Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlemcl GIF version

Theorem caucvgprlemcl 6831
 Description: Lemma for caucvgpr 6837. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
Hypotheses
Ref Expression
caucvgpr.f (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
Assertion
Ref Expression
caucvgprlemcl (𝜑𝐿P)
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑙   𝑢,𝐹,𝑗   𝑛,𝐹,𝑘   𝑗,𝑘,𝐿   𝑘,𝑛
Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝐿(𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlemcl
Dummy variables 𝑠 𝑎 𝑐 𝑑 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . . 4 (𝜑𝐹:NQ)
2 caucvgpr.cau . . . 4 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3 caucvgpr.bnd . . . . 5 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
4 fveq2 5205 . . . . . . 7 (𝑗 = 𝑎 → (𝐹𝑗) = (𝐹𝑎))
54breq2d 3803 . . . . . 6 (𝑗 = 𝑎 → (𝐴 <Q (𝐹𝑗) ↔ 𝐴 <Q (𝐹𝑎)))
65cbvralv 2550 . . . . 5 (∀𝑗N 𝐴 <Q (𝐹𝑗) ↔ ∀𝑎N 𝐴 <Q (𝐹𝑎))
73, 6sylib 131 . . . 4 (𝜑 → ∀𝑎N 𝐴 <Q (𝐹𝑎))
8 caucvgpr.lim . . . . 5 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
9 opeq1 3576 . . . . . . . . . . . . 13 (𝑗 = 𝑎 → ⟨𝑗, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
109eceq1d 6172 . . . . . . . . . . . 12 (𝑗 = 𝑎 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
1110fveq2d 5209 . . . . . . . . . . 11 (𝑗 = 𝑎 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
1211oveq2d 5555 . . . . . . . . . 10 (𝑗 = 𝑎 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1312, 4breq12d 3804 . . . . . . . . 9 (𝑗 = 𝑎 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)))
1413cbvrexv 2551 . . . . . . . 8 (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎))
1514a1i 9 . . . . . . 7 (𝑙Q → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)))
1615rabbiia 2564 . . . . . 6 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} = {𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)}
174, 11oveq12d 5557 . . . . . . . . . 10 (𝑗 = 𝑎 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1817breq1d 3801 . . . . . . . . 9 (𝑗 = 𝑎 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢))
1918cbvrexv 2551 . . . . . . . 8 (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢)
2019a1i 9 . . . . . . 7 (𝑢Q → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢))
2120rabbiia 2564 . . . . . 6 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} = {𝑢Q ∣ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}
2216, 21opeq12i 3581 . . . . 5 ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = ⟨{𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)}, {𝑢Q ∣ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
238, 22eqtri 2076 . . . 4 𝐿 = ⟨{𝑙Q ∣ ∃𝑎N (𝑙 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q (𝐹𝑎)}, {𝑢Q ∣ ∃𝑎N ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
241, 2, 7, 23caucvgprlemm 6823 . . 3 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿)))
25 ssrab2 3052 . . . . . 6 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ⊆ Q
26 nqex 6518 . . . . . . 7 Q ∈ V
2726elpw2 3938 . . . . . 6 ({𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ 𝒫 Q ↔ {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ⊆ Q)
2825, 27mpbir 138 . . . . 5 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ 𝒫 Q
29 ssrab2 3052 . . . . . 6 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ⊆ Q
3026elpw2 3938 . . . . . 6 ({𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q ↔ {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ⊆ Q)
3129, 30mpbir 138 . . . . 5 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q
32 opelxpi 4403 . . . . 5 (({𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ 𝒫 Q ∧ {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ 𝒫 Q) → ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ ∈ (𝒫 Q × 𝒫 Q))
