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Type | Label | Description |
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Statement | ||
Theorem | 1idpr 7301 | 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) |
⊢ (𝐴 ∈ P → (𝐴 ·P 1P) = 𝐴) | ||
Theorem | ltnqpr 7302* | We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 19-Jun-2021.) |
⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ 〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉)) | ||
Theorem | ltnqpri 7303* | We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.) |
⊢ (𝐴 <Q 𝐵 → 〈{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}〉) | ||
Theorem | ltpopr 7304 | Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 7305. (Contributed by Jim Kingdon, 15-Dec-2019.) |
⊢ <P Po P | ||
Theorem | ltsopr 7305 | Positive real 'less than' is a weak linear order (in the sense of df-iso 4157). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.) |
⊢ <P Or P | ||
Theorem | ltaddpr 7306 | The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴<P (𝐴 +P 𝐵)) | ||
Theorem | ltexprlemell 7307* | Element in lower cut of the constructed difference. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 21-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵)))) | ||
Theorem | ltexprlemelu 7308* | Element in upper cut of the constructed difference. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 21-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵)))) | ||
Theorem | ltexprlemm 7309* | Our constructed difference is inhabited. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶))) | ||
Theorem | ltexprlemopl 7310* | The lower cut of our constructed difference is open. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 21-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐶)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) | ||
Theorem | ltexprlemlol 7311* | The lower cut of our constructed difference is lower. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 21-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ ((𝐴<P 𝐵 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶)) → 𝑞 ∈ (1st ‘𝐶))) | ||
Theorem | ltexprlemopu 7312* | The upper cut of our constructed difference is open. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 21-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶))) | ||
Theorem | ltexprlemupu 7313* | The upper cut of our constructed difference is upper. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 21-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)) → 𝑟 ∈ (2nd ‘𝐶))) | ||
Theorem | ltexprlemrnd 7314* | Our constructed difference is rounded. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶))))) | ||
Theorem | ltexprlemdisj 7315* | Our constructed difference is disjoint. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐶) ∧ 𝑞 ∈ (2nd ‘𝐶))) | ||
Theorem | ltexprlemloc 7316* | Our constructed difference is located. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶)))) | ||
Theorem | ltexprlempr 7317* | Our constructed difference is a positive real. Lemma for ltexpri 7322. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → 𝐶 ∈ P) | ||
Theorem | ltexprlemfl 7318* | Lemma for ltexpri 7322. One directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st ‘𝐵)) | ||
Theorem | ltexprlemrl 7319* | Lemma for ltexpri 7322. Reverse directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → (1st ‘𝐵) ⊆ (1st ‘(𝐴 +P 𝐶))) | ||
Theorem | ltexprlemfu 7320* | Lemma for ltexpri 7322. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵)) | ||
Theorem | ltexprlemru 7321* | Lemma for ltexpri 7322. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.) |
⊢ 𝐶 = 〈{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}〉 ⇒ ⊢ (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶))) | ||
Theorem | ltexpri 7322* | Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) |
⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | ||
Theorem | addcanprleml 7323 | Lemma for addcanprg 7325. (Contributed by Jim Kingdon, 25-Dec-2019.) |
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st ‘𝐵) ⊆ (1st ‘𝐶)) | ||
Theorem | addcanprlemu 7324 | Lemma for addcanprg 7325. (Contributed by Jim Kingdon, 25-Dec-2019.) |
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐶)) | ||
Theorem | addcanprg 7325 | Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)) | ||
Theorem | lteupri 7326* | The difference from ltexpri 7322 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.) |
