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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mulcom 9901 | Alias for ax-mulcom 9879, for naming consistency with mulcomi 9925. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | addass 9902 | Alias for ax-addass 9880, for naming consistency with addassi 9927. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) | ||
Theorem | mulass 9903 | Alias for ax-mulass 9881, for naming consistency with mulassi 9928. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) | ||
Theorem | adddi 9904 | Alias for ax-distr 9882, for naming consistency with adddii 9929. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) | ||
Theorem | recn 9905 | A real number is a complex number. (Contributed by NM, 10-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | ||
Theorem | reex 9906 | The real numbers form a set. See also reexALT 11702. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ℝ ∈ V | ||
Theorem | reelprrecn 9907 | Reals are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ ℝ ∈ {ℝ, ℂ} | ||
Theorem | cnelprrecn 9908 | Complex numbers are a subset of the pair of real and complex numbers (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ ℂ ∈ {ℝ, ℂ} | ||
Theorem | elimne0 9909 | Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.) |
⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0 | ||
Theorem | adddir 9910 | Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))) | ||
Theorem | 0cn 9911 | 0 is a complex number. See also 0cnALT 10149. (Contributed by NM, 19-Feb-2005.) |
⊢ 0 ∈ ℂ | ||
Theorem | 0cnd 9912 | 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℂ) | ||
Theorem | c0ex 9913 | 0 is a set (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 0 ∈ V | ||
Theorem | 1ex 9914 | 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 1 ∈ V | ||
Theorem | cnre 9915* | Alias for ax-cnre 9888, for naming consistency. (Contributed by NM, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) | ||
Theorem | mulid1 9916 | 1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | ||
Theorem | mulid2 9917 | Identity law for multiplication. Note: see mulid1 9916 for commuted version. (Contributed by NM, 8-Oct-1999.) |
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | ||
Theorem | 1re 9918 | 1 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 9873, by exploiting properties of the imaginary unit i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
⊢ 1 ∈ ℝ | ||
Theorem | 0re 9919 | 0 is a real number. See also 0reALT 10257. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) |
⊢ 0 ∈ ℝ | ||
Theorem | 0red 9920 | 0 is a real number, deductive form. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 0 ∈ ℝ) | ||
Theorem | mulid1i 9921 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · 1) = 𝐴 | ||
Theorem | mulid2i 9922 | Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (1 · 𝐴) = 𝐴 | ||
Theorem | addcli 9923 | Closure law for addition. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 + 𝐵) ∈ ℂ | ||
Theorem | mulcli 9924 | Closure law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 · 𝐵) ∈ ℂ | ||
Theorem | mulcomi 9925 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴) | ||
Theorem | mulcomli 9926 | Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 · 𝐵) = 𝐶 ⇒ ⊢ (𝐵 · 𝐴) = 𝐶 | ||
Theorem | addassi 9927 | Associative law for addition. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)) | ||
Theorem | mulassi 9928 | Associative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) | ||
Theorem | adddii 9929 | Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)) | ||
Theorem | adddiri 9930 | 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 9931 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ∈ ℂ | ||
Theorem | readdcli 9932 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 + 𝐵) ∈ ℝ | ||
Theorem | remulcli 9933 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 · 𝐵) ∈ ℝ | ||
Theorem | 1red 9934 | 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℝ) | ||
Theorem | 1cnd 9935 | 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (𝜑 → 1 ∈ ℂ) | ||
Theorem | mulid1d 9936 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 1) = 𝐴) | ||
Theorem | mulid2d 9937 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (1 · 𝐴) = 𝐴) | ||
Theorem | addcld 9938 | Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) | ||
Theorem | mulcld 9939 | Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ) | ||
Theorem | mulcomd 9940 | Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | addassd 9941 | 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 9942 | 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 9943 | 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 9944 | 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 9945 | 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 9946 | 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 9947 | Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) | ||
Theorem | readdcld 9948 | 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 9949 | 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 9950 | Plus infinity. |
class +∞ | ||
Syntax | cmnf 9951 | Minus infinity. |
class -∞ | ||
Syntax | cxr 9952 | The set of extended reals (includes plus and minus infinity). |
class ℝ* | ||
Syntax | clt 9953 | 'Less than' predicate (extended to include the extended reals). |
class < | ||
Syntax | cle 9954 | Extend wff notation to include the 'less than or equal to' relation. |
class ≤ | ||
Definition | df-pnf 9955 |
Define plus infinity. Note that the definition is arbitrary, requiring
only that +∞ be a set not in ℝ and different from -∞
(df-mnf 9956). 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 9960,
mnfnre 9961, and pnfnemnf 9973.
