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Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaucvgprlemlim 7501* Lemma for caucvgpr 7502. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))    &   𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩       (𝜑 → ∀𝑥Q𝑗N𝑘N (𝑗 <N 𝑘 → (⟨{𝑙𝑙 <Q (𝐹𝑘)}, {𝑢 ∣ (𝐹𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑥}, {𝑢𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹𝑘) +Q 𝑥) <Q 𝑢}⟩)))

Theoremcaucvgpr 7502* 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 7482 and caucvgprpr 7532. Reading cauappcvgpr 7482 first (the simplest of the three) might help understanding the other two.

(Contributed by Jim Kingdon, 18-Jun-2020.)

(𝜑𝐹:NQ)    &   (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))    &   (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))       (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}⟩)))

Theoremcaucvgprprlemk 7503* Lemma for caucvgprpr 7532. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)
(𝜑𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)       (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

Theoremcaucvgprprlemloccalc 7504* Lemma for caucvgprpr 7532. Rearranging some expressions for caucvgprprlemloc 7523. (Contributed by Jim Kingdon, 8-Feb-2021.)
(𝜑𝑆 <Q 𝑇)    &   (𝜑𝑌Q)    &   (𝜑 → (𝑆 +Q 𝑌) = 𝑇)    &   (𝜑𝑋Q)    &   (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)    &   (𝜑𝑀N)    &   (𝜑 → (*Q‘[⟨𝑀, 1o⟩] ~Q ) <Q 𝑋)       (𝜑 → (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)

Theoremcaucvgprprlemell 7505* Lemma for caucvgprpr 7532. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
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}⟩}⟩       (𝑋 ∈ (1st𝐿) ↔ (𝑋Q ∧ ∃𝑏N ⟨{𝑝𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑏)))

Theoremcaucvgprprlemelu 7506* Lemma for caucvgprpr 7532. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
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}⟩}⟩       (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}⟩))

Theoremcaucvgprprlemcbv 7507* Lemma for caucvgprpr 7532. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))       (𝜑 → ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))

Theoremcaucvgprprlemval 7508* Lemma for caucvgprpr 7532. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))       ((𝜑𝐴 <N 𝐵) → ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)))

Theoremcaucvgprprlemnkltj 7509* Lemma for caucvgprpr 7532. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐾 <N 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

Theoremcaucvgprprlemnkeqj 7510* Lemma for caucvgprpr 7532. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐾 = 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

Theoremcaucvgprprlemnjltk 7511* Lemma for caucvgprpr 7532. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       ((𝜑𝐽 <N 𝐾) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

Theoremcaucvgprprlemnkj 7512* Lemma for caucvgprpr 7532. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑𝐾N)    &   (𝜑𝐽N)    &   (𝜑𝑆Q)       (𝜑 → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))

Theoremcaucvgprprlemnbj 7513* Lemma for caucvgprpr 7532. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑𝐵N)    &   (𝜑𝐽N)       (𝜑 → ¬ (((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝐽))

Theoremcaucvgprprlemml 7514* Lemma for caucvgprpr 7532. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑 → ∃𝑠Q 𝑠 ∈ (1st𝐿))

Theoremcaucvgprprlemmu 7515* Lemma for caucvgprpr 7532. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑 → ∃𝑡Q 𝑡 ∈ (2nd𝐿))

Theoremcaucvgprprlemm 7516* Lemma for caucvgprpr 7532. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑡Q 𝑡 ∈ (2nd𝐿)))

Theoremcaucvgprprlemopl 7517* Lemma for caucvgprpr 7532. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       ((𝜑𝑠 ∈ (1st𝐿)) → ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)))

Theoremcaucvgprprlemlol 7518* Lemma for caucvgprpr 7532. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       ((𝜑𝑠 <Q 𝑡𝑡 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))

Theoremcaucvgprprlemopu 7519* Lemma for caucvgprpr 7532. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       ((𝜑𝑡 ∈ (2nd𝐿)) → ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))

Theoremcaucvgprprlemupu 7520* Lemma for caucvgprpr 7532. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       ((𝜑𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)) → 𝑡 ∈ (2nd𝐿))

Theoremcaucvgprprlemrnd 7521* Lemma for caucvgprpr 7532. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿))) ∧ ∀𝑡Q (𝑡 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))))

Theoremcaucvgprprlemdisj 7522* Lemma for caucvgprpr 7532. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))

Theoremcaucvgprprlemloc 7523* Lemma for caucvgprpr 7532. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑 → ∀𝑠Q𝑡Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))

Theoremcaucvgprprlemcl 7524* Lemma for caucvgprpr 7532. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑𝐿P)

