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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | adddiri 7901 | Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)) | ||
Theorem | recni 7902 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ∈ ℂ | ||
Theorem | readdcli 7903 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 + 𝐵) ∈ ℝ | ||
Theorem | remulcli 7904 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 · 𝐵) ∈ ℝ | ||
Theorem | 1red 7905 | 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℝ) | ||
Theorem | 1cnd 7906 | 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℂ) | ||
Theorem | mulid1d 7907 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 1) = 𝐴) | ||
Theorem | mulid2d 7908 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (1 · 𝐴) = 𝐴) | ||
Theorem | addcld 7909 | Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) | ||
Theorem | mulcld 7910 | Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ) | ||
Theorem | mulcomd 7911 | Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | addassd 7912 | Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | mulassd 7913 | Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | adddid 7914 | Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | ||
Theorem | adddird 7915 | Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))) | ||
Theorem | adddirp1d 7916 | Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵)) | ||
Theorem | joinlmuladdmuld 7917 | Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) | ||
Theorem | recnd 7918 | Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | readdcld 7919 | Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) | ||
Theorem | remulcld 7920 | Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ) | ||
Syntax | cpnf 7921 | Plus infinity. |
class +∞ | ||
Syntax | cmnf 7922 | Minus infinity. |
class -∞ | ||
Syntax | cxr 7923 | The set of extended reals (includes plus and minus infinity). |
class ℝ* | ||
Syntax | clt 7924 | 'Less than' predicate (extended to include the extended reals). |
class < | ||
Syntax | cle 7925 | Extend wff notation to include the 'less than or equal to' relation. |
class ≤ | ||
Definition | df-pnf 7926 |
Define plus infinity. Note that the definition is arbitrary, requiring
only that +∞ be a set not in ℝ and different from -∞
(df-mnf 7927). We use 𝒫 ∪ ℂ to make it independent of the
construction of ℂ, and Cantor's Theorem will
show that it is
different from any member of ℂ and therefore
ℝ. See pnfnre 7931
and mnfnre 7932, and we'll also be able to prove +∞ ≠ -∞.
A simpler possibility is to define +∞ as ℂ and -∞ as {ℂ}, but that approach requires the Axiom of Regularity to show that +∞ and -∞ are different from each other and from all members of ℝ. (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.) |
⊢ +∞ = 𝒫 ∪ ℂ | ||
Definition | df-mnf 7927 | Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that -∞ be a set not in ℝ and different from +∞ (see mnfnre 7932). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.) |
⊢ -∞ = 𝒫 +∞ | ||
Definition | df-xr 7928 | Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.) |
⊢ ℝ* = (ℝ ∪ {+∞, -∞}) | ||
Definition | df-ltxr 7929* | Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, <ℝ is primitive and not necessarily a relation on ℝ. (Contributed by NM, 13-Oct-2005.) |
⊢ < = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) | ||
Definition | df-le 7930 | Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.) |
⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < ) | ||
Theorem | pnfnre 7931 | Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
⊢ +∞ ∉ ℝ | ||
Theorem | mnfnre 7932 | Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
⊢ -∞ ∉ ℝ | ||
Theorem | ressxr 7933 | The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) |
⊢ ℝ ⊆ ℝ* | ||
Theorem | rexpssxrxp 7934 | The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*) | ||
Theorem | rexr 7935 | A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | ||
Theorem | 0xr 7936 | Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.) |
⊢ 0 ∈ ℝ* | ||
Theorem | renepnf 7937 | No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | ||
Theorem | renemnf 7938 | No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | ||
Theorem | rexrd 7939 | A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ*) | ||
Theorem | renepnfd 7940 | No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≠ +∞) | ||
Theorem | renemnfd 7941 | No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≠ -∞) | ||
Theorem | pnfxr 7942 | Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) |
⊢ +∞ ∈ ℝ* | ||
Theorem | pnfex 7943 | Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ +∞ ∈ V | ||
Theorem | pnfnemnf 7944 | Plus and minus infinity are different elements of ℝ*. (Contributed by NM, 14-Oct-2005.) |
⊢ +∞ ≠ -∞ | ||
Theorem | mnfnepnf 7945 | Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -∞ ≠ +∞ | ||
Theorem | mnfxr 7946 | Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ -∞ ∈ ℝ* | ||
Theorem | rexri 7947 | A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ∈ ℝ* | ||
Theorem | 1xr 7948 | 1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ 1 ∈ ℝ* | ||
Theorem | renfdisj 7949 | The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (ℝ ∩ {+∞, -∞}) = ∅ | ||
Theorem | ltrelxr 7950 | 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ < ⊆ (ℝ* × ℝ*) | ||
Theorem | ltrel 7951 | 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.) |
⊢ Rel < | ||
