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

Theorem caucvgpr 6780
 Description: A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). This proof (including its lemmas) is similar to the proofs of cauappcvgpr 6760 and caucvgprpr 6810. Reading cauappcvgpr 6760 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-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))
Assertion
Ref Expression
caucvgpr (𝜑 → ∃𝑦P𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑘,𝑛,𝑙,𝑢,𝑥,𝑦   𝜑,𝑗,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑢,𝑛,𝑙)   𝐴(𝑥,𝑦,𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgpr
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . 3 (𝜑𝐹:NQ)
2 caucvgpr.cau . . 3 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3 caucvgpr.bnd . . 3 (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
4 opeq1 3549 . . . . . . . . . . 11 (𝑧 = 𝑗 → ⟨𝑧, 1𝑜⟩ = ⟨𝑗, 1𝑜⟩)
54eceq1d 6142 . . . . . . . . . 10 (𝑧 = 𝑗 → [⟨𝑧, 1𝑜⟩] ~Q = [⟨𝑗, 1𝑜⟩] ~Q )
65fveq2d 5182 . . . . . . . . 9 (𝑧 = 𝑗 → (*Q‘[⟨𝑧, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ))
76oveq2d 5528 . . . . . . . 8 (𝑧 = 𝑗 → (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
8 fveq2 5178 . . . . . . . 8 (𝑧 = 𝑗 → (𝐹𝑧) = (𝐹𝑗))
97, 8breq12d 3777 . . . . . . 7 (𝑧 = 𝑗 → ((𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧) ↔ (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
109cbvrexv 2534 . . . . . 6 (∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧) ↔ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
1110a1i 9 . . . . 5 (𝑙Q → (∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧) ↔ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1211rabbiia 2547 . . . 4 {𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)} = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
138, 6oveq12d 5530 . . . . . . . 8 (𝑧 = 𝑗 → ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) = ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
1413breq1d 3774 . . . . . . 7 (𝑧 = 𝑗 → (((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢))
1514cbvrexv 2534 . . . . . 6 (∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢)
1615a1i 9 . . . . 5 (𝑢Q → (∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢))
1716rabbiia 2547 . . . 4 {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢} = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
1812, 17opeq12i 3554 . . 3 ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
191, 2, 3, 18caucvgprlemcl 6774 . 2 (𝜑 → ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ ∈ P)
201, 2, 3, 18caucvgprlemlim 6779 . 2 (𝜑 → ∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
21 oveq1 5519 . . . . . . . 8 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ → (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) = (⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩))
2221breq2d 3776 . . . . . . 7 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩)))
23 breq1 3767 . . . . . . 7 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ → (𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩ ↔ ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩))
2422, 23anbi12d 442 . . . . . 6 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ → ((⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩) ↔ (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
2524imbi2d 219 . . . . 5 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ → ((𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)) ↔ (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩))))
2625rexralbidv 2350 . . . 4 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ → (∃𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)) ↔ ∃𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩))))
2726ralbidv 2326 . . 3 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ → (∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)) ↔ ∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩))))
2827rspcev 2656 . 2 ((⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ ∈ P ∧ ∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧N (𝑙 +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧N ((𝐹𝑧) +Q (*Q‘[⟨𝑧, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩))) → ∃𝑦P𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
2919, 20, 28syl2anc 391 1 (𝜑 → ∃𝑦P𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6370
 Copyright terms: Public domain W3C validator