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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   <N clti 7103   ~Q ceq 7107  Qcnq 7108  1Qc1q 7109   +Q cplq 7110  *Qcrq 7112   <Q cltq 7113  Pcnp 7119   +P cpp 7121  <P cltp 7123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-2o 6318  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-mpq 7173  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-mqqs 7178  df-1nqqs 7179  df-rq 7180  df-ltnqqs 7181  df-enq0 7252  df-nq0 7253  df-0nq0 7254  df-plq0 7255  df-mq0 7256  df-inp 7294  df-iplp 7296  df-iltp 7298
This theorem is referenced by:  caucvgprprlemm  7524
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