⊢ (𝐴<P 𝐵 → ∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | ||
Theorem | ltaprlem 7327 | Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) |
⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) | ||
Theorem | ltaprg 7328 | Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) | ||
Theorem | prplnqu 7329* | Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.) |
⊢ (𝜑 → 𝑋 ∈ P) & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝐴 ∈ (2nd ‘(𝑋 +P 〈{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}〉))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴) | ||
Theorem | addextpr 7330 | Strong extensionality of addition (ordering version). This is similar to addext 8238 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.) |
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → (𝐴<P 𝐶 ∨ 𝐵<P 𝐷))) | ||
Theorem | recexprlemell 7331* | Membership in the lower cut of 𝐵. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))) | ||
Theorem | recexprlemelu 7332* | Membership in the upper cut of 𝐵. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐶 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))) | ||
Theorem | recexprlemm 7333* | 𝐵 is inhabited. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵))) | ||
Theorem | recexprlemopl 7334* | The lower cut of 𝐵 is open. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 28-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) | ||
Theorem | recexprlemlol 7335* | The lower cut of 𝐵 is lower. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 28-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)) → 𝑞 ∈ (1st ‘𝐵))) | ||
Theorem | recexprlemopu 7336* | The upper cut of 𝐵 is open. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 28-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) | ||
Theorem | recexprlemupu 7337* | The upper cut of 𝐵 is upper. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 28-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵))) | ||
Theorem | recexprlemrnd 7338* | 𝐵 is rounded. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))) | ||
Theorem | recexprlemdisj 7339* | 𝐵 is disjoint. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐵) ∧ 𝑞 ∈ (2nd ‘𝐵))) | ||
Theorem | recexprlemloc 7340* | 𝐵 is located. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))) | ||
Theorem | recexprlempr 7341* | 𝐵 is a positive real. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → 𝐵 ∈ P) | ||
Theorem | recexprlem1ssl 7342* | The lower cut of one is a subset of the lower cut of 𝐴 ·P 𝐵. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵))) | ||
Theorem | recexprlem1ssu 7343* | The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵))) | ||
Theorem | recexprlemss1l 7344* | The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P)) | ||
Theorem | recexprlemss1u 7345* | The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P)) | ||
Theorem | recexprlemex 7346* | 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7347. (Contributed by Jim Kingdon, 27-Dec-2019.) |
⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 ⇒ ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P) | ||
Theorem | recexpr 7347* | The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·P 𝑥) = 1P) | ||
Theorem | aptiprleml 7348 | Lemma for aptipr 7350. (Contributed by Jim Kingdon, 28-Jan-2020.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵)) | ||
Theorem | aptiprlemu 7349 | Lemma for aptipr 7350. (Contributed by Jim Kingdon, 28-Jan-2020.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴)) | ||
Theorem | aptipr 7350 | Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵) | ||
Theorem | ltmprr 7351 | Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵)) | ||
Theorem | archpr 7352* | For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer 𝑥 is embedded into the reals as described at nnprlu 7262. (Contributed by Jim Kingdon, 22-Apr-2020.) |
⊢ (𝐴 ∈ P → ∃𝑥 ∈ N 𝐴<P 〈{𝑙 ∣ 𝑙 <Q [〈𝑥, 1o〉] ~Q }, {𝑢 ∣ [〈𝑥, 1o〉] ~Q <Q 𝑢}〉) | ||
Theorem | caucvgprlemcanl 7353* | Lemma for cauappcvgprlemladdrl 7366. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.) |
⊢ (𝜑 → 𝐿 ∈ P) & ⊢ (𝜑 → 𝑆 ∈ Q) & ⊢ (𝜑 → 𝑅 ∈ Q) & ⊢ (𝜑 → 𝑄 ∈ Q) ⇒ ⊢ (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}〉)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)))) | ||
Theorem | cauappcvgprlemm 7354* | Lemma for cauappcvgpr 7371. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))) | ||
Theorem | cauappcvgprlemopl 7355* | Lemma for cauappcvgpr 7371. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) | ||
