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Theorem caucvgprprlemelu 7589
 Description: Lemma for caucvgprpr 7615. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
Hypothesis
Ref Expression
caucvgprprlemell.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
caucvgprprlemelu (𝑋 ∈ (2nd𝐿) ↔ (𝑋Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩))
Distinct variable groups:   𝐹,𝑏   𝐹,𝑙,𝑟   𝑢,𝐹,𝑟   𝑋,𝑏,𝑝   𝑋,𝑙,𝑟,𝑝   𝑢,𝑋,𝑝   𝑋,𝑞,𝑏   𝑞,𝑙,𝑟   𝑢,𝑞
Allowed substitution hints:   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑟,𝑞,𝑝,𝑏,𝑙)

Proof of Theorem caucvgprprlemelu
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 breq2 3969 . . . . . . 7 (𝑢 = 𝑋 → (𝑝 <Q 𝑢𝑝 <Q 𝑋))
21abbidv 2275 . . . . . 6 (𝑢 = 𝑋 → {𝑝𝑝 <Q 𝑢} = {𝑝𝑝 <Q 𝑋})
3 breq1 3968 . . . . . . 7 (𝑢 = 𝑋 → (𝑢 <Q 𝑞𝑋 <Q 𝑞))
43abbidv 2275 . . . . . 6 (𝑢 = 𝑋 → {𝑞𝑢 <Q 𝑞} = {𝑞𝑋 <Q 𝑞})
52, 4opeq12d 3749 . . . . 5 (𝑢 = 𝑋 → ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩)
65breq2d 3977 . . . 4 (𝑢 = 𝑋 → (((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}, {𝑞𝑋 <Q 𝑞}⟩))
76rexbidv 2458 . . 3 (𝑢 = 𝑋 → (∃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}, {𝑞𝑋 <Q 𝑞}⟩))
8 caucvgprprlemell.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}⟩}⟩
98fveq2i 5468 . . . 4 (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}⟩}⟩)
10 nqex 7266 . . . . . 6 Q ∈ V
1110rabex 4108 . . . . 5 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ V
1210rabex 4108 . . . . 5 {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
1311, 12op2nd 6089 . . . 4 (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}⟩}
149, 13eqtri 2178 . . 3 (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}⟩}
157, 14elrab2 2871 . 2 (𝑋 ∈ (2nd𝐿) ↔ (𝑋Q ∧ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩))
16 fveq2 5465 . . . . . . 7 (𝑟 = 𝑎 → (𝐹𝑟) = (𝐹𝑎))
17 opeq1 3741 . . . . . . . . . . . 12 (𝑟 = 𝑎 → ⟨𝑟, 1o⟩ = ⟨𝑎, 1o⟩)
1817eceq1d 6509 . . . . . . . . . . 11 (𝑟 = 𝑎 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
1918fveq2d 5469 . . . . . . . . . 10 (𝑟 = 𝑎 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
2019breq2d 3977 . . . . . . . . 9 (𝑟 = 𝑎 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
2120abbidv 2275 . . . . . . . 8 (𝑟 = 𝑎 → {𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )})
2219breq1d 3975 . . . . . . . . 9 (𝑟 = 𝑎 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞))
2322abbidv 2275 . . . . . . . 8 (𝑟 = 𝑎 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞})
2421, 23opeq12d 3749 . . . . . . 7 (𝑟 = 𝑎 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)
2516, 24oveq12d 5836 . . . . . 6 (𝑟 = 𝑎 → ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹𝑎) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩))
2625breq1d 3975 . . . . 5 (𝑟 = 𝑎 → (((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩ ↔ ((𝐹𝑎) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩))
2726cbvrexv 2681 . . . 4 (∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩ ↔ ∃𝑎N ((𝐹𝑎) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩)
28 fveq2 5465 . . . . . . 7 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
29 opeq1 3741 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ⟨𝑎, 1o⟩ = ⟨𝑏, 1o⟩)
3029eceq1d 6509 . . . . . . . . . . 11 (𝑎 = 𝑏 → [⟨𝑎, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
3130fveq2d 5469 . . . . . . . . . 10 (𝑎 = 𝑏 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
3231breq2d 3977 . . . . . . . . 9 (𝑎 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
3332abbidv 2275 . . . . . . . 8 (𝑎 = 𝑏 → {𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )})
3431breq1d 3975 . . . . . . . . 9 (𝑎 = 𝑏 → ((*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞))
3534abbidv 2275 . . . . . . . 8 (𝑎 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞})
3633, 35opeq12d 3749 . . . . . . 7 (𝑎 = 𝑏 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)
3728, 36oveq12d 5836 . . . . . 6 (𝑎 = 𝑏 → ((𝐹𝑎) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩))
3837breq1d 3975 . . . . 5 (𝑎 = 𝑏 → (((𝐹𝑎) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩ ↔ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩))
3938cbvrexv 2681 . . . 4 (∃𝑎N ((𝐹𝑎) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩ ↔ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩)
4027, 39bitri 183 . . 3 (∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩ ↔ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩)
4140anbi2i 453 . 2 ((𝑋Q ∧ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩) ↔ (𝑋Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩))
4215, 41bitri 183 1 (𝑋 ∈ (2nd𝐿) ↔ (𝑋Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑋}, {𝑞𝑋 <Q 𝑞}⟩))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128  {cab 2143  ∃wrex 2436  {crab 2439  ⟨cop 3563   class class class wbr 3965  ‘cfv 5167  (class class class)co 5818  2nd c2nd 6081  1oc1o 6350  [cec 6471  Ncnpi 7175   ~Q ceq 7182  Qcnq 7183   +Q cplq 7185  *Qcrq 7187
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