2928fveq2i 5428 . . . . . . . . . . . 12 (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
30 rec1nq 7223 . . . . . . . . . . . 12 (*Q‘1Q) = 1Q
3129, 30eqtr3i 2163 . . . . . . . . . . 11 (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
3231breq2i 3941 . . . . . . . . . 10 (𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑝 <Q 1Q)
3332abbii 2256 . . . . . . . . 9 {𝑝𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑝𝑝 <Q 1Q}
3431breq1i 3940 . . . . . . . . . 10 ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞 ↔ 1Q <Q 𝑞)
3534abbii 2256 . . . . . . . . 9 {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ 1Q <Q 𝑞}
3633, 35opeq12i 3714 . . . . . . . 8 ⟨{𝑝𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩
3736oveq2i 5789 . . . . . . 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 7389 . . . . . . 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 410 . . . . . 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 3964 . . . . 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 5425 . . . . . . . 8 (𝑟 = 1o → (𝐹𝑟) = (𝐹‘1o))
43 opeq1 3709 . . . . . . . . . . . . 13 (𝑟 = 1o → ⟨𝑟, 1o⟩ = ⟨1o, 1o⟩)
4443eceq1d 6469 . . . . . . . . . . . 12 (𝑟 = 1o → [⟨𝑟, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
4544fveq2d 5429 . . . . . . . . . . 11 (𝑟 = 1o → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
4645breq2d 3945 . . . . . . . . . 10 (𝑟 = 1o → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
4746abbidv 2258 . . . . . . . . 9 (𝑟 = 1o → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
4845breq1d 3943 . . . . . . . . . 10 (𝑟 = 1o → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞))
4948abbidv 2258 . . . . . . . . 9 (𝑟 = 1o → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞})
5047, 49opeq12d 3717 . . . . . . . 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 5796 . . . . . . 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 3943 . . . . . 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 2790 . . . . 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 409 . . . 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 3937 . . . . . . . . 9 (𝑢 = (𝑥 +Q 1Q) → (𝑝 <Q 𝑢𝑝 <Q (𝑥 +Q 1Q)))
5655abbidv 2258 . . . . . . . 8 (𝑢 = (𝑥 +Q 1Q) → {𝑝𝑝 <Q 𝑢} = {𝑝𝑝 <Q (𝑥 +Q 1Q)})
57 breq1 3936 . . . . . . . . 9 (𝑢 = (𝑥 +Q 1Q) → (𝑢 <Q 𝑞 ↔ (𝑥 +Q 1Q) <Q 𝑞))
5857abbidv 2258 . . . . . . . 8 (𝑢 = (𝑥 +Q 1Q) → {𝑞𝑢 <Q 𝑞} = {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞})
5956, 58opeq12d 3717 . . . . . . 7 (𝑢 = (𝑥 +Q 1Q) → ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
6059breq2d 3945 . . . . . 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 2439 . . . . 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 5428 . . . . . 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 7191 . . . . . . . 8 Q ∈ V
6564rabex 4076 . . . . . . 7 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
6664rabex 4076 . . . . . . 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 6049 . . . . . 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 2161 . . . . 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 2844 . . . 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 414 . . 3 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝑥 +Q 1Q) ∈ (2nd𝐿))
71 eleq1 2203 . . . 4 (𝑡 = (𝑥 +Q 1Q) → (𝑡 ∈ (2nd𝐿) ↔ (𝑥 +Q 1Q) ∈ (2nd𝐿)))
7271rspcev 2790 . . 3 (((𝑥 +Q 1Q) ∈ Q ∧ (𝑥 +Q 1Q) ∈ (2nd𝐿)) → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
7311, 70, 72syl2anc 409 . 2 ((𝜑 ∧ (𝑥Q𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
747, 73rexlimddv 2555 1 (𝜑 → ∃𝑡Q 𝑡 ∈ (2nd𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  {crab 2421  ⟨cop 3531   class class class wbr 3933  ⟶wf 5123  ‘cfv 5127  (class class class)co 5778  1st c1st 6040  2nd c2nd 6041  1oc1o 6310  [cec 6431  Ncnpi 7100
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