Theorem | cauappcvgprlemlol 7356* | Lemma for cauappcvgpr 7371. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿)) | ||
Theorem | cauappcvgprlemopu 7357* | Lemma for cauappcvgpr 7371. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿))) | ||
Theorem | cauappcvgprlemupu 7358* | Lemma for cauappcvgpr 7371. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ (2nd ‘𝐿)) | ||
Theorem | cauappcvgprlemrnd 7359* | Lemma for cauappcvgpr 7371. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿))))) | ||
Theorem | cauappcvgprlemdisj 7360* | Lemma for cauappcvgpr 7371. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) | ||
Theorem | cauappcvgprlemloc 7361* | Lemma for cauappcvgpr 7371. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))) | ||
Theorem | cauappcvgprlemcl 7362* | Lemma for cauappcvgpr 7371. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → 𝐿 ∈ P) | ||
Theorem | cauappcvgprlemladdfu 7363* | Lemma for cauappcvgprlemladd 7367. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)) ⊆ (2nd ‘〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}〉)) | ||
Theorem | cauappcvgprlemladdfl 7364* | Lemma for cauappcvgprlemladd 7367. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (1st ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)) ⊆ (1st ‘〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}〉)) | ||
Theorem | cauappcvgprlemladdru 7365* | Lemma for cauappcvgprlemladd 7367. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (2nd ‘〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}〉) ⊆ (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) | ||
Theorem | cauappcvgprlemladdrl 7366* | Lemma for cauappcvgprlemladd 7367. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (1st ‘〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}〉) ⊆ (1st ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) | ||
Theorem | cauappcvgprlemladd 7367* | Lemma for cauappcvgpr 7371. This takes 𝐿 and offsets it by the positive fraction 𝑆. (Contributed by Jim Kingdon, 23-Jun-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉) = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}〉) | ||
Theorem | cauappcvgprlem1 7368* | Lemma for cauappcvgpr 7371. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝑅 ∈ Q) ⇒ ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (𝐹‘𝑄)}, {𝑢 ∣ (𝐹‘𝑄) <Q 𝑢}〉<P (𝐿 +P 〈{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}〉)) | ||
Theorem | cauappcvgprlem2 7369* | Lemma for cauappcvgpr 7371. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝑅 ∈ Q) ⇒ ⊢ (𝜑 → 𝐿<P 〈{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}〉) | ||
Theorem | cauappcvgprlemlim 7370* | Lemma for cauappcvgpr 7371. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (〈{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}〉<P (𝐿 +P 〈{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}〉) ∧ 𝐿<P 〈{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}〉)) | ||
Theorem | cauappcvgpr 7371* |
A Cauchy approximation has a limit. A Cauchy approximation, here
𝐹, is similar to a Cauchy sequence but
is indexed by the desired
tolerance (that is, how close together terms needs to be) rather than
by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p.
(varies) with a few differences such as that we are proving the
existence of a limit without anything about how fast it converges
(that is, mere existence instead of existence, in HoTT terms), and
that the codomain of 𝐹 is Q rather than P. 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 caucvgpr 7391 and caucvgprpr 7421 but is somewhat simpler, so reading this one first may help understanding the other two. (Contributed by Jim Kingdon, 19-Jun-2020.) |
⊢ (𝜑 → 𝐹:Q⟶Q) & ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞)))) & ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ P ∀𝑞 ∈ Q ∀𝑟 ∈ Q (〈{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}〉<P (𝑦 +P 〈{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}〉) ∧ 𝑦<P 〈{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}〉)) | ||
Theorem | archrecnq 7372* | Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.) |
⊢ (𝐴 ∈ Q → ∃𝑗 ∈ N (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝐴) | ||
Theorem | archrecpr 7373* | Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.) |
⊢ (𝐴 ∈ P → ∃𝑗 ∈ N 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑗, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑗, 1o〉] ~Q ) <Q 𝑢}〉<P 𝐴) | ||
