P. 132 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rpcnd 13101	A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	rpgt0d 13102	A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 < 𝐴)
Theorem	rpge0d 13103	A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 ≤ 𝐴)
Theorem	rpne0d 13104	A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	rpregt0d 13105	A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	rprege0d 13106	A positive real is real and greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
Theorem	rprene0d 13107	A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
Theorem	rpcnne0d 13108	A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
Theorem	rpreccld 13109	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+)
Theorem	rprecred 13110	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
Theorem	rphalfcld 13111	Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ+)
Theorem	reclt1d 13112	The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
Theorem	recgt1d 13113	The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ (1 / 𝐴) < 1))
Theorem	rpaddcld 13114	Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ+)
Theorem	rpmulcld 13115	Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ+)
Theorem	rpdivcld 13116	Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ+)
Theorem	ltrecd 13117	The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)))
Theorem	lerecd 13118	The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))
Theorem	ltrec1d 13119	Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → (1 / 𝐴) < 𝐵)    ⇒   ⊢ (𝜑 → (1 / 𝐵) < 𝐴)
Theorem	lerec2d 13120	Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 ≤ (1 / 𝐵))    ⇒   ⊢ (𝜑 → 𝐵 ≤ (1 / 𝐴))
Theorem	lediv2ad 13121	Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))
Theorem	ltdiv2d 13122	Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 / 𝐵) < (𝐶 / 𝐴)))
Theorem	lediv2d 13123	Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
Theorem	ledivdivd 13124	Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴 / 𝐵) ≤ (𝐶 / 𝐷))    ⇒   ⊢ (𝜑 → (𝐷 / 𝐶) ≤ (𝐵 / 𝐴))
Theorem	divge1 13125	The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 1 ≤ (𝐵 / 𝐴))
Theorem	divlt1lt 13126	A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) < 1 ↔ 𝐴 < 𝐵))
Theorem	divle1le 13127	A real number divided by a positive real number is less than or equal to 1 iff the real number is less than or equal to the positive real number. (Contributed by AV, 29-Jun-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ 𝐵))
Theorem	ledivge1le 13128	If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 1 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐴 / 𝐶) ≤ 𝐵))
Theorem	ge0p1rpd 13129	A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+)
Theorem	rerpdivcld 13130	Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
Theorem	ltsubrpd 13131	Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐴)
Theorem	ltaddrpd 13132	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵))
Theorem	ltaddrp2d 13133	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐴))
Theorem	ltmulgt11d 13134	Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐵 · 𝐴)))
Theorem	ltmulgt12d 13135	Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐴 · 𝐵)))
Theorem	gt0divd 13136	Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵)))
Theorem	ge0divd 13137	Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵)))
Theorem	rpgecld 13138	A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ+)
Theorem	divge0d 13139	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 / 𝐵))
Theorem	ltmul1d 13140	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
Theorem	ltmul2d 13141	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))
Theorem	lemul1d 13142	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
Theorem	lemul2d 13143	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
Theorem	ltdiv1d 13144	Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)))
Theorem	lediv1d 13145	Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)))
Theorem	ltmuldivd 13146	'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶)))
Theorem	ltmuldiv2d 13147	'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐶 · 𝐴) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶)))
Theorem	lemuldivd 13148	'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶)))
Theorem	lemuldiv2d 13149	'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐶 · 𝐴) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶)))
Theorem	ltdivmuld 13150	'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 / 𝐶) < 𝐵 ↔ 𝐴 < (𝐶 · 𝐵)))
Theorem	ltdivmul2d 13151	'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 / 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 · 𝐶)))
Theorem	ledivmuld 13152	'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐶 · 𝐵)))
Theorem	ledivmul2d 13153	'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 · 𝐶)))
Theorem	ltmul1dd 13154	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐶))
Theorem	ltmul2dd 13155	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 · 𝐴) < (𝐶 · 𝐵))
Theorem	ltdiv1dd 13156	Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) < (𝐵 / 𝐶))
