P. 112 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)

Type	Label	Description
Statement
Axiom	ax-pre-lttrn 11101	Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn 11077. Note: The more general version for extended reals is axlttrn 11205. Normally new proofs would use lttr 11209. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))
Axiom	ax-pre-ltadd 11102	Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by Theorem axpre-ltadd 11078. Normally new proofs would use axltadd 11206. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))
Axiom	ax-pre-mulgt0 11103	The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by Theorem axpre-mulgt0 11079. Normally new proofs would use axmulgt0 11207. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 <ℝ 𝐴 ∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵)))
Axiom	ax-pre-sup 11104*	A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by Theorem axpre-sup 11080. Note: Normally new proofs would use axsup 11208. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
Axiom	ax-addf 11105	Addition is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first-order or second-order statement (see https://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific addcl 11108 should be used. Note that uses of ax-addf 11105 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) in place of +, from which this axiom (with the defined operation in place of +) follows as a theorem. This axiom is justified by Theorem axaddf 11056. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
⊢ + :(ℂ × ℂ)⟶ℂ
Axiom	ax-mulf 11106	Multiplication is an operation on the complex numbers. This axiom tells us that · is defined only on complex numbers which is analogous to the way that other operations are defined, for example see subf 11382 or eff 16004. However, while Metamath can handle this axiom, if we wish to work with weaker complex number axioms, we can avoid it by using the less specific mulcl 11110. Note that uses of ax-mulf 11106 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11121. This axiom is justified by Theorem axmulf 11057. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
⊢ · :(ℂ × ℂ)⟶ℂ
5.2  Derive the basic properties from the field axioms
5.2.1  Some deductions from the field axioms for complex numbers
Theorem	cnex 11107	Alias for ax-cnex 11082. See also cnexALT 12899. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℂ ∈ V
Theorem	addcl 11108	Alias for ax-addcl 11086, for naming consistency with addcli 11138. Use this theorem instead of ax-addcl 11086 or axaddcl 11062. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Theorem	readdcl 11109	Alias for ax-addrcl 11087, for naming consistency with readdcli 11147. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	mulcl 11110	Alias for ax-mulcl 11088, for naming consistency with mulcli 11139. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Theorem	remulcl 11111	Alias for ax-mulrcl 11089, for naming consistency with remulcli 11148. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Theorem	mulcom 11112	Alias for ax-mulcom 11090, for naming consistency with mulcomi 11140. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addass 11113	Alias for ax-addass 11091, for naming consistency with addassi 11142. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	mulass 11114	Alias for ax-mulass 11092, for naming consistency with mulassi 11143. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	adddi 11115	Alias for ax-distr 11093, for naming consistency with adddii 11144. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Theorem	recn 11116	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
Theorem	reex 11117	The real numbers form a set. See also reexALT 12897. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℝ ∈ V
Theorem	reelprrecn 11118	Reals are a subset of the pair of real and complex numbers. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ℝ ∈ {ℝ, ℂ}
Theorem	cnelprrecn 11119	Complex numbers are a subset of the pair of real and complex numbers . (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ ℂ ∈ {ℝ, ℂ}
Theorem	mpoaddf 11120*	Addition is an operation on complex numbers. Version of ax-addf 11105 using maps-to notation, proved from the axioms of set theory and ax-addcl 11086. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)):(ℂ × ℂ)⟶ℂ
Theorem	mpomulf 11121*	Multiplication is an operation on complex numbers. Version of ax-mulf 11106 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11088. (Contributed by GG, 16-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
Theorem	elimne0 11122	Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.)
⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
Theorem	adddir 11123	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 11124	Zero is a complex number. See also 0cnALT 11368. (Contributed by NM, 19-Feb-2005.)
⊢ 0 ∈ ℂ
Theorem	0cnd 11125	Zero is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℂ)
Theorem	c0ex 11126	Zero is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ∈ V
Theorem	1cnd 11127	One is a complex number, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℂ)
Theorem	1ex 11128	One is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 1 ∈ V
Theorem	cnre 11129*	Alias for ax-cnre 11099, for naming consistency. (Contributed by NM, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	mulrid 11130	The number 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	mullid 11131	Identity law for multiplication. See mulrid 11130 for commuted version. (Contributed by NM, 8-Oct-1999.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	1re 11132	The number 1 is real. 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 11084, 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	1red 11133	The number 1 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℝ)
Theorem	0re 11134	The number 0 is real. Remark: the first step could also be ax-icn 11085. See also 0reALT 11478. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 11-Oct-2022.)
