P. 114 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)

Type	Label	Description
Statement
Axiom	ax-pre-lttri 11301	Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by Theorem axpre-lttri 11277. Note: The more general version for extended reals is axlttri 11408. Normally new proofs would use xrlttri 13295. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
Axiom	ax-pre-lttrn 11302	Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by Theorem axpre-lttrn 11278. Note: The more general version for extended reals is axlttrn 11409. Normally new proofs would use lttr 11413. (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 11303	Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by Theorem axpre-ltadd 11279. Normally new proofs would use axltadd 11410. (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 11304	The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by Theorem axpre-mulgt0 11280. Normally new proofs would use axmulgt0 11411. (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 11305*	A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by Theorem axpre-sup 11281. Note: Normally new proofs would use axsup 11412. (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 11306	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 11309 should be used. Note that uses of ax-addf 11306 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 11257. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)
⊢ + :(ℂ × ℂ)⟶ℂ
Axiom	ax-mulf 11307	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 11586 or eff 16276. 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 11311. Note that uses of ax-mulf 11307 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11322. This axiom is justified by Theorem axmulf 11258. (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 11308	Alias for ax-cnex 11283. See also cnexALT 13141. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℂ ∈ V
Theorem	addcl 11309	Alias for ax-addcl 11287, for naming consistency with addcli 11342. Use this theorem instead of ax-addcl 11287 or axaddcl 11263. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
Theorem	readdcl 11310	Alias for ax-addrcl 11288, for naming consistency with readdcli 11351. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
Theorem	mulcl 11311	Alias for ax-mulcl 11289, for naming consistency with mulcli 11343. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Theorem	remulcl 11312	Alias for ax-mulrcl 11290, for naming consistency with remulcli 11352. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Theorem	mulcom 11313	Alias for ax-mulcom 11291, for naming consistency with mulcomi 11344. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addass 11314	Alias for ax-addass 11292, for naming consistency with addassi 11346. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	mulass 11315	Alias for ax-mulass 11293, for naming consistency with mulassi 11347. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	adddi 11316	Alias for ax-distr 11294, for naming consistency with adddii 11348. (Contributed by NM, 10-Mar-2008.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Theorem	recn 11317	A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
Theorem	reex 11318	The real numbers form a set. See also reexALT 13139. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ ℝ ∈ V
Theorem	reelprrecn 11319	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 11320	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 11321*	Addition is an operation on complex numbers. Version of ax-addf 11306 using maps-to notation, proved from the axioms of set theory and ax-addcl 11287. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)):(ℂ × ℂ)⟶ℂ
Theorem	mpomulf 11322*	Multiplication is an operation on complex numbers. Version of ax-mulf 11307 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11289. (Contributed by GG, 16-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
Theorem	elimne0 11323	Hypothesis for weak deduction theorem to eliminate 𝐴 ≠ 0. (Contributed by NM, 15-May-1999.)
⊢ if(𝐴 ≠ 0, 𝐴, 1) ≠ 0
Theorem	adddir 11324	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 11325	Zero is a complex number. See also 0cnALT 11572. (Contributed by NM, 19-Feb-2005.)
⊢ 0 ∈ ℂ
Theorem	0cnd 11326	Zero is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℂ)
Theorem	c0ex 11327	Zero is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 0 ∈ V
Theorem	0elpr01 11328	0 is an element of {0, 1}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 ∈ {0, 1}
Theorem	1cnd 11329	One is a complex number, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℂ)
Theorem	1ex 11330	One is a set. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ 1 ∈ V
Theorem	1elpr01 11331	1 is an element of {0, 1}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1 ∈ {0, 1}
Theorem	cnre 11332*	Alias for ax-cnre 11300, for naming consistency. (Contributed by NM, 3-Jan-2013.)
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	mulrid 11333	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 11334	Identity law for multiplication. See mulrid 11333 for commuted version. (Contributed by NM, 8-Oct-1999.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	1re 11335	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 11285, 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 11336	The number 1 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 1 ∈ ℝ)
Theorem	0re 11337	The number 0 is real. Remark: the first step could also be ax-icn 11286. See also 0reALT 11682. (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 11338	The number 0 is real, deduction form. (Contributed by David A. Wheeler, 6-Dec-2018.)