Theorem | lerelxr 7952 | 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ ≤ ⊆ (ℝ* × ℝ*) | ||
Theorem | lerel 7953 | 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ Rel ≤ | ||
Theorem | xrlenlt 7954 | 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
Theorem | ltxrlt 7955 | The standard less-than <ℝ and the extended real less-than < are identical when restricted to the non-extended reals ℝ. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) | ||
Theorem | axltirr 7956 | Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7856 with ordering on the extended reals. New proofs should use ltnr 7966 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.) |
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | ||
Theorem | axltwlin 7957 | Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7857 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵))) | ||
Theorem | axlttrn 7958 | Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7858 with ordering on the extended reals. New proofs should use lttr 7963 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | axltadd 7959 | Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7860 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵))) | ||
Theorem | axapti 7960 | Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7859 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) → 𝐴 = 𝐵) | ||
Theorem | axmulgt0 7961 | The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7861 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))) | ||
Theorem | axsuploc 7962* | An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 7865 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.) |
⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦)))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | ||
Theorem | lttr 7963 | Alias for axlttrn 7958, for naming consistency with lttri 7994. New proofs should generally use this instead of ax-pre-lttrn 7858. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | mulgt0 7964 | The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | ||
Theorem | lenlt 7965 | 'Less than or equal to' expressed in terms of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
Theorem | ltnr 7966 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | ||
Theorem | ltso 7967 | 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
⊢ < Or ℝ | ||
Theorem | gtso 7968 | 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
⊢ ◡ < Or ℝ | ||
Theorem | lttri3 7969 | Tightness of real apartness. (Contributed by NM, 5-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | ||
Theorem | letri3 7970 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | ||
Theorem | ltleletr 7971 | Transitive law, weaker form of (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶. (Contributed by AV, 14-Oct-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
Theorem | letr 7972 | Transitive law. (Contributed by NM, 12-Nov-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
Theorem | leid 7973 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | ||
Theorem | ltne 7974 | 'Less than' implies not equal. See also ltap 8522 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | ||
Theorem | ltnsym 7975 | 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | ||
Theorem | eqlelt 7976 | Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵))) | ||
Theorem | ltle 7977 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | ||
Theorem | lelttr 7978 | Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | ltletr 7979 | Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | ltnsym2 7980 | 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (𝐴 < 𝐵 ∧ 𝐵 < 𝐴)) | ||
Theorem | eqle 7981 | Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | ||
Theorem | ltnri 7982 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ ¬ 𝐴 < 𝐴 | ||
Theorem | eqlei 7983 | Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵) | ||
Theorem | eqlei2 7984 | Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴) | ||
Theorem | gtneii 7985 | 'Less than' implies not equal. See also gtapii 8523 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐵 ≠ 𝐴 | ||
Theorem | ltneii 7986 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
Theorem | lttri3i 7987 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | ||
Theorem | letri3i 7988 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)) | ||
Theorem | ltnsymi 7989 | 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴) | ||
Theorem | lenlti 7990 | 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴) | ||
Theorem | ltlei 7991 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵) | ||
Theorem | ltleii 7992 | 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 ≤ 𝐵 | ||
Theorem | ltnei 7993 | 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 ≠ 𝐴) | ||
Theorem | lttri 7994 | 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) | ||
Theorem | lelttri 7995 | 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) | ||
Theorem | ltletri 7996 | 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶) | ||
Theorem | letri 7997 | 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) | ||
Theorem | le2tri3i 7998 | Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) | ||
Theorem | mulgt0i 7999 | The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)) | ||
Theorem | mulgt0ii 8000 | The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 0 < 𝐴 & ⊢ 0 < 𝐵 ⇒ ⊢ 0 < (𝐴 · 𝐵) |
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