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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eluzp1p1 12001 | Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘(𝑀 + 1))) | ||
Theorem | eluzaddi 12002 | Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) |
⊢ 𝑀 ∈ ℤ & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) | ||
Theorem | eluzsubi 12003 | Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) |
⊢ 𝑀 ∈ ℤ & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) | ||
Theorem | eluzadd 12004 | Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) | ||
Theorem | eluzsub 12005 | Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) | ||
Theorem | subeluzsub 12006 | Membership of a difference in an earlier upper set of integers. (Contributed by AV, 10-May-2022.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → (𝑀 − 𝐾) ∈ (ℤ≥‘(𝑀 − 𝑁))) | ||
Theorem | uzm1 12007 | Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ≥‘𝑀))) | ||
Theorem | uznn0sub 12008 | The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) | ||
Theorem | uzin 12009 | Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((ℤ≥‘𝑀) ∩ (ℤ≥‘𝑁)) = (ℤ≥‘if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) | ||
Theorem | uzp1 12010 | Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))) | ||
Theorem | nn0uz 12011 | Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ ℕ0 = (ℤ≥‘0) | ||
Theorem | nnuz 12012 | Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
⊢ ℕ = (ℤ≥‘1) | ||
Theorem | elnnuz 12013 | A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.) |
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | ||
Theorem | elnn0uz 12014 | A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.) |
⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0)) | ||
Theorem | eluz2nn 12015 | An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.) |
⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℕ) | ||
Theorem | eluzge2nn0 12016 | If an integer is greater than or equal to 2, then it is a nonnegative integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV, 3-Nov-2018.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ0) | ||
Theorem | eluz2n0 12017 | An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 0) | ||
Theorem | uzuzle23 12018 | An integer in the upper set of integers starting at 3 is element of the upper set of integers starting at 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
⊢ (𝐴 ∈ (ℤ≥‘3) → 𝐴 ∈ (ℤ≥‘2)) | ||
Theorem | eluzge3nn 12019 | If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | ||
Theorem | uz3m2nn 12020 | An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 12053. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) | ||
Theorem | 1eluzge0 12021 | 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ 1 ∈ (ℤ≥‘0) | ||
Theorem | 2eluzge0 12022 | 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) |
⊢ 2 ∈ (ℤ≥‘0) | ||
Theorem | 2eluzge1 12023 | 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ 2 ∈ (ℤ≥‘1) | ||
Theorem | uznnssnn 12024 | The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.) |
⊢ (𝑁 ∈ ℕ → (ℤ≥‘𝑁) ⊆ ℕ) | ||
Theorem | raluz 12025* | Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.) |
⊢ (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑))) | ||
Theorem | raluz2 12026* | Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.) |
⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑))) | ||
Theorem | rexuz 12027* | Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.) |
⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) | ||
Theorem | rexuz2 12028* | Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.) |
⊢ (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) | ||
Theorem | 2rexuz 12029* | Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.) |
⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑)) | ||
Theorem | peano2uz 12030 | Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀)) | ||
Theorem | peano2uzs 12031 | Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → (𝑁 + 1) ∈ 𝑍) | ||
Theorem | peano2uzr 12032 | Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀)) | ||
Theorem | uzaddcl 12033 | Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.) |
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ≥‘𝑀)) | ||
Theorem | nn0pzuz 12034 | The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ≥‘𝑍)) | ||
Theorem | uzind4 12035* | Induction on the upper set of integers that starts at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.) |
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) | ||
Theorem | uzind4ALT 12036* | Induction on the upper set of integers that starts at an integer 𝑀. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 12035 or uzind4ALT 12036 may be used; see comment for nnind 11377. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) & ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) | ||
Theorem | uzind4s 12037* | Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.) |
⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑) & ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑) | ||
Theorem | uzind4s2 12038* | Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 12037 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.) |
⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑) & ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑) | ||
Theorem | uzind4i 12039* | Induction on the upper integers that start at 𝑀. The first four give us the substitution instances we need, and the last two are the basis and the induction step. This is a stronger version of uzind4 12035 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ≥‘𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 11980). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.) |
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) | ||
Theorem | uzind4iOLD 12040* | Obsolete version of uzind4i 12039 as of 13-Jul-2022. (Contributed by NM, 4-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑀 ∈ ℤ & ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) | ||
Theorem | uzwo 12041* | Well-ordering principle: any nonempty subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.) |
⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) | ||
Theorem | uzwo2 12042* | Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. (Contributed by NM, 8-Oct-2005.) |
⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃!𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) | ||
Theorem | nnwo 12043* | Well-ordering principle: any nonempty set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.) |
⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | ||
Theorem | nnwof 12044* | Well-ordering principle: any nonempty set of positive integers has a least element. This version allows 𝑥 and 𝑦 to be present in 𝐴 as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | ||
Theorem | nnwos 12045* | Well-ordering principle: any nonempty set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ ℕ 𝜑 → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦))) | ||
Theorem | indstr 12046* | Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑)) ⇒ ⊢ (𝑥 ∈ ℕ → 𝜑) | ||
Theorem | eluznn0 12047 | Membership in a nonnegative upper set of integers implies membership in ℕ0. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ0) | ||
Theorem | eluznn 12048 | Membership in a positive upper set of integers implies membership in ℕ. (Contributed by JJ, 1-Oct-2018.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) | ||
Theorem | eluz2b1 12049 | Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.) |
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁)) | ||
Theorem | eluz2gt1 12050 | An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁) | ||
Theorem | eluz2b2 12051 | Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.) |
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) | ||
Theorem | eluz2b3 12052 | Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.) |
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) | ||
Theorem | uz2m1nn 12053 | One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.) |
⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ) | ||
Theorem | 1nuz2 12054 | 1 is not in (ℤ≥‘2). (Contributed by Paul Chapman, 21-Nov-2012.) |
⊢ ¬ 1 ∈ (ℤ≥‘2) | ||
Theorem | elnn1uz2 12055 | A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | ||
Theorem | uz2mulcl 12056 | Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.) |
⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑀 · 𝑁) ∈ (ℤ≥‘2)) | ||
Theorem | indstr2 12057* | Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 21-Nov-2012.) |
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜒 & ⊢ (𝑥 ∈ (ℤ≥‘2) → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑)) ⇒ ⊢ (𝑥 ∈ ℕ → 𝜑) | ||
Theorem | uzinfi 12058 | Extract the lower bound of an upper set of integers as its infimum. (Contributed by NM, 7-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
⊢ 𝑀 ∈ ℤ ⇒ ⊢ inf((ℤ≥‘𝑀), ℝ, < ) = 𝑀 | ||
Theorem | nninf 12059 | The infimum of the set of positive integers is one. (Contributed by NM, 16-Jun-2005.) (Revised by AV, 5-Sep-2020.) |
⊢ inf(ℕ, ℝ, < ) = 1 | ||
Theorem | nn0inf 12060 | The infimum of the set of nonnegative integers is zero. (Contributed by NM, 16-Jun-2005.) (Revised by AV, 5-Sep-2020.) |
⊢ inf(ℕ0, ℝ, < ) = 0 | ||
Theorem | infssuzle 12061 | The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 5-Sep-2020.) |
⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝐴 ∈ 𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴) | ||
Theorem | infssuzcl 12062 | The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 5-Sep-2020.) |
⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆) | ||
Theorem | ublbneg 12063* | The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦) | ||
Theorem | eqreznegel 12064* | Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ (𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴}) | ||
Theorem | supminf 12065* | The supremum of a bounded-above set of reals is the negation of the infimum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.) ( Revised by AV, 13-Sep-2020.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) = -inf({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < )) | ||
Theorem | lbzbi 12066* | If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.) |
⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) | ||
Theorem | zsupss 12067* | Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 10337.) (Contributed by Mario Carneiro, 21-Apr-2015.) |
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | ||
Theorem | suprzcl2 12068* | The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 11792 avoids ax-pre-sup 10337.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴) | ||
Theorem | suprzub 12069* | The supremum of a bounded-above set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) |
⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < )) | ||
Theorem | uzsupss 12070* | Any bounded subset of an upper set of integers has a supremum. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 21-Apr-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ 𝑍 ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝑍 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝑍 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | ||
Theorem | nn01to3 12071 | A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)) | ||
Theorem | nn0ge2m1nnALT 12072 | Alternate proof of nn0ge2m1nn 11694: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 11981, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 11694. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ) | ||
Theorem | uzwo3 12073* | Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 12042 allows the lower bound 𝐵 to be any real number. See also nnwo 12043 and nnwos 12045. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 27-Sep-2020.) |
⊢ ((𝐵 ∈ ℝ ∧ (𝐴 ⊆ {𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧} ∧ 𝐴 ≠ ∅)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) | ||
Theorem | zmin 12074* | There is a unique smallest integer greater than or equal to a given real number. (Contributed by NM, 12-Nov-2004.) (Revised by Mario Carneiro, 13-Jun-2014.) |
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦))) | ||
Theorem | zmax 12075* | There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ ∀𝑦 ∈ ℤ (𝑦 ≤ 𝐴 → 𝑦 ≤ 𝑥))) | ||
Theorem | zbtwnre 12076* | There is a unique integer between a real number and the number plus one. Exercise 5 of [Apostol] p. 28. (Contributed by NM, 13-Nov-2004.) |
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝐴 ≤ 𝑥 ∧ 𝑥 < (𝐴 + 1))) | ||
Theorem | rebtwnz 12077* | There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) | ||
Syntax | cq 12078 | Extend class notation to include the class of rationals. |
class ℚ | ||
Definition | df-q 12079 | Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 12080 for the relation "is rational." (Contributed by NM, 8-Jan-2002.) |
⊢ ℚ = ( / “ (ℤ × ℕ)) | ||
Theorem | elq 12080* | Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.) |
⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | ||
Theorem | qmulz 12081* | If 𝐴 is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐴 ∈ ℚ → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ) | ||
Theorem | znq 12082 | The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ) | ||
Theorem | qre 12083 | A rational number is a real number. (Contributed by NM, 14-Nov-2002.) |
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | ||
Theorem | zq 12084 | An integer is a rational number. (Contributed by NM, 9-Jan-2002.) |
⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | ||
Theorem | zssq 12085 | The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.) |
⊢ ℤ ⊆ ℚ | ||
Theorem | nn0ssq 12086 | The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.) |
⊢ ℕ0 ⊆ ℚ | ||
Theorem | nnssq 12087 | The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.) |
⊢ ℕ ⊆ ℚ | ||
Theorem | qssre 12088 | The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.) |
⊢ ℚ ⊆ ℝ | ||
Theorem | qsscn 12089 | The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
⊢ ℚ ⊆ ℂ | ||
Theorem | qex 12090 | The set of rational numbers exists. See also qexALT 12093. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ℚ ∈ V | ||
Theorem | nnq 12091 | A positive integer is rational. (Contributed by NM, 17-Nov-2004.) |
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ) | ||
Theorem | qcn 12092 | A rational number is a complex number. (Contributed by NM, 2-Aug-2004.) |
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | ||
Theorem | qexALT 12093 | Alternate proof of qex 12090. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ℚ ∈ V | ||
Theorem | qaddcl 12094 | Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ) | ||
Theorem | qnegcl 12095 | Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ) | ||
Theorem | qmulcl 12096 | Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ) | ||
Theorem | qsubcl 12097 | Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ) | ||
Theorem | qreccl 12098 | Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ) | ||
Theorem | qdivcl 12099 | Closure of division of rationals. (Contributed by NM, 3-Aug-2004.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ) | ||
Theorem | qrevaddcl 12100 | Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.) |
⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ)) |
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