⊢ 0 ∈ ℝ
Theorem	0red 11135	The number 0 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℝ)
Theorem	mulridi 11136	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 1) = 𝐴
Theorem	mullidi 11137	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (1 · 𝐴) = 𝐴
Theorem	addcli 11138	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℂ
Theorem	mulcli 11139	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℂ
Theorem	mulcomi 11140	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)
Theorem	mulcomli 11141	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ (𝐵 · 𝐴) = 𝐶
Theorem	addassi 11142	Associative law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
Theorem	mulassi 11143	Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Theorem	adddii 11144	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 11145	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 11146	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	readdcli 11147	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℝ
Theorem	remulcli 11148	Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℝ
Theorem	mulridd 11149	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 1) = 𝐴)
Theorem	mullidd 11150	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐴)
Theorem	addcld 11151	Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
Theorem	mulcld 11152	Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
Theorem	mulcomd 11153	Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addassd 11154	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 11155	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 11156	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 11157	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 11158	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 11159	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 11160	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	readdcld 11161	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 11162	Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ)
5.2.2  Infinity and the extended real number system
Syntax	cpnf 11163	Plus infinity.
class +∞
Syntax	cmnf 11164	Minus infinity.
class -∞
Syntax	cxr 11165	The set of extended reals (includes plus and minus infinity).
class ℝ*
Syntax	clt 11166	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 11167	Extend wff notation to include the 'less than or equal to' relation.
class ≤
Definition	df-pnf 11168	Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in ℝ and different from -∞ (df-mnf 11169). 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 11173, mnfnre 11175, and pnfnemnf 11187. 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 11169	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 11175 and pnfnemnf 11187). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
⊢ -∞ = 𝒫 +∞
Definition	df-xr 11170	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 11171*	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 11172	Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 11219 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)
⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
Theorem	pnfnre 11173	Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
⊢ +∞ ∉ ℝ
Theorem	pnfnre2 11174	Plus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ ¬ +∞ ∈ ℝ
Theorem	mnfnre 11175	Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
⊢ -∞ ∉ ℝ
Theorem	ressxr 11176	The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
⊢ ℝ ⊆ ℝ*
Theorem	rexpssxrxp 11177	The Cartesian product of standard reals are a subset of the Cartesian product of extended reals. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
Theorem	rexr 11178	A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
Theorem	0xr 11179	Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
⊢ 0 ∈ ℝ*
Theorem	renepnf 11180	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 11181	No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
Theorem	rexrd 11182	A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	renepnfd 11183	No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≠ +∞)
Theorem	renemnfd 11184	No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≠ -∞)
Theorem	pnfex 11185	Plus infinity exists. (Contributed by David A. Wheeler, 8-Dec-2018.) (Revised by Steven Nguyen, 7-Dec-2022.)
⊢ +∞ ∈ V
Theorem	pnfxr 11186	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	pnfnemnf 11187	Plus and minus infinity are different elements of ℝ*. (Contributed by NM, 14-Oct-2005.)
⊢ +∞ ≠ -∞
Theorem	mnfnepnf 11188	Minus and plus infinity are different. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -∞ ≠ +∞
Theorem	mnfxr 11189	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 11190	A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ∈ ℝ*
Theorem	1xr 11191	1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 1 ∈ ℝ*
Theorem	renfdisj 11192	The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (ℝ ∩ {+∞, -∞}) = ∅
Theorem	ltrelxr 11193	"Less than" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ < ⊆ (ℝ* × ℝ*)
Theorem	ltrel 11194	"Less than" is a relation. (Contributed by NM, 14-Oct-2005.)
⊢ Rel <
Theorem	lerelxr 11195	"Less than or equal to" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ ≤ ⊆ (ℝ* × ℝ*)
Theorem	lerel 11196	"Less than or equal to" is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ Rel ≤
Theorem	xrlenlt 11197	"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 11198	"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 11199	"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	xrltnled 11200	'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))