⊢ (𝜑 → 0 ∈ ℝ)
Theorem	pr01ssre 11339	The pair {0, 1} is a subset of ℝ. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ {0, 1} ⊆ ℝ
Theorem	mulridi 11340	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (𝐴 · 1) = 𝐴
Theorem	mullidi 11341	Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
⊢ 𝐴 ∈ ℂ    ⇒   ⊢ (1 · 𝐴) = 𝐴
Theorem	addcli 11342	Closure law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℂ
Theorem	mulcli 11343	Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℂ
Theorem	mulcomi 11344	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)
Theorem	mulcomli 11345	Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ (𝐵 · 𝐴) = 𝐶
Theorem	addassi 11346	Associative law for addition. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
Theorem	mulassi 11347	Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    ⇒   ⊢ ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Theorem	adddii 11348	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 11349	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 11350	A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	readdcli 11351	Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 + 𝐵) ∈ ℝ
Theorem	remulcli 11352	Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℝ
Theorem	mulridd 11353	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 1) = 𝐴)
Theorem	mullidd 11354	Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐴)
Theorem	addcld 11355	Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ)
Theorem	mulcld 11356	Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
Theorem	mulcomd 11357	Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	addassd 11358	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 11359	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 11360	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 11361	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 11362	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 11363	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 11364	Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	readdcld 11365	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 11366	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 11367	Plus infinity.
class +∞
Syntax	cmnf 11368	Minus infinity.
class -∞
Syntax	cxr 11369	The set of extended reals (includes plus and minus infinity).
class ℝ*
Syntax	clt 11370	'Less than' predicate (extended to include the extended reals).
class <
Syntax	cle 11371	Extend wff notation to include the 'less than or equal to' relation.
class ≤
Definition	df-pnf 11372	Define plus infinity. Note that the definition is arbitrary, requiring only that +∞ be a set not in ℝ and different from -∞ (df-mnf 11373). 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 11377, mnfnre 11379, and pnfnemnf 11391. 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 11373	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 11379 and pnfnemnf 11391). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)
⊢ -∞ = 𝒫 +∞
Definition	df-xr 11374	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 11375*	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 11376	Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 11423 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)
⊢ ≤ = ((ℝ* × ℝ*) ∖ ◡ < )
Theorem	pnfnre 11377	Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
⊢ +∞ ∉ ℝ
Theorem	pnfnre2 11378	Plus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ ¬ +∞ ∈ ℝ
Theorem	mnfnre 11379	Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
⊢ -∞ ∉ ℝ
Theorem	ressxr 11380	The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)
⊢ ℝ ⊆ ℝ*
Theorem	rexpssxrxp 11381	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 11382	A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)
⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
Theorem	0xr 11383	Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)
⊢ 0 ∈ ℝ*
Theorem	renepnf 11384	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 11385	No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
Theorem	rexrd 11386	A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	renepnfd 11387	No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≠ +∞)
Theorem	renemnfd 11388	No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≠ -∞)
Theorem	pnfex 11389	Plus infinity exists. (Contributed by David A. Wheeler, 8-Dec-2018.) (Revised by Steven Nguyen, 7-Dec-2022.)
⊢ +∞ ∈ V
Theorem	pnfxr 11390	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 11391	Plus and minus infinity are different elements of ℝ*. (Contributed by NM, 14-Oct-2005.)
⊢ +∞ ≠ -∞
Theorem	mnfnepnf 11392	Minus and plus infinity are different. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -∞ ≠ +∞
Theorem	mnfxr 11393	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 11394	A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ∈ ℝ*
Theorem	1xr 11395	1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ 1 ∈ ℝ*
Theorem	renfdisj 11396	The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (ℝ ∩ {+∞, -∞}) = ∅
Theorem	ltrelxr 11397	"Less than" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ < ⊆ (ℝ* × ℝ*)
Theorem	ltrel 11398	"Less than" is a relation. (Contributed by NM, 14-Oct-2005.)
⊢ Rel <
Theorem	lerelxr 11399	"Less than or equal to" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ ≤ ⊆ (ℝ* × ℝ*)
Theorem	lerel 11400	"Less than or equal to" is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ Rel ≤