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Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecidapd 7501 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (A · (1 / A)) = 1)
 
Theoremrecidap2d 7502 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → ((1 / A) · A) = 1)
 
Theoremrecrecapd 7503 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (1 / (1 / A)) = A)
 
Theoremdividapd 7504 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (A / A) = 1)
 
Theoremdiv0apd 7505 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φA # 0)       (φ → (0 / A) = 0)
 
Theoremapmul1 7506 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
((A B (𝐶 𝐶 # 0)) → (A # B ↔ (A · 𝐶) # (B · 𝐶)))
 
Theoremdivclapd 7507 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (A / B) ℂ)
 
Theoremdivcanap1d 7508 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A / B) · B) = A)
 
Theoremdivcanap2d 7509 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (B · (A / B)) = A)
 
Theoremdivrecapd 7510 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (A / B) = (A · (1 / B)))
 
Theoremdivrecap2d 7511 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (A / B) = ((1 / B) · A))
 
Theoremdivcanap3d 7512 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((B · A) / B) = A)
 
Theoremdivcanap4d 7513 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A · B) / B) = A)
 
Theoremdiveqap0d 7514 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)    &   (φ → (A / B) = 0)       (φA = 0)
 
Theoremdiveqap1d 7515 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)    &   (φ → (A / B) = 1)       (φA = B)
 
Theoremdiveqap1ad 7516 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7424. Generalization of diveqap1d 7515. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A / B) = 1 ↔ A = B))
 
Theoremdiveqap0ad 7517 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7403. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → ((A / B) = 0 ↔ A = 0))
 
Theoremdivap1d 7518 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)    &   (φA # B)       (φ → (A / B) # 1)
 
Theoremdivap0bd 7519 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (A # 0 ↔ (A / B) # 0))
 
Theoremdivnegapd 7520 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → -(A / B) = (-A / B))
 
Theoremdivneg2apd 7521 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → -(A / B) = (A / -B))
 
Theoremdiv2negapd 7522 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φB # 0)       (φ → (-A / -B) = (A / B))
 
Theoremdivap0d 7523 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → (A / B) # 0)
 
Theoremrecdivapd 7524 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → (1 / (A / B)) = (B / A))
 
Theoremrecdivap2d 7525 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → ((1 / A) / B) = (1 / (A · B)))
 
Theoremdivcanap6d 7526 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → ((A / B) · (B / A)) = 1)
 
Theoremddcanapd 7527 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)       (φ → (A / (A / B)) = B)
 
Theoremrec11apd 7528 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φB # 0)    &   (φ → (1 / A) = (1 / B))       (φA = B)
 
Theoremdivmulapd 7529 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)       (φ → ((A / B) = 𝐶 ↔ (B · 𝐶) = A))
 
Theoremdiv32apd 7530 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)       (φ → ((A / B) · 𝐶) = (A · (𝐶 / B)))
 
Theoremdiv13apd 7531 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)       (φ → ((A / B) · 𝐶) = ((𝐶 / B) · A))
 
Theoremdivdiv32apd 7532 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A / B) / 𝐶) = ((A / 𝐶) / B))
 
Theoremdivcanap5d 7533 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((𝐶 · A) / (𝐶 · B)) = (A / B))
 
Theoremdivcanap5rd 7534 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A · 𝐶) / (B · 𝐶)) = (A / B))
 
Theoremdivcanap7d 7535 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A / 𝐶) / (B / 𝐶)) = (A / B))
 
Theoremdmdcanapd 7536 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((B / 𝐶) · (A / B)) = (A / 𝐶))
 
Theoremdmdcanap2d 7537 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A / B) · (B / 𝐶)) = (A / 𝐶))
 
Theoremdivdivap1d 7538 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → ((A / B) / 𝐶) = (A / (B · 𝐶)))
 
Theoremdivdivap2d 7539 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB # 0)    &   (φ𝐶 # 0)       (φ → (A / (B / 𝐶)) = ((A · 𝐶) / B))
 
Theoremdivmulap2d 7540 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A / 𝐶) = BA = (𝐶 · B)))
 
Theoremdivmulap3d 7541 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A / 𝐶) = BA = (B · 𝐶)))
 
Theoremdivassapd 7542 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A · B) / 𝐶) = (A · (B / 𝐶)))
 
