P. 77 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prodge0 7601	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.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (0 < A ∧ 0 ≤ (A · B))) → 0 ≤ B)
Theorem	prodge02 7602	Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (0 < B ∧ 0 ≤ (A · B))) → 0 ≤ A)
Theorem	ltmul2 7603	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A < B ↔ (𝐶 · A) < (𝐶 · B)))
Theorem	lemul2 7604	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A ≤ B ↔ (𝐶 · A) ≤ (𝐶 · B)))
Theorem	lemul1a 7605	Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 21-Feb-2005.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ A ≤ B) → (A · 𝐶) ≤ (B · 𝐶))
Theorem	lemul2a 7606	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ A ≤ B) → (𝐶 · A) ≤ (𝐶 · B))
Theorem	ltmul12a 7607	Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
⊢ ((((A ∈ ℝ ∧ B ∈ ℝ) ∧ (0 ≤ A ∧ A < B)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 < 𝐷))) → (A · 𝐶) < (B · 𝐷))
Theorem	lemul12b 7608	Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
⊢ ((((A ∈ ℝ ∧ 0 ≤ A) ∧ B ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 ≤ 𝐷))) → ((A ≤ B ∧ 𝐶 ≤ 𝐷) → (A · 𝐶) ≤ (B · 𝐷)))
Theorem	lemul12a 7609	Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
⊢ ((((A ∈ ℝ ∧ 0 ≤ A) ∧ B ∈ ℝ) ∧ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) ∧ 𝐷 ∈ ℝ)) → ((A ≤ B ∧ 𝐶 ≤ 𝐷) → (A · 𝐶) ≤ (B · 𝐷)))
Theorem	mulgt1 7610	The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (1 < A ∧ 1 < B)) → 1 < (A · B))
Theorem	ltmulgt11 7611	Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 0 < A) → (1 < B ↔ A < (A · B)))
Theorem	ltmulgt12 7612	Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 0 < A) → (1 < B ↔ A < (B · A)))
Theorem	lemulge11 7613	Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (0 ≤ A ∧ 1 ≤ B)) → A ≤ (A · B))
Theorem	lemulge12 7614	Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧ (0 ≤ A ∧ 1 ≤ B)) → A ≤ (B · A))
Theorem	ltdiv1 7615	Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A < B ↔ (A / 𝐶) < (B / 𝐶)))
Theorem	lediv1 7616	Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A ≤ B ↔ (A / 𝐶) ≤ (B / 𝐶)))
Theorem	gt0div 7617	Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 0 < B) → (0 < A ↔ 0 < (A / B)))
Theorem	ge0div 7618	Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 0 < B) → (0 ≤ A ↔ 0 ≤ (A / B)))
Theorem	divgt0 7619	The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
⊢ (((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B)) → 0 < (A / B))
Theorem	divge0 7620	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 < B)) → 0 ≤ (A / B))
Theorem	ltmuldiv 7621	'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A · 𝐶) < B ↔ A < (B / 𝐶)))
Theorem	ltmuldiv2 7622	'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · A) < B ↔ A < (B / 𝐶)))
Theorem	ltdivmul 7623	'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A / 𝐶) < B ↔ A < (𝐶 · B)))
Theorem	ledivmul 7624	'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A / 𝐶) ≤ B ↔ A ≤ (𝐶 · B)))
Theorem	ltdivmul2 7625	'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A / 𝐶) < B ↔ A < (B · 𝐶)))
Theorem	lt2mul2div 7626	'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
⊢ (((A ∈ ℝ ∧ (B ∈ ℝ ∧ 0 < B)) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((A · B) < (𝐶 · 𝐷) ↔ (A / 𝐷) < (𝐶 / B)))
Theorem	ledivmul2 7627	'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A / 𝐶) ≤ B ↔ A ≤ (B · 𝐶)))
Theorem	lemuldiv 7628	'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A · 𝐶) ≤ B ↔ A ≤ (B / 𝐶)))
Theorem	lemuldiv2 7629	'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · A) ≤ B ↔ A ≤ (B / 𝐶)))
Theorem	ltrec 7630	The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B)) → (A < B ↔ (1 / B) < (1 / A)))
Theorem	lerec 7631	The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B)) → (A ≤ B ↔ (1 / B) ≤ (1 / A)))
Theorem	lt2msq1 7632	Lemma for lt2msq 7633. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ B ∈ ℝ ∧ A < B) → (A · A) < (B · B))
Theorem	lt2msq 7633	Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → (A < B ↔ (A · A) < (B · B)))
Theorem	ltdiv2 7634	Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
⊢ (((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A < B ↔ (𝐶 / B) < (𝐶 / A)))
Theorem	ltrec1 7635	Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