Theoremcaucvgprprlemclphr 7525* Lemma for caucvgprpr 7532. The putative limit is a positive real. Like caucvgprprlemcl 7524 but without a distinct variable constraint between 𝜑 and 𝑟. (Contributed by Jim Kingdon, 19-Jun-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑𝐿P)

Theoremcaucvgprprlemexbt 7526* Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩    &   (𝜑𝑄Q)    &   (𝜑𝑇P)    &   (𝜑 → (𝐿 +P ⟨{𝑝𝑝 <Q 𝑄}, {𝑞𝑄 <Q 𝑞}⟩)<P 𝑇)       (𝜑 → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q 𝑄}, {𝑞𝑄 <Q 𝑞}⟩)<P 𝑇)

Theoremcaucvgprprlemexb 7527* Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩    &   (𝜑𝑄P)    &   (𝜑𝑅N)       (𝜑 → (((𝐿 +P 𝑄) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑅) +P 𝑄) → ∃𝑏N (((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑅, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1o⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹𝑅) +P 𝑄)))

Theoremcaucvgprprlemaddq 7528* Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩    &   (𝜑𝑋P)    &   (𝜑𝑄P)    &   (𝜑 → ∃𝑟N (𝑋 +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝑟) +P 𝑄))       (𝜑𝑋<P (𝐿 +P 𝑄))

Theoremcaucvgprprlem1 7529* Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩    &   (𝜑𝑄P)    &   (𝜑𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)       (𝜑 → (𝐹𝐾)<P (𝐿 +P 𝑄))

Theoremcaucvgprprlem2 7530* Lemma for caucvgprpr 7532. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩    &   (𝜑𝑄P)    &   (𝜑𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)       (𝜑𝐿<P ((𝐹𝐾) +P 𝑄))

Theoremcaucvgprprlemlim 7531* Lemma for caucvgprpr 7532. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)
(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))    &   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}⟩}⟩       (𝜑 → ∀𝑥P𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹𝑘) +P 𝑥))))

Theoremcaucvgprpr 7532* A Cauchy sequence of positive reals 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 given value 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This is similar to caucvgpr 7502 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7482) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.)

(𝜑𝐹:NP)    &   (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}⟩))))    &   (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))       (𝜑 → ∃𝑦P𝑥P𝑗N𝑘N (𝑗 <N 𝑘 → ((𝐹𝑘)<P (𝑦 +P 𝑥) ∧ 𝑦<P ((𝐹𝑘) +P 𝑥))))

Theoremsuplocexprlemell 7533* Lemma for suplocexpr 7545. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝐵 (1st𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (1st𝑥))

Theoremsuplocexprlem2b 7534 Lemma for suplocexpr 7545. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})

Theoremsuplocexprlemss 7535* Lemma for suplocexpr 7545. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑𝐴P)

Theoremsuplocexprlemml 7536* Lemma for suplocexpr 7545. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑 → ∃𝑠Q 𝑠 (1st𝐴))

Theoremsuplocexprlemrl 7537* Lemma for suplocexpr 7545. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑 → ∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))

Theoremsuplocexprlemmu 7538* Lemma for suplocexpr 7545. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∃𝑠Q 𝑠 ∈ (2nd𝐵))

Theoremsuplocexprlemru 7539* Lemma for suplocexpr 7545. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵))))

Theoremsuplocexprlemdisj 7540* Lemma for suplocexpr 7545. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)))

Theoremsuplocexprlemloc 7541* Lemma for suplocexpr 7545. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))

Theoremsuplocexprlemex 7542* Lemma for suplocexpr 7545. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑𝐵P)

Theoremsuplocexprlemub 7543* Lemma for suplocexpr 7545. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → ∀𝑦𝐴 ¬ 𝐵<P 𝑦)

Theoremsuplocexprlemlub 7544* Lemma for suplocexpr 7545. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))    &   𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩       (𝜑 → (𝑦<P 𝐵 → ∃𝑧𝐴 𝑦<P 𝑧))

Theoremsuplocexpr 7545* An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)    &   (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))       (𝜑 → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))

Definitiondf-enr 7546* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)
~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))}

Definitiondf-nr 7547 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)
R = ((P × P) / ~R )

Definitiondf-plr 7548* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
+R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))}

Definitiondf-mr 7549* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}

Definitiondf-ltr 7550* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.)
<R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}

Definitiondf-0r 7551 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)
0R = [⟨1P, 1P⟩] ~R

Definitiondf-1r 7552 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)
1R = [⟨(1P +P 1P), 1P⟩] ~R

Definitiondf-m1r 7553 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.)
-1R = [⟨1P, (1P +P 1P)⟩] ~R

Theoremenrbreq 7554 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → (⟨𝐴, 𝐵⟩ ~R𝐶, 𝐷⟩ ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))