Theoremdiv12apd 7543 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → (A · (B / 𝐶)) = (B · (A / 𝐶)))
 
Theoremdiv23apd 7544 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A · B) / 𝐶) = ((A / 𝐶) · B))
 
Theoremdivdirapd 7545 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((A + B) / 𝐶) = ((A / 𝐶) + (B / 𝐶)))
 
Theoremdivsubdirapd 7546 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)       (φ → ((AB) / 𝐶) = ((A / 𝐶) − (B / 𝐶)))
 
Theoremdiv11apd 7547 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐶 # 0)    &   (φ → (A / 𝐶) = (B / 𝐶))       (φA = B)
 
Theoremrerecclapd 7548 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℝ)    &   (φA # 0)       (φ → (1 / A) ℝ)
 
Theoremredivclapd 7549 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
(φA ℝ)    &   (φB ℝ)    &   (φB # 0)       (φ → (A / B) ℝ)
 
Theoremmvllmulapd 7550 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
(φA ℂ)    &   (φB ℂ)    &   (φA # 0)    &   (φ → (A · B) = 𝐶)       (φB = (𝐶 / A))
 
3.3.9  Ordering on reals (cont.)
 
Theoremltp1 7551 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
(A ℝ → A < (A + 1))
 
Theoremlep1 7552 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
(A ℝ → A ≤ (A + 1))
 
Theoremltm1 7553 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
(A ℝ → (A − 1) < A)
 
Theoremlem1 7554 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
(A ℝ → (A − 1) ≤ A)
 
Theoremletrp1 7555 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
((A B AB) → A ≤ (B + 1))
 
Theoremp1le 7556 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
((A B (A + 1) ≤ B) → AB)
 
Theoremrecgt0 7557 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((A 0 < A) → 0 < (1 / A))
 
Theoremprodgt0gt0 7558 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7559 for the same theorem with 0 < A replaced by the weaker condition 0 ≤ A. (Contributed by Jim Kingdon, 29-Feb-2020.)
(((A B ℝ) (0 < A 0 < (A · B))) → 0 < B)
 
Theoremprodgt0 7559 Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((A B ℝ) (0 ≤ A 0 < (A · B))) → 0 < B)
 
Theoremprodgt02 7560 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
(((A B ℝ) (0 ≤ B 0 < (A · B))) → 0 < A)
 
Theoremprodge0 7561 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 B ℝ) (0 < A 0 ≤ (A · B))) → 0 ≤ B)
 
Theoremprodge02 7562 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
(((A B ℝ) (0 < B 0 ≤ (A · B))) → 0 ≤ A)
 
Theoremltmul2 7563 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 B (𝐶 0 < 𝐶)) → (A < B ↔ (𝐶 · A) < (𝐶 · B)))
 
Theoremlemul2 7564 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
((A B (𝐶 0 < 𝐶)) → (AB ↔ (𝐶 · A) ≤ (𝐶 · B)))
 
Theoremlemul1a 7565 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 B (𝐶 0 ≤ 𝐶)) AB) → (A · 𝐶) ≤ (B · 𝐶))
 
Theoremlemul2a 7566 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
(((A B (𝐶 0 ≤ 𝐶)) AB) → (𝐶 · A) ≤ (𝐶 · B))
 
Theoremltmul12a 7567 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
((((A B ℝ) (0 ≤ A A < B)) ((𝐶 𝐷 ℝ) (0 ≤ 𝐶 𝐶 < 𝐷))) → (A · 𝐶) < (B · 𝐷))
 
Theoremlemul12b 7568 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
((((A 0 ≤ A) B ℝ) (𝐶 (𝐷 0 ≤ 𝐷))) → ((AB 𝐶𝐷) → (A · 𝐶) ≤ (B · 𝐷)))
 
Theoremlemul12a 7569 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
((((A 0 ≤ A) B ℝ) ((𝐶 0 ≤ 𝐶) 𝐷 ℝ)) → ((AB 𝐶𝐷) → (A · 𝐶) ≤ (B · 𝐷)))
 
Theoremmulgt1 7570 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
(((A B ℝ) (1 < A 1 < B)) → 1 < (A · B))
 
Theoremltmulgt11 7571 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((A B 0 < A) → (1 < BA < (A · B)))
 
Theoremltmulgt12 7572 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((A B 0 < A) → (1 < BA < (B · A)))
 
