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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivcanap3zi 7801 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcanap4zi 7802 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremrec11api 7803 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 # 0 ∧ 𝐵 # 0) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremdivclapi 7804 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℂ
 
Theoremdivcanap2i 7805 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐵 · (𝐴 / 𝐵)) = 𝐴
 
Theoremdivcanap1i 7806 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 / 𝐵) · 𝐵) = 𝐴
 
Theoremdivrecapi 7807 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))
 
Theoremdivcanap3i 7808 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐵 · 𝐴) / 𝐵) = 𝐴
 
Theoremdivcanap4i 7809 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0       ((𝐴 · 𝐵) / 𝐵) = 𝐴
 
Theoremdivap0i 7810 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0       (𝐴 / 𝐵) # 0
 
Theoremrec11apii 7811 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0       ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵)
 
Theoremdivassapzi 7812 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐶 # 0 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
Theoremdivmulapzi 7813 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐵 # 0 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))
 
Theoremdivdirapzi 7814 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐶 # 0 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
Theoremdivdiv23apzi 7815 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐵 # 0 ∧ 𝐶 # 0) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
Theoremdivmulapi 7816 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐵 # 0       ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴)
 
Theoremdivdiv32api 7817 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐵 # 0    &   𝐶 # 0       ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)
 
Theoremdivassapi 7818 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))
 
Theoremdivdirapi 7819 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))
 
Theoremdiv23api 7820 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)
 
Theoremdiv11api 7821 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵)
 
Theoremdivmuldivapi 7822 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0       ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))
 
Theoremdivmul13api 7823 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0       ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐶 / 𝐵) · (𝐴 / 𝐷))
 
Theoremdivadddivapi 7824 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0       ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷))
 
Theoremdivdivdivapi 7825 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0    &   𝐶 # 0       ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))
 
Theoremrerecclapzi 7826 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ       (𝐴 # 0 → (1 / 𝐴) ∈ ℝ)
 
Theoremrerecclapi 7827 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐴 # 0       (1 / 𝐴) ∈ ℝ
 
Theoremredivclapzi 7828 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremredivclapi 7829 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐵 # 0       (𝐴 / 𝐵) ∈ ℝ
 
Theoremdiv1d 7830 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 / 1) = 𝐴)
 
Theoremrecclapd 7831 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) ∈ ℂ)
 
Theoremrecap0d 7832 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) # 0)
 
Theoremrecidapd 7833 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremrecidap2d 7834 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → ((1 / 𝐴) · 𝐴) = 1)
 
Theoremrecrecapd 7835 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 7836 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (𝐴 / 𝐴) = 1)
 
Theoremdiv0apd 7837 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → (0 / 𝐴) = 0)
 
Theoremapmul1 7838 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
Theoremdivclapd 7839 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremdivcanap1d 7840 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdivcanap2d 7841 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivrecapd 7842 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 7843 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 7844 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcanap4d 7845 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremdiveqap0d 7846 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 7847 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 7848 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7755. Generalization of diveqap1d 7847. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))
 
Theoremdiveqap0ad 7849 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7734. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
 
Theoremdivap1d 7850 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 7851 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 7852 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 7853 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 7854 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 # 0)       (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
Theoremdivap0d 7855 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 7856 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 7857 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 7858 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 7859 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 7860 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 7861 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 7862 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 7863 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 7864 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 7865 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 7866 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 7867 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 7868 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 7869 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 7870 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 7871 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 7872 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 7873 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 7874 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 7875 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 7876 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 7877 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 7878 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 7879 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)
 
Theoremdivmuldivapd 7880 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐷 # 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)))
 
Theoremrerecclapd 7881 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) ∈ ℝ)
 
Theoremredivclapd 7882 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremmvllmulapd 7883 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 7884 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
 
Theoremlep1 7885 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
(𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1))
 
Theoremltm1 7886 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
(𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)
 
Theoremlem1 7887 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴)
 
Theoremletrp1 7888 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → 𝐴 ≤ (𝐵 + 1))
 
Theoremp1le 7889 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴𝐵)
 
Theoremrecgt0 7890 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 ∈ ℝ ∧ 0 < 𝐴) → 0 < (1 / 𝐴))
 
Theoremprodgt0gt0 7891 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7892 for the same theorem with 0 < 𝐴 replaced by the weaker condition 0 ≤ 𝐴. (Contributed by Jim Kingdon, 29-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵)
 
Theoremprodgt0 7892 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 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵)
 
Theoremprodgt02 7893 Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐵 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐴)
 
Theoremprodge0 7894 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)
 
Theoremprodge02 7895 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)
 
Theoremltmul2 7896 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)))
 
Theoremlemul2 7897 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)) → (𝐴𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
 
Theoremlemul1a 7898 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)) ∧ 𝐴𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))
 
Theoremlemul2a 7899 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)) ∧ 𝐴𝐵) → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))
 
Theoremltmul12a 7900 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 ≤ 𝐶𝐶 < 𝐷))) → (𝐴 · 𝐶) < (𝐵 · 𝐷))
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