Theoremenrer 7555 The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
~R Er (P × P)

Theoremenreceq 7556 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))

Theoremenrex 7557 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)
~R ∈ V

Theoremltrelsr 7558 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
<R ⊆ (R × R)

((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨(𝐴 +P 𝐹), (𝐵 +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩))

Theoremmulcmpblnrlemg 7560 Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))))

Theoremmulcmpblnr 7561 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)
((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)), ((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹))⟩ ~R ⟨((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)), ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅))⟩))

Theoremprsrlem1 7562* Decomposing signed reals into positive reals. Lemma for addsrpr 7565 and mulsrpr 7566. (Contributed by Jim Kingdon, 30-Dec-2019.)
(((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔))))

Theoremaddsrmo 7563* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))

Theoremmulsrmo 7564* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))

Theoremaddsrpr 7565 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )

Theoremmulsrpr 7566 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )

Theoremltsrprg 7567 Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))

Theoremgt0srpr 7568 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
(0R <R [⟨𝐴, 𝐵⟩] ~R𝐵<P 𝐴)

Theorem0nsr 7569 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)
¬ ∅ ∈ R

Theorem0r 7570 The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.)
0RR

Theorem1sr 7571 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.)
1RR

Theoremm1r 7572 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.)
-1RR

Theoremaddclsr 7573 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)

Theoremmulclsr 7574 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)
((𝐴R𝐵R) → (𝐴 ·R 𝐵) ∈ R)

Theoremaddcomsrg 7575 Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝐴R𝐵R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴))

Theoremaddasssrg 7576 Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝐴R𝐵R𝐶R) → ((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶)))

Theoremmulcomsrg 7577 Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝐴R𝐵R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴))

Theoremmulasssrg 7578 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
((𝐴R𝐵R𝐶R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)))

Theoremdistrsrg 7579 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
((𝐴R𝐵R𝐶R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))

Theoremm1p1sr 7580 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
(-1R +R 1R) = 0R

Theoremm1m1sr 7581 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
(-1R ·R -1R) = 1R

Theoremlttrsr 7582* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
((𝑓R𝑔RR) → ((𝑓 <R 𝑔𝑔 <R ) → 𝑓 <R ))

Theoremltposr 7583 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
<R Po R

Theoremltsosr 7584 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
<R Or R

Theorem0lt1sr 7585 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
0R <R 1R

Theorem1ne0sr 7586 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
¬ 1R = 0R

Theorem0idsr 7587 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)
(𝐴R → (𝐴 +R 0R) = 𝐴)

Theorem1idsr 7588 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
(𝐴R → (𝐴 ·R 1R) = 𝐴)

Theorem00sr 7589 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
(𝐴R → (𝐴 ·R 0R) = 0R)

Theoremltasrg 7590 Ordering property of addition. (Contributed by NM, 10-May-1996.)
((𝐴R𝐵R𝐶R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))

Theorempn0sr 7591 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
(𝐴R → (𝐴 +R (𝐴 ·R -1R)) = 0R)

Theoremnegexsr 7592* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)
(𝐴R → ∃𝑥R (𝐴 +R 𝑥) = 0R)

Theoremrecexgt0sr 7593* The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
(0R <R 𝐴 → ∃𝑥R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R))

Theoremrecexsrlem 7594* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.)
(0R <R 𝐴 → ∃𝑥R (𝐴 ·R 𝑥) = 1R)

Theoremaddgt0sr 7595 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.)
((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵))

Theoremltadd1sr 7596 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴R𝐴 <R (𝐴 +R 1R))

Theoremltm1sr 7597 Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
(𝐴R → (𝐴 +R -1R) <R 𝐴)

Theoremmulgt0sr 7598 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))

Theoremaptisr 7599 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴R𝐵R ∧ ¬ (𝐴 <R 𝐵𝐵 <R 𝐴)) → 𝐴 = 𝐵)

Theoremmulextsr1lem 7600 Lemma for mulextsr1 7601. (Contributed by Jim Kingdon, 17-Feb-2020.)
(((𝑋P𝑌P) ∧ (𝑍P𝑊P) ∧ (𝑈P𝑉P)) → ((((𝑋 ·P 𝑈) +P (𝑌 ·P 𝑉)) +P ((𝑍 ·P 𝑉) +P (𝑊 ·P 𝑈)))<P (((𝑋 ·P 𝑉) +P (𝑌 ·P 𝑈)) +P ((𝑍 ·P 𝑈) +P (𝑊 ·P 𝑉))) → ((𝑋 +P 𝑊)<P (𝑌 +P 𝑍) ∨ (𝑍 +P 𝑌)<P (𝑊 +P 𝑋))))

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