Theoremlemulge11 7573 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
(((A B ℝ) (0 ≤ A 1 ≤ B)) → A ≤ (A · B))
 
Theoremlemulge12 7574 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
(((A B ℝ) (0 ≤ A 1 ≤ B)) → A ≤ (B · A))
 
Theoremltdiv1 7575 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 B (𝐶 0 < 𝐶)) → (A < B ↔ (A / 𝐶) < (B / 𝐶)))
 
Theoremlediv1 7576 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
((A B (𝐶 0 < 𝐶)) → (AB ↔ (A / 𝐶) ≤ (B / 𝐶)))
 
Theoremgt0div 7577 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
((A B 0 < B) → (0 < A ↔ 0 < (A / B)))
 
Theoremge0div 7578 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
((A B 0 < B) → (0 ≤ A ↔ 0 ≤ (A / B)))
 
Theoremdivgt0 7579 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
(((A 0 < A) (B 0 < B)) → 0 < (A / B))
 
Theoremdivge0 7580 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
(((A 0 ≤ A) (B 0 < B)) → 0 ≤ (A / B))
 
Theoremltmuldiv 7581 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B (𝐶 0 < 𝐶)) → ((A · 𝐶) < BA < (B / 𝐶)))
 
Theoremltmuldiv2 7582 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((A B (𝐶 0 < 𝐶)) → ((𝐶 · A) < BA < (B / 𝐶)))
 
Theoremltdivmul 7583 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) < BA < (𝐶 · B)))
 
Theoremledivmul 7584 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) ≤ BA ≤ (𝐶 · B)))
 
Theoremltdivmul2 7585 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) < BA < (B · 𝐶)))
 
Theoremlt2mul2div 7586 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
(((A (B 0 < B)) (𝐶 (𝐷 0 < 𝐷))) → ((A · B) < (𝐶 · 𝐷) ↔ (A / 𝐷) < (𝐶 / B)))
 
Theoremledivmul2 7587 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
((A B (𝐶 0 < 𝐶)) → ((A / 𝐶) ≤ BA ≤ (B · 𝐶)))
 
Theoremlemuldiv 7588 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
((A B (𝐶 0 < 𝐶)) → ((A · 𝐶) ≤ BA ≤ (B / 𝐶)))
 
Theoremlemuldiv2 7589 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
((A B (𝐶 0 < 𝐶)) → ((𝐶 · A) ≤ BA ≤ (B / 𝐶)))
 
Theoremltrec 7590 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((A 0 < A) (B 0 < B)) → (A < B ↔ (1 / B) < (1 / A)))
 
Theoremlerec 7591 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 0 < A) (B 0 < B)) → (AB ↔ (1 / B) ≤ (1 / A)))
 
Theoremlt2msq1 7592 Lemma for lt2msq 7593. (Contributed by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) B A < B) → (A · A) < (B · B))
 
Theoremlt2msq 7593 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 0 ≤ A) (B 0 ≤ B)) → (A < B ↔ (A · A) < (B · B)))
 
Theoremltdiv2 7594 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
(((A 0 < A) (B 0 < B) (𝐶 0 < 𝐶)) → (A < B ↔ (𝐶 / B) < (𝐶 / A)))
 
Theoremltrec1 7595 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
(((A 0 < A) (B 0 < B)) → ((1 / A) < B ↔ (1 / B) < A))
 
Theoremlerec2 7596 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
(((A 0 < A) (B 0 < B)) → (A ≤ (1 / B) ↔ B ≤ (1 / A)))
 
Theoremledivdiv 7597 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
((((A 0 < A) (B 0 < B)) ((𝐶 0 < 𝐶) (𝐷 0 < 𝐷))) → ((A / B) ≤ (𝐶 / 𝐷) ↔ (𝐷 / 𝐶) ≤ (B / A)))
 
Theoremlediv2 7598 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
(((A 0 < A) (B 0 < B) (𝐶 0 < 𝐶)) → (AB ↔ (𝐶 / B) ≤ (𝐶 / A)))
 
Theoremltdiv23 7599 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
((A (B 0 < B) (𝐶 0 < 𝐶)) → ((A / B) < 𝐶 ↔ (A / 𝐶) < B))
 
Theoremlediv23 7600 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
((A (B 0 < B) (𝐶 0 < 𝐶)) → ((A / B) ≤ 𝐶 ↔ (A / 𝐶) ≤ B))
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