Theorem	lediv1dd 13157	Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) ≤ (𝐵 / 𝐶))
Theorem	lediv12ad 13158	Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶))
Theorem	mul2lt0rlt0 13159	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴)
Theorem	mul2lt0rgt0 13160	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ ((𝜑 ∧ 0 < 𝐵) → 𝐴 < 0)
Theorem	mul2lt0llt0 13161	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ ((𝜑 ∧ 𝐴 < 0) → 0 < 𝐵)
Theorem	mul2lt0lgt0 13162	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 2-Oct-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐵 < 0)
Theorem	mul2lt0bi 13163	If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) < 0 ↔ ((𝐴 < 0 ∧ 0 < 𝐵) ∨ (0 < 𝐴 ∧ 𝐵 < 0))))
Theorem	prodge0rd 13164	Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Revised by AV, 9-Jul-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵))    ⇒   ⊢ (𝜑 → 0 ≤ 𝐵)
Theorem	prodge0ld 13165	Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by AV, 9-Jul-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵))    ⇒   ⊢ (𝜑 → 0 ≤ 𝐴)
Theorem	ltdiv23d 13166	Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴 / 𝐵) < 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) < 𝐵)
Theorem	lediv23d 13167	Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴 / 𝐵) ≤ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) ≤ 𝐵)
Theorem	lt2mul2divd 13168	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵)))
Theorem	nnledivrp 13169	Division of a positive integer by a positive number is less than or equal to the integer iff the number is greater than or equal to 1. (Contributed by AV, 19-Jun-2021.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → (1 ≤ 𝐵 ↔ (𝐴 / 𝐵) ≤ 𝐴))
Theorem	nn0ledivnn 13170	Division of a nonnegative integer by a positive integer is less than or equal to the integer. (Contributed by AV, 19-Jun-2021.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≤ 𝐴)
Theorem	addlelt 13171	If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.)
⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → ((𝑀 + 𝐴) ≤ 𝑁 → 𝑀 < 𝑁))
5.5.2  Infinity and the extended real number system (cont.)
Syntax	cxne 13172	Extend class notation to include the negative of an extended real.
class -𝑒𝐴
Syntax	cxad 13173	Extend class notation to include addition of extended reals.
class +𝑒
Syntax	cxmu 13174	Extend class notation to include multiplication of extended reals.
class ·e
Definition	df-xneg 13175	Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)
⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
Definition	df-xadd 13176*	Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
Definition	df-xmul 13177*	Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ·e = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))))
Theorem	ltxr 13178	The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
Theorem	elxr 13179	Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
Theorem	xrnemnf 13180	An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
Theorem	xrnepnf 13181	An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
Theorem	xrltnr 13182	The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)
⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
Theorem	ltpnf 13183	Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
Theorem	ltpnfd 13184	Any (finite) real is less than plus infinity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 < +∞)
Theorem	0ltpnf 13185	Zero is less than plus infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 < +∞
Theorem	mnflt 13186	Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
Theorem	mnfltd 13187	Minus infinity is less than any (finite) real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → -∞ < 𝐴)
Theorem	mnflt0 13188	Minus infinity is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -∞ < 0
Theorem	mnfltpnf 13189	Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)
⊢ -∞ < +∞
Theorem	mnfltxr 13190	Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)
⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴)
Theorem	pnfnlt 13191	No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)
⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴)
Theorem	nltmnf 13192	No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)
⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞)
Theorem	pnfge 13193	Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)
⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞)
Theorem	xnn0n0n1ge2b 13194	An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by AV, 5-Apr-2021.)
⊢ (𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁))
Theorem	0lepnf 13195	0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 0 ≤ +∞
Theorem	xnn0ge0 13196	An extended nonnegative integer is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 10-Dec-2020.)
⊢ (𝑁 ∈ ℕ0* → 0 ≤ 𝑁)
Theorem	mnfle 13197	Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)
⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴)
Theorem	mnfled 13198	Minus infinity is less than or equal to any extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → -∞ ≤ 𝐴)
Theorem	xrltnsym 13199	Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴))
Theorem	xrltnsym2 13200	'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵 ∧ 𝐵 < 𝐴))