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 9956 | 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 9961 and pnfnemnf 9973). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.) |
⊢ -∞ = 𝒫 +∞ | ||
Definition | df-xr 9957 | 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 9958* | 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 9959 | Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 10003 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.) |
⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < ) | ||
Theorem | pnfnre 9960 | Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
⊢ +∞ ∉ ℝ | ||
Theorem | mnfnre 9961 | Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
⊢ -∞ ∉ ℝ | ||
Theorem | ressxr 9962 | The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) |
⊢ ℝ ⊆ ℝ* | ||
Theorem | rexpssxrxp 9963 | 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 9964 | A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | ||
Theorem | 0xr 9965 | Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.) |
⊢ 0 ∈ ℝ* | ||
Theorem | renepnf 9966 | 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 9967 | No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | ||
Theorem | rexrd 9968 | A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ*) | ||
Theorem | renepnfd 9969 | No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≠ +∞) | ||
Theorem | renemnfd 9970 | No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≠ -∞) | ||
Theorem | pnfxr 9971 | 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 9972 | Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ +∞ ∈ V | ||
Theorem | pnfnemnf 9973 | Plus and minus infinity are different elements of ℝ*. (Contributed by NM, 14-Oct-2005.) |
⊢ +∞ ≠ -∞ | ||
Theorem | mnfnepnf 9974 | Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -∞ ≠ +∞ | ||
Theorem | mnfxr 9975 | 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 9976 | A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 ∈ ℝ* | ||
Theorem | renfdisj 9977 | The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (ℝ ∩ {+∞, -∞}) = ∅ | ||
Theorem | ltrelxr 9978 | 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ < ⊆ (ℝ* × ℝ*) | ||
Theorem | ltrel 9979 | 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.) |
⊢ Rel < | ||
Theorem | lerelxr 9980 | 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ ≤ ⊆ (ℝ* × ℝ*) | ||
Theorem | lerel 9981 | 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ Rel ≤ | ||
Theorem | xrlenlt 9982 | '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 | xrlenltd 9983 | 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
Theorem | xrltnle 9984 | 'Less than' expressed in terms of 'less than or equal to', for extended reals. (Contributed by NM, 6-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | ||
Theorem | xrnltled 9985 | 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → ¬ 𝐵 < 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | ssxr 9986 | The three (non-exclusive) possibilities implied by a subset of extended reals. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) | ||
Theorem | ltxrlt 9987 | 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 | axlttri 9988 | Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri 9889 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | axlttrn 9989 | Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn 9890 with ordering on the extended reals. New proofs should use lttr 9993 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 9990 | Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-ltadd 9891 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 | axmulgt0 9991 | The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 9892 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))) | ||
Theorem | axsup 9992* | A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-sup 9893 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | ||
Theorem | lttr 9993 | Alias for axlttrn 9989, for naming consistency with lttri 10042. New proofs should generally use this instead of ax-pre-lttrn 9890. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | mulgt0 9994 | 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 9995 | 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
Theorem | ltnle 9996 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | ||
Theorem | ltso 9997 | 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
⊢ < Or ℝ | ||
Theorem | gtso 9998 | 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
⊢ ◡ < Or ℝ | ||
Theorem | lttri2 9999 | Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | lttri3 10000 | Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
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