⊢ (((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B)) → ((1 / A) < B ↔ (1 / B) < A))
Theorem	lerec2 7636	Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
⊢ (((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B)) → (A ≤ (1 / B) ↔ B ≤ (1 / A)))
Theorem	ledivdiv 7637	Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
⊢ ((((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B)) ∧ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((A / B) ≤ (𝐶 / 𝐷) ↔ (𝐷 / 𝐶) ≤ (B / A)))
Theorem	lediv2 7638	Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
⊢ (((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A ≤ B ↔ (𝐶 / B) ≤ (𝐶 / A)))
Theorem	ltdiv23 7639	Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
⊢ ((A ∈ ℝ ∧ (B ∈ ℝ ∧ 0 < B) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A / B) < 𝐶 ↔ (A / 𝐶) < B))
Theorem	lediv23 7640	Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
⊢ ((A ∈ ℝ ∧ (B ∈ ℝ ∧ 0 < B) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A / B) ≤ 𝐶 ↔ (A / 𝐶) ≤ B))
Theorem	lediv12a 7641	Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
⊢ ((((A ∈ ℝ ∧ B ∈ ℝ) ∧ (0 ≤ A ∧ A ≤ B)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 < 𝐶 ∧ 𝐶 ≤ 𝐷))) → (A / 𝐷) ≤ (B / 𝐶))
Theorem	lediv2a 7642	Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((((A ∈ ℝ ∧ 0 < A) ∧ (B ∈ ℝ ∧ 0 < B) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ A ≤ B) → (𝐶 / B) ≤ (𝐶 / A))
Theorem	reclt1 7643	The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)
⊢ ((A ∈ ℝ ∧ 0 < A) → (A < 1 ↔ 1 < (1 / A)))
Theorem	recgt1 7644	The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)
⊢ ((A ∈ ℝ ∧ 0 < A) → (1 < A ↔ (1 / A) < 1))
Theorem	recgt1i 7645	The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)
⊢ ((A ∈ ℝ ∧ 1 < A) → (0 < (1 / A) ∧ (1 / A) < 1))
Theorem	recp1lt1 7646	Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (A / (1 + A)) < 1)
Theorem	recreclt 7647	Given a positive number A, construct a new positive number less than both A and 1. (Contributed by NM, 28-Dec-2005.)
⊢ ((A ∈ ℝ ∧ 0 < A) → ((1 / (1 + (1 / A))) < 1 ∧ (1 / (1 + (1 / A))) < A))
Theorem	le2msq 7648	The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → (A ≤ B ↔ (A · A) ≤ (B · B)))
Theorem	msq11 7649	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → ((A · A) = (B · B) ↔ A = B))
Theorem	ledivp1 7650	Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
⊢ (((A ∈ ℝ ∧ 0 ≤ A) ∧ (B ∈ ℝ ∧ 0 ≤ B)) → ((A / (B + 1)) · B) ≤ A)
Theorem	squeeze0 7651*	If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
⊢ ((A ∈ ℝ ∧ 0 ≤ A ∧ ∀x ∈ ℝ (0 < x → A < x)) → A = 0)
Theorem	ltp1i 7652	A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
⊢ A ∈ ℝ    ⇒   ⊢ A < (A + 1)
Theorem	recgt0i 7653	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
⊢ A ∈ ℝ    ⇒   ⊢ (0 < A → 0 < (1 / A))
Theorem	recgt0ii 7654	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
⊢ A ∈ ℝ    &   ⊢ 0 < A    ⇒   ⊢ 0 < (1 / A)
Theorem	prodgt0i 7655	Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 < (A · B)) → 0 < B)
Theorem	prodge0i 7656	Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 < A ∧ 0 ≤ (A · B)) → 0 ≤ B)
Theorem	divgt0i 7657	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 < A ∧ 0 < B) → 0 < (A / B))
Theorem	divge0i 7658	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 < B) → 0 ≤ (A / B))
Theorem	ltreci 7659	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 < A ∧ 0 < B) → (A < B ↔ (1 / B) < (1 / A)))
Theorem	lereci 7660	The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 < A ∧ 0 < B) → (A ≤ B ↔ (1 / B) ≤ (1 / A)))
Theorem	lt2msqi 7661	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 ≤ B) → (A < B ↔ (A · A) < (B · B)))
Theorem	le2msqi 7662	The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 ≤ B) → (A ≤ B ↔ (A · A) ≤ (B · B)))
Theorem	msq11i 7663	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    ⇒   ⊢ ((0 ≤ A ∧ 0 ≤ B) → ((A · A) = (B · B) ↔ A = B))
Theorem	divgt0i2i 7664	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 0 < B    ⇒   ⊢ (0 < A → 0 < (A / B))
Theorem	ltrecii 7665	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 0 < A    &   ⊢ 0 < B    ⇒   ⊢ (A < B ↔ (1 / B) < (1 / A))
Theorem	divgt0ii 7666	The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 0 < A    &   ⊢ 0 < B    ⇒   ⊢ 0 < (A / B)
Theorem	ltmul1i 7667	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A < B ↔ (A · 𝐶) < (B · 𝐶)))
