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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltapi 7601 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐵 # 𝐴)
 
Theoremgtapd 7602 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵 # 𝐴)
 
Theoremltapd 7603 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴 # 𝐵)
 
Theoremleltapd 7604 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 < 𝐵𝐵 # 𝐴))
 
Theoremap0gt0 7605 A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ 0 < 𝐴))
 
Theoremap0gt0d 7606 A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 # 0)       (𝜑 → 0 < 𝐴)
 
Theoremsubap0d 7607 Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 𝐵)       (𝜑 → (𝐴𝐵) # 0)
 
3.3.7  Reciprocals
 
Theoremrecextlem1 7608 Lemma for recexap 7610. (Contributed by Eric Schmidt, 23-May-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵)))
 
Theoremrecexaplem2 7609 Lemma for recexap 7610. (Contributed by Jim Kingdon, 20-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0)
 
Theoremrecexap 7610* Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1)
 
Theoremmulap0 7611 The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 · 𝐵) # 0)
 
Theoremmulap0b 7612 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0))
 
Theoremmulap0i 7613 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0       (𝐴 · 𝐵) # 0
 
Theoremmulap0bd 7614 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0))
 
Theoremmulap0d 7615 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 · 𝐵) # 0)
 
Theoremmulap0bad 7616 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7615 and consequence of mulap0bd 7614. (Contributed by Jim Kingdon, 24-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) # 0)       (𝜑𝐴 # 0)
 
Theoremmulap0bbd 7617 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7615 and consequence of mulap0bd 7614. (Contributed by Jim Kingdon, 24-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) # 0)       (𝜑𝐵 # 0)
 
Theoremmulcanapd 7618 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremmulcanap2d 7619 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremmulcanapad 7620 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 7618. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)    &   (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremmulcanap2ad 7621 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 7619. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)    &   (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremmulcanap 7622 Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremmulcanap2 7623 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremmulcanapi 7624 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0       ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)
 
Theoremmuleqadd 7625 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 + 𝐵) ↔ ((𝐴 − 1) · (𝐵 − 1)) = 1))
 
Theoremreceuap 7626* Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)
 
3.3.8  Division
 
Syntaxcdiv 7627 Extend class notation to include division.
class /
 
Definitiondf-div 7628* Define division. Theorem divmulap 7630 relates it to multiplication, and divclap 7633 and redivclap 7683 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
/ = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
 
Theoremdivvalap 7629* Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
 
Theoremdivmulap 7630 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴))
 
Theoremdivmulap2 7631 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐶 · 𝐵)))
 
Theoremdivmulap3 7632 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵𝐴 = (𝐵 · 𝐶)))
 
Theoremdivclap 7633 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremrecclap 7634 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ)
 
Theoremdivcanap2 7635 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
Theoremdivcanap1 7636 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
Theoremdiveqap0 7637 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
 
Theoremdivap0b 7638 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0))
 
Theoremdivap0 7639 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / 𝐵) # 0)
 
Theoremrecap0 7640 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) # 0)
 
Theoremrecidap 7641 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremrecidap2 7642 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1)
 
Theoremdivrecap 7643 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
Theoremdivrecap2 7644 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
 
Theoremdivassap 7645 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
Theoremdiv23ap 7646 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))
 
Theoremdiv32ap 7647 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))
 
Theoremdiv13ap 7648 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))
 
Theoremdiv12ap 7649 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))
 
Theoremdivdirap 7650 Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
Theoremdivcanap3 7651 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
Theoremdivcanap4 7652 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
Theoremdiv11ap 7653 One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremdividap 7654 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1)
 
Theoremdiv0ap 7655 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0)
 
Theoremdiv1 7656 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴)
 
Theorem1div1e1 7657 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
(1 / 1) = 1
 
Theoremdiveqap1 7658 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))
 
Theoremdivnegap 7659 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))
 
Theoremdivsubdirap 7660 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))
 
Theoremrecrecap 7661 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (1 / 𝐴)) = 𝐴)
 
Theoremrec11ap 7662 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremrec11rap 7663 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴))
 
Theoremdivmuldivap 7664 Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 · 𝐵) / (𝐶 · 𝐷)))
 
Theoremdivdivdivap 7665 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)))
 
Theoremdivcanap5 7666 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))
 
Theoremdivmul13ap 7667 Swap the denominators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐵 / 𝐶) · (𝐴 / 𝐷)))
 
Theoremdivmul24ap 7668 Swap the numerators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 / 𝐷) · (𝐵 / 𝐶)))
 
Theoremdivmuleqap 7669 Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) = (𝐵 / 𝐷) ↔ (𝐴 · 𝐷) = (𝐵 · 𝐶)))
 
Theoremrecdivap 7670 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
 
Theoremdivcanap6 7671 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
 
Theoremdivdiv32ap 7672 Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
Theoremdivcanap7 7673 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵))
 
Theoremdmdcanap 7674 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · (𝐶 / 𝐴)) = (𝐶 / 𝐵))
 
Theoremdivdivap1 7675 Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))
 
Theoremdivdivap2 7676 Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))
 
Theoremrecdivap2 7677 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
 
Theoremddcanap 7678 Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
 
Theoremdivadddivap 7679 Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) + (𝐵 / 𝐷)) = (((𝐴 · 𝐷) + (𝐵 · 𝐶)) / (𝐶 · 𝐷)))
 
Theoremdivsubdivap 7680 Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) − (𝐵 / 𝐷)) = (((𝐴 · 𝐷) − (𝐵 · 𝐶)) / (𝐶 · 𝐷)))
 
Theoremconjmulap 7681 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝑃 ∈ ℂ ∧ 𝑃 # 0) ∧ (𝑄 ∈ ℂ ∧ 𝑄 # 0)) → (((1 / 𝑃) + (1 / 𝑄)) = 1 ↔ ((𝑃 − 1) · (𝑄 − 1)) = 1))
 
Theoremrerecclap 7682 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ)
 
Theoremredivclap 7683 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremeqneg 7684 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 = -𝐴𝐴 = 0))
 
Theoremeqnegd 7685 A complex number equals its negative iff it is zero. Deduction form of eqneg 7684. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = -𝐴𝐴 = 0))
 
Theoremeqnegad 7686 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 7684. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 = -𝐴)       (𝜑𝐴 = 0)
 
Theoremdiv2negap 7687 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
Theoremdivneg2ap 7688 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
 
Theoremrecclapzi 7689 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (1 / 𝐴) ∈ ℂ)
 
Theoremrecap0apzi 7690 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (1 / 𝐴) # 0)
 
Theoremrecidapzi 7691 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ       (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1)
 
Theoremdiv1i 7692 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)
𝐴 ∈ ℂ       (𝐴 / 1) = 𝐴
 
Theoremeqnegi 7693 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 = -𝐴𝐴 = 0)
 
Theoremrecclapi 7694 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)
𝐴 ∈ ℂ    &   𝐴 # 0       (1 / 𝐴) ∈ ℂ
 
Theoremrecidapi 7695 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (𝐴 · (1 / 𝐴)) = 1
 
Theoremrecrecapi 7696 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (1 / (1 / 𝐴)) = 𝐴
 
Theoremdividapi 7697 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)
𝐴 ∈ ℂ    &   𝐴 # 0       (𝐴 / 𝐴) = 1
 
Theoremdiv0api 7698 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)
𝐴 ∈ ℂ    &   𝐴 # 0       (0 / 𝐴) = 0
 
Theoremdivclapzi 7699 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ)
 
Theoremdivcanap1zi 7700 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
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