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Theorem caucvgprprlemopu 6854
Description: Lemma for caucvgprpr 6867. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-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
caucvgprprlemopu ((𝜑𝑡 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹,𝑟,𝑠   𝐿,𝑠   𝑝,𝑙,𝑞,𝑡,𝑟,𝑠   𝑢,𝑝,𝑞,𝑡   𝜑,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑡,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑡,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑡,𝑘,𝑛,𝑞,𝑝)   𝐿(𝑢,𝑡,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemopu
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.lim . . . . 5 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}⟩}⟩
21caucvgprprlemelu 6841 . . . 4 (𝑡 ∈ (2nd𝐿) ↔ (𝑡Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))
32simprbi 264 . . 3 (𝑡 ∈ (2nd𝐿) → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
43adantl 266 . 2 ((𝜑𝑡 ∈ (2nd𝐿)) → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
5 simprr 492 . . . . 5 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)
6 caucvgprpr.f . . . . . . . . 9 (𝜑𝐹:NP)
76ffvelrnda 5329 . . . . . . . 8 ((𝜑𝑏N) → (𝐹𝑏) ∈ P)
8 recnnpr 6703 . . . . . . . . 9 (𝑏N → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
98adantl 266 . . . . . . . 8 ((𝜑𝑏N) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
10 addclpr 6692 . . . . . . . 8 (((𝐹𝑏) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
117, 9, 10syl2anc 397 . . . . . . 7 ((𝜑𝑏N) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
1211ad2ant2r 486 . . . . . 6 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
132simplbi 263 . . . . . . . 8 (𝑡 ∈ (2nd𝐿) → 𝑡Q)
1413ad2antlr 466 . . . . . . 7 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → 𝑡Q)
15 nqprlu 6702 . . . . . . 7 (𝑡Q → ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ∈ P)
1614, 15syl 14 . . . . . 6 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ∈ P)
17 ltdfpr 6661 . . . . . 6 ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ∈ P) → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ↔ ∃𝑠Q (𝑠 ∈ (2nd ‘((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))))
1812, 16, 17syl2anc 397 . . . . 5 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩ ↔ ∃𝑠Q (𝑠 ∈ (2nd ‘((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩))))
195, 18mpbid 139 . . . 4 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → ∃𝑠Q (𝑠 ∈ (2nd ‘((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)))
20 simpr 107 . . . . . . . 8 ((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) → 𝑠Q)
2112adantr 265 . . . . . . . 8 ((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
22 nqpru 6707 . . . . . . . 8 ((𝑠Q ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P) → (𝑠 ∈ (2nd ‘((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ↔ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
2320, 21, 22syl2anc 397 . . . . . . 7 ((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) → (𝑠 ∈ (2nd ‘((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ↔ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
24 vex 2577 . . . . . . . . 9 𝑠 ∈ V
25 breq1 3794 . . . . . . . . 9 (𝑝 = 𝑠 → (𝑝 <Q 𝑡𝑠 <Q 𝑡))
26 ltnqex 6704 . . . . . . . . . 10 {𝑝𝑝 <Q 𝑡} ∈ V
27 gtnqex 6705 . . . . . . . . . 10 {𝑞𝑡 <Q 𝑞} ∈ V
2826, 27op1st 5800 . . . . . . . . 9 (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) = {𝑝𝑝 <Q 𝑡}
2924, 25, 28elab2 2712 . . . . . . . 8 (𝑠 ∈ (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) ↔ 𝑠 <Q 𝑡)
3029a1i 9 . . . . . . 7 ((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) → (𝑠 ∈ (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩) ↔ 𝑠 <Q 𝑡))
3123, 30anbi12d 450 . . . . . 6 ((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) → ((𝑠 ∈ (2nd ‘((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ↔ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)))
3231biimpd 136 . . . . 5 ((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) → ((𝑠 ∈ (2nd ‘((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)))
3332reximdva 2438 . . . 4 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → (∃𝑠Q (𝑠 ∈ (2nd ‘((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) ∧ 𝑠 ∈ (1st ‘⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → ∃𝑠Q (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)))
3419, 33mpd 13 . . 3 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → ∃𝑠Q (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡))
35 simprr 492 . . . . . 6 (((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) ∧ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → 𝑠 <Q 𝑡)
36 simplr 490 . . . . . . 7 (((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) ∧ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → 𝑠Q)
37 simplrl 495 . . . . . . . . 9 ((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) → 𝑏N)
3837adantr 265 . . . . . . . 8 (((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) ∧ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → 𝑏N)
39 simprl 491 . . . . . . . 8 (((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) ∧ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩)
40 fveq2 5205 . . . . . . . . . . 11 (𝑟 = 𝑏 → (𝐹𝑟) = (𝐹𝑏))
41 opeq1 3576 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑏 → ⟨𝑟, 1𝑜⟩ = ⟨𝑏, 1𝑜⟩)
4241eceq1d 6172 . . . . . . . . . . . . . . 15 (𝑟 = 𝑏 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑏, 1𝑜⟩] ~Q )
4342fveq2d 5209 . . . . . . . . . . . . . 14 (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))
4443breq2d 3803 . . . . . . . . . . . . 13 (𝑟 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
4544abbidv 2171 . . . . . . . . . . . 12 (𝑟 = 𝑏 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )})
4643breq1d 3801 . . . . . . . . . . . . 13 (𝑟 = 𝑏 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞))
4746abbidv 2171 . . . . . . . . . . . 12 (𝑟 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞})
4845, 47opeq12d 3584 . . . . . . . . . . 11 (𝑟 = 𝑏 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)
4940, 48oveq12d 5557 . . . . . . . . . 10 (𝑟 = 𝑏 → ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
5049breq1d 3801 . . . . . . . . 9 (𝑟 = 𝑏 → (((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ↔ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
5150rspcev 2673 . . . . . . . 8 ((𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩) → ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩)
5238, 39, 51syl2anc 397 . . . . . . 7 (((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) ∧ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩)
531caucvgprprlemelu 6841 . . . . . . 7 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
5436, 52, 53sylanbrc 402 . . . . . 6 (((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) ∧ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → 𝑠 ∈ (2nd𝐿))
5535, 54jca 294 . . . . 5 (((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) ∧ (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡)) → (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))
5655ex 112 . . . 4 ((((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) ∧ 𝑠Q) → ((((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡) → (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿))))
5756reximdva 2438 . . 3 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → (∃𝑠Q (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩ ∧ 𝑠 <Q 𝑡) → ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿))))
5834, 57mpd 13 . 2 (((𝜑𝑡 ∈ (2nd𝐿)) ∧ (𝑏N ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑡}, {𝑞𝑡 <Q 𝑞}⟩)) → ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))
594, 58rexlimddv 2454 1 ((𝜑𝑡 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  {crab 2327  cop 3405   class class class wbr 3791  wf 4925  cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   <N clti 6430   ~Q ceq 6434  Qcnq 6435   +Q cplq 6437  *Qcrq 6439   <Q cltq 6440  Pcnp 6446   +P cpp 6448  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623  df-iltp 6625
This theorem is referenced by:  caucvgprprlemrnd  6856
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