Theorem	ltdiv1i 7668	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A < B ↔ (A / 𝐶) < (B / 𝐶)))
Theorem	ltmuldivi 7669	'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → ((A · 𝐶) < B ↔ A < (B / 𝐶)))
Theorem	ltmul2i 7670	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A < B ↔ (𝐶 · A) < (𝐶 · B)))
Theorem	lemul1i 7671	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A ≤ B ↔ (A · 𝐶) ≤ (B · 𝐶)))
Theorem	lemul2i 7672	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (A ≤ B ↔ (𝐶 · A) ≤ (𝐶 · B)))
Theorem	ltdiv23i 7673	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((0 < B ∧ 0 < 𝐶) → ((A / B) < 𝐶 ↔ (A / 𝐶) < B))
Theorem	ltdiv23ii 7674	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < B    &   ⊢ 0 < 𝐶    ⇒   ⊢ ((A / B) < 𝐶 ↔ (A / 𝐶) < B)
Theorem	ltmul1ii 7675	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < 𝐶    ⇒   ⊢ (A < B ↔ (A · 𝐶) < (B · 𝐶))
Theorem	ltdiv1ii 7676	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
⊢ A ∈ ℝ    &   ⊢ B ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < 𝐶    ⇒   ⊢ (A < B ↔ (A / 𝐶) < (B / 𝐶))
Theorem	ltp1d 7677	A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → A < (A + 1))
Theorem	lep1d 7678	A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → A ≤ (A + 1))
Theorem	ltm1d 7679	A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → (A − 1) < A)
Theorem	lem1d 7680	A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    ⇒   ⊢ (φ → (A − 1) ≤ A)
Theorem	recgt0d 7681	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → 0 < A)    ⇒   ⊢ (φ → 0 < (1 / A))
Theorem	divgt0d 7682	The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 0 < A)    &   ⊢ (φ → 0 < B)    ⇒   ⊢ (φ → 0 < (A / B))
Theorem	mulgt1d 7683	The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 1 < A)    &   ⊢ (φ → 1 < B)    ⇒   ⊢ (φ → 1 < (A · B))
Theorem	lemulge11d 7684	Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 0 ≤ A)    &   ⊢ (φ → 1 ≤ B)    ⇒   ⊢ (φ → A ≤ (A · B))
Theorem	lemulge12d 7685	Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (φ → A ∈ ℝ)    &   ⊢ (φ → B ∈ ℝ)    &   ⊢ (φ → 0 ≤ A)    &   ⊢ (φ → 1 ≤ B)    ⇒   ⊢ (φ → A ≤ (B · A))
Theorem	lemul1ad 7686	Multiplication of both sides of 'less than or equal to' by 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 → (A · 𝐶) ≤ (B · 𝐶))
Theorem	lemul2ad 7687	Multiplication of both sides of 'less than or equal to' by 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 · A) ≤ (𝐶 · B))
Theorem	ltmul12ad 7688	Comparison of product of two positive 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 → 0 ≤ 𝐶)    &   ⊢ (φ → 𝐶 < 𝐷)    ⇒   ⊢ (φ → (A · 𝐶) < (B · 𝐷))
Theorem	lemul12ad 7689	Comparison of product 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 → 0 ≤ 𝐶)    &   ⊢ (φ → A ≤ B)    &   ⊢ (φ → 𝐶 ≤ 𝐷)    ⇒   ⊢ (φ → (A · 𝐶) ≤ (B · 𝐷))
Theorem	lemul12bd 7690	Comparison of product 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 → 0 ≤ 𝐷)    &   ⊢ (φ → A ≤ B)    &   ⊢ (φ → 𝐶 ≤ 𝐷)    ⇒   ⊢ (φ → (A · 𝐶) ≤ (B · 𝐷))
3.3.10  Imaginary and complex number properties
Theorem	crap0 7691	The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → ((A # 0 ∨ B # 0) ↔ (A + (i · B)) # 0))
Theorem	creur 7692*	The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (A ∈ ℂ → ∃!x ∈ ℝ ∃y ∈ ℝ A = (x + (i · y)))
Theorem	creui 7693*	The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (A ∈ ℂ → ∃!y ∈ ℝ ∃x ∈ ℝ A = (x + (i · y)))
Theorem	cju 7694*	The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
⊢ (A ∈ ℂ → ∃!x ∈ ℂ ((A + x) ∈ ℝ ∧ (i · (A − x)) ∈ ℝ))
3.4  Integer sets
3.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 7695	Extend class notation to include the class of positive integers.
class ℕ
Definition	df-inn 7696*	Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 7697 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
⊢ ℕ = ∩ {x ∣ (1 ∈ x ∧ ∀y ∈ x (y + 1) ∈ x)}
Theorem	dfnn2 7697*	Definition of the set of positive integers. Another name for df-inn 7696. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ ℕ = ∩ {x ∣ (1 ∈ x ∧ ∀y ∈ x (y + 1) ∈ x)}
Theorem	peano5nni 7698*	Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ((1 ∈ A ∧ ∀x ∈ A (x + 1) ∈ A) → ℕ ⊆ A)
Theorem	nnssre 7699	The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
⊢ ℕ ⊆ ℝ
Theorem	nnsscn 7700	The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℕ ⊆ ℂ