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Theorem caucvgpr 6653
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 fractions which converge to zero (which is not a positive real). (Contributed by Jim Kingdon, 18-Jun-2020.)
Hypotheses
Ref Expression
caucvgpr.f (φ𝐹:NQ)
caucvgpr.cau (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (φ𝑗 N A <Q (𝐹𝑗))
Assertion
Ref Expression
caucvgpr (φy P x Q 𝑗 N 𝑘 N (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 (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3 caucvgpr.bnd . . 3 (φ𝑗 N A <Q (𝐹𝑗))
4 opeq1 3540 . . . . . . . . . . 11 (z = 𝑗 → ⟨z, 1𝑜⟩ = ⟨𝑗, 1𝑜⟩)
54eceq1d 6078 . . . . . . . . . 10 (z = 𝑗 → [⟨z, 1𝑜⟩] ~Q = [⟨𝑗, 1𝑜⟩] ~Q )
65fveq2d 5125 . . . . . . . . 9 (z = 𝑗 → (*Q‘[⟨z, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ))
76oveq2d 5471 . . . . . . . 8 (z = 𝑗 → (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
8 fveq2 5121 . . . . . . . 8 (z = 𝑗 → (𝐹z) = (𝐹𝑗))
97, 8breq12d 3768 . . . . . . 7 (z = 𝑗 → ((𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z) ↔ (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
109cbvrexv 2528 . . . . . 6 (z N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z) ↔ 𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
1110a1i 9 . . . . 5 (𝑙 Q → (z N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z) ↔ 𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1211rabbiia 2541 . . . 4 {𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)} = {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
138, 6oveq12d 5473 . . . . . . . 8 (z = 𝑗 → ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) = ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
1413breq1d 3765 . . . . . . 7 (z = 𝑗 → (((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u))
1514cbvrexv 2528 . . . . . 6 (z N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u)
1615a1i 9 . . . . 5 (u Q → (z N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u))
1716rabbiia 2541 . . . 4 {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u} = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
1812, 17opeq12i 3545 . . 3 ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
191, 2, 3, 18caucvgprlemcl 6647 . 2 (φ → ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ P)
201, 2, 3, 18caucvgprlemlim 6652 . 2 (φx Q 𝑗 N 𝑘 N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {u ∣ (𝐹𝑘) <Q u}⟩<P (⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩) ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q x)}, {u ∣ ((𝐹𝑘) +Q x) <Q u}⟩)))
21 oveq1 5462 . . . . . . . 8 (y = ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ → (y +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩) = (⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩))
2221breq2d 3767 . . . . . . 7 (y = ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((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 (⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩)))
23 breq1 3758 . . . . . . 7 (y = ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ → (y<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q x)}, {u ∣ ((𝐹𝑘) +Q x) <Q u}⟩ ↔ ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((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 = ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((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 (⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩) ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q x)}, {u ∣ ((𝐹𝑘) +Q x) <Q u}⟩)))
2524imbi2d 219 . . . . 5 (y = ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((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 (⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩) ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q x)}, {u ∣ ((𝐹𝑘) +Q x) <Q u}⟩))))
2625rexralbidv 2344 . . . 4 (y = ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ → (𝑗 N 𝑘 N (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 N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {u ∣ (𝐹𝑘) <Q u}⟩<P (⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩) ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q x)}, {u ∣ ((𝐹𝑘) +Q x) <Q u}⟩))))
2726ralbidv 2320 . . 3 (y = ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ → (x Q 𝑗 N 𝑘 N (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}⟩)) ↔ x Q 𝑗 N 𝑘 N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {u ∣ (𝐹𝑘) <Q u}⟩<P (⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩) ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q x)}, {u ∣ ((𝐹𝑘) +Q x) <Q u}⟩))))
2827rspcev 2650 . 2 ((⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ P x Q 𝑗 N 𝑘 N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {u ∣ (𝐹𝑘) <Q u}⟩<P (⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩ +P ⟨{𝑙𝑙 <Q x}, {ux <Q u}⟩) ⟨{𝑙 Qz N (𝑙 +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q (𝐹z)}, {u Qz N ((𝐹z) +Q (*Q‘[⟨z, 1𝑜⟩] ~Q )) <Q u}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q x)}, {u ∣ ((𝐹𝑘) +Q x) <Q u}⟩))) → y P x Q 𝑗 N 𝑘 N (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 (φy P x Q 𝑗 N 𝑘 N (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 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  {crab 2304  cop 3370   class class class wbr 3755  wf 4841  cfv 4845  (class class class)co 5455  1𝑜c1o 5933  [cec 6040  Ncnpi 6256   <N clti 6259   ~Q ceq 6263  Qcnq 6264   +Q cplq 6266  *Qcrq 6268   <Q cltq 6269  Pcnp 6275   +P cpp 6277  <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451  df-iltp 6453
This theorem is referenced by: (None)
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