3328, 31, 32mp2an 410 . . . 4 ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ ∈ (𝒫 Q × 𝒫 Q)
348, 33eqeltri 2126 . . 3 𝐿 ∈ (𝒫 Q × 𝒫 Q)
3524, 34jctil 299 . 2 (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿))))
361, 2, 7, 23caucvgprlemrnd 6828 . . 3 (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))))
37 breq1 3794 . . . . . . 7 (𝑛 = 𝑐 → (𝑛 <N 𝑘𝑐 <N 𝑘))
38 fveq2 5205 . . . . . . . . 9 (𝑛 = 𝑐 → (𝐹𝑛) = (𝐹𝑐))
39 opeq1 3576 . . . . . . . . . . . 12 (𝑛 = 𝑐 → ⟨𝑛, 1𝑜⟩ = ⟨𝑐, 1𝑜⟩)
4039eceq1d 6172 . . . . . . . . . . 11 (𝑛 = 𝑐 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝑐, 1𝑜⟩] ~Q )
4140fveq2d 5209 . . . . . . . . . 10 (𝑛 = 𝑐 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))
4241oveq2d 5555 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
4338, 42breq12d 3804 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
4438, 41oveq12d 5557 . . . . . . . . 9 (𝑛 = 𝑐 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
4544breq2d 3803 . . . . . . . 8 (𝑛 = 𝑐 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
4643, 45anbi12d 450 . . . . . . 7 (𝑛 = 𝑐 → (((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))))
4737, 46imbi12d 227 . . . . . 6 (𝑛 = 𝑐 → ((𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) ↔ (𝑐 <N 𝑘 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))))
48 breq2 3795 . . . . . . 7 (𝑘 = 𝑑 → (𝑐 <N 𝑘𝑐 <N 𝑑))
49 fveq2 5205 . . . . . . . . . 10 (𝑘 = 𝑑 → (𝐹𝑘) = (𝐹𝑑))
5049oveq1d 5554 . . . . . . . . 9 (𝑘 = 𝑑 → ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) = ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))
5150breq2d 3803 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ↔ (𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
5249breq1d 3801 . . . . . . . 8 (𝑘 = 𝑑 → ((𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ↔ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))
5351, 52anbi12d 450 . . . . . . 7 (𝑘 = 𝑑 → (((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∧ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))))
5448, 53imbi12d 227 . . . . . 6 (𝑘 = 𝑑 → ((𝑐 <N 𝑘 → ((𝐹𝑐) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))) ↔ (𝑐 <N 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∧ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q ))))))
5547, 54cbvral2v 2558 . . . . 5 (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) ↔ ∀𝑐N𝑑N (𝑐 <N 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∧ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))))
562, 55sylib 131 . . . 4 (𝜑 → ∀𝑐N𝑑N (𝑐 <N 𝑑 → ((𝐹𝑐) <Q ((𝐹𝑑) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )) ∧ (𝐹𝑑) <Q ((𝐹𝑐) +Q (*Q‘[⟨𝑐, 1𝑜⟩] ~Q )))))
571, 56, 7, 23caucvgprlemdisj 6829 . . 3 (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
581, 2, 7, 23caucvgprlemloc 6830 . . 3 (𝜑 → ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))
5936, 57, 583jca 1095 . 2 (𝜑 → ((∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))) ∧ ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) ∧ ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿)))))
60 elnp1st2nd 6631 . 2 (𝐿P ↔ ((𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐿))) ∧ ((∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑟Q (𝑠 <Q 𝑟𝑟 ∈ (1st𝐿))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)))) ∧ ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) ∧ ∀𝑠Q𝑟Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st𝐿) ∨ 𝑟 ∈ (2nd𝐿))))))
6135, 59, 60sylanbrc 402 1 (𝜑𝐿P)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∨ wo 639   ∧ w3a 896   = wceq 1259   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324  {crab 2327   ⊆ wss 2944  𝒫 cpw 3386  ⟨cop 3405   class class class wbr 3791   × cxp 4370  ⟶wf 4925  ‘cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  1𝑜c1o 6024  [cec 6134  Ncnpi 6427
 Copyright terms: Public domain W3C validator