Theorem | caucvgprlemk 7374 | Lemma for caucvgpr 7391. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.) |
⊢ (𝜑 → 𝐽 <N 𝐾) & ⊢ (𝜑 → (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑄) ⇒ ⊢ (𝜑 → (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑄) | ||
Theorem | caucvgprlemnkj 7375* | Lemma for caucvgpr 7391. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → ¬ ((𝑆 +Q (*Q‘[〈𝐾, 1o〉] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[〈𝐽, 1o〉] ~Q )) <Q 𝑆)) | ||
Theorem | caucvgprlemnbj 7376* | Lemma for caucvgpr 7391. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → 𝐵 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) ⇒ ⊢ (𝜑 → ¬ (((𝐹‘𝐵) +Q (*Q‘[〈𝐵, 1o〉] ~Q )) +Q (*Q‘[〈𝐽, 1o〉] ~Q )) <Q (𝐹‘𝐽)) | ||
Theorem | caucvgprlemm 7377* | Lemma for caucvgpr 7391. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))) | ||
Theorem | caucvgprlemopl 7378* | Lemma for caucvgpr 7391. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) | ||
Theorem | caucvgprlemlol 7379* | Lemma for caucvgpr 7391. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿)) | ||
Theorem | caucvgprlemopu 7380* | Lemma for caucvgpr 7391. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿))) | ||
Theorem | caucvgprlemupu 7381* | Lemma for caucvgpr 7391. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ (2nd ‘𝐿)) | ||
Theorem | caucvgprlemrnd 7382* | Lemma for caucvgpr 7391. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿))))) | ||
Theorem | caucvgprlemdisj 7383* | Lemma for caucvgpr 7391. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) | ||
Theorem | caucvgprlemloc 7384* | Lemma for caucvgpr 7391. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))) | ||
Theorem | caucvgprlemcl 7385* | Lemma for caucvgpr 7391. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → 𝐿 ∈ P) | ||
Theorem | caucvgprlemladdfu 7386* | Lemma for caucvgpr 7391. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → (2nd ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉)) ⊆ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) +Q 𝑆) <Q 𝑢}) | ||
Theorem | caucvgprlemladdrl 7387* | Lemma for caucvgpr 7391. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ (𝜑 → {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}〉))) | ||
Theorem | caucvgprlem1 7388* | Lemma for caucvgpr 7391. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝐽 <N 𝐾) & ⊢ (𝜑 → (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑄) ⇒ ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (𝐹‘𝐾)}, {𝑢 ∣ (𝐹‘𝐾) <Q 𝑢}〉<P (𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}〉)) | ||
Theorem | caucvgprlem2 7389* | Lemma for caucvgpr 7391. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 & ⊢ (𝜑 → 𝑄 ∈ Q) & ⊢ (𝜑 → 𝐽 <N 𝐾) & ⊢ (𝜑 → (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑄) ⇒ ⊢ (𝜑 → 𝐿<P 〈{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}〉) | ||
Theorem | caucvgprlemlim 7390* | Lemma for caucvgpr 7391. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) & ⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[〈𝑗, 1o〉] ~Q )) <Q 𝑢}〉 ⇒ ⊢ (𝜑 → ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (〈{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}〉<P (𝐿 +P 〈{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}〉) ∧ 𝐿<P 〈{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}〉))) | ||
Theorem | caucvgpr 7391* |
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 7371 and caucvgprpr 7421. Reading cauappcvgpr 7371 first (the simplest of the three) might help understanding the other two. (Contributed by Jim Kingdon, 18-Jun-2020.) |
⊢ (𝜑 → 𝐹:N⟶Q) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[〈𝑛, 1o〉] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[〈𝑛, 1o〉] ~Q ))))) & ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗)) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ P ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (〈{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}〉<P (𝑦 +P 〈{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}〉) ∧ 𝑦<P 〈{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}〉))) | ||
Theorem | caucvgprprlemk 7392* | Lemma for caucvgprpr 7421. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.) |
⊢ (𝜑 → 𝐽 <N 𝐾) & ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑢}〉<P 𝑄) ⇒ ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐾, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑢}〉<P 𝑄) | ||
Theorem | caucvgprprlemloccalc 7393* | Lemma for caucvgprpr 7421. Rearranging some expressions for caucvgprprlemloc 7412. (Contributed by Jim Kingdon, 8-Feb-2021.) |
⊢ (𝜑 → 𝑆 <Q 𝑇) & ⊢ (𝜑 → 𝑌 ∈ Q) & ⊢ (𝜑 → (𝑆 +Q 𝑌) = 𝑇) & ⊢ (𝜑 → 𝑋 ∈ Q) & ⊢ (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌) & ⊢ (𝜑 → 𝑀 ∈ N) & ⊢ (𝜑 → (*Q‘[〈𝑀, 1o〉] ~Q ) <Q 𝑋) ⇒ ⊢ (𝜑 → (〈{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[〈𝑀, 1o〉] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[〈𝑀, 1o〉] ~Q )) <Q 𝑢}〉 +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑀, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑀, 1o〉] ~Q ) <Q 𝑢}〉)<P 〈{𝑙 ∣ 𝑙 <Q 𝑇}, {𝑢 ∣ 𝑇 <Q 𝑢}〉) | ||
Theorem | caucvgprprlemell 7394* | Lemma for caucvgprpr 7421. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.) |
⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑟 ∈ N 〈{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[〈𝑟, 1o〉] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[〈𝑟, 1o〉] ~Q )) <Q 𝑞}〉<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑟, 1o〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑟, 1o〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉}〉 ⇒ ⊢ (𝑋 ∈ (1st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N 〈{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[〈𝑏, 1o〉] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[〈𝑏, 1o〉] ~Q )) <Q 𝑞}〉<P (𝐹‘𝑏))) | ||
Theorem | caucvgprprlemelu 7395* | Lemma for caucvgprpr 7421. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.) |
⊢ 𝐿 = 〈{𝑙 ∈ Q ∣ ∃𝑟 ∈ N 〈{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[〈𝑟, 1o〉] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[〈𝑟, 1o〉] ~Q )) <Q 𝑞}〉<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑟, 1o〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑟, 1o〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}〉}〉 ⇒ ⊢ (𝑋 ∈ (2nd ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝑏, 1o〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝑏, 1o〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}〉)) | ||
Theorem | caucvgprprlemcbv 7396* | Lemma for caucvgprpr 7421. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.) |
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉)))) ⇒ ⊢ (𝜑 → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑎, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑎, 1o〉] ~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑎, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑎, 1o〉] ~Q ) <Q 𝑢}〉)))) | ||
Theorem | caucvgprprlemval 7397* | Lemma for caucvgprpr 7421. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.) |
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉)))) ⇒ ⊢ ((𝜑 ∧ 𝐴 <N 𝐵) → ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝐴, 1o〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝐴, 1o〉] ~Q ) <Q 𝑞}〉) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝐴, 1o〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝐴, 1o〉] ~Q ) <Q 𝑞}〉))) | ||
Theorem | caucvgprprlemnkltj 7398* | Lemma for caucvgprpr 7421. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉)))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → ¬ (〈{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[〈𝐾, 1o〉] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[〈𝐾, 1o〉] ~Q )) <Q 𝑞}〉<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}〉)) | ||
Theorem | caucvgprprlemnkeqj 7399* | Lemma for caucvgprpr 7421. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉)))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ¬ (〈{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[〈𝐾, 1o〉] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[〈𝐾, 1o〉] ~Q )) <Q 𝑞}〉<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}〉)) | ||
Theorem | caucvgprprlemnjltk 7400* | Lemma for caucvgprpr 7421. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.) |
⊢ (𝜑 → 𝐹:N⟶P) & ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝑛, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝑛, 1o〉] ~Q ) <Q 𝑢}〉)))) & ⊢ (𝜑 → 𝐾 ∈ N) & ⊢ (𝜑 → 𝐽 ∈ N) & ⊢ (𝜑 → 𝑆 ∈ Q) ⇒ ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → ¬ (〈{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[〈𝐾, 1o〉] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[〈𝐾, 1o〉] ~Q )) <Q 𝑞}〉<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P 〈{𝑝 ∣ 𝑝 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑞 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑞}〉)<P 〈{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}〉)) |
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