HomeTheorem List (p. 90 of 169)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | negap0d 8901 | The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → -𝐴 # 0) | ||
| Theorem | ltleap 8902 | Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵))) | ||
| Theorem | ltap 8903 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴) | ||
| Theorem | gtapii 8904 | 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐵 # 𝐴 | ||
| Theorem | ltapii 8905 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 # 𝐵 | ||
| Theorem | ltapi 8906 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 # 𝐴) | ||
| Theorem | gtapd 8907 | 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 # 𝐴) | ||
| Theorem | ltapd 8908 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 # 𝐵) | ||
| Theorem | leltapd 8909 | ≤ implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴)) | ||
| Theorem | ap0gt0 8910 | 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)) | ||
| Theorem | ap0gt0d 8911 | A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → 0 < 𝐴) | ||
| Theorem | apsub1 8912 | Subtraction respects apartness. Analogue of subcan2 8494 for apartness. (Contributed by Jim Kingdon, 6-Jan-2022.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 − 𝐶) # (𝐵 − 𝐶))) | ||
| Theorem | subap0 8913 | Two numbers being apart is equivalent to their difference being apart from zero. (Contributed by Jim Kingdon, 25-Dec-2022.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) # 0 ↔ 𝐴 # 𝐵)) | ||
| Theorem | subap0d 8914 | Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) # 0) | ||
| Theorem | cnstab 8915 | Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 839. This theorem for real numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim Kingdon, 1-Aug-2023.) (Proof shortened by BJ, 15-Aug-2024.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → STAB 𝐴 = 𝐵) | ||
| Theorem | aprcl 8916 | Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.) |
| ⊢ (𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) | ||
| Theorem | apsscn 8917* | The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.) |
| ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ⊆ ℂ | ||
| Theorem | lt0ap0 8918 | A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 # 0) | ||
| Theorem | lt0ap0d 8919 | A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 0) ⇒ ⊢ (𝜑 → 𝐴 # 0) | ||
| Theorem | aptap 8920 | Complex apartness (as defined at df-ap 8852) is a tight apartness (as defined at df-tap 7560). (Contributed by Jim Kingdon, 16-Feb-2025.) |
| ⊢ # TAp ℂ | ||
| Theorem | recextlem1 8921 | Lemma for recexap 8923. (Contributed by Eric Schmidt, 23-May-2007.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵))) | ||
| Theorem | recexaplem2 8922 | Lemma for recexap 8923. (Contributed by Jim Kingdon, 20-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0) | ||
| Theorem | recexap 8923* | Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1) | ||
| Theorem | mulap0 8924 | 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) | ||
| Theorem | mulap0b 8925 | 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)) | ||
| Theorem | mulap0i 8926 | 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 | ||
| Theorem | mulap0bd 8927 | The product of two numbers apart from zero is apart from zero. Exercise 11.11 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0)) | ||
| Theorem | mulap0d 8928 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) # 0) | ||
| Theorem | mulap0bad 8929 | A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8928 and consequence of mulap0bd 8927. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) # 0) ⇒ ⊢ (𝜑 → 𝐴 # 0) | ||
| Theorem | mulap0bbd 8930 | A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8928 and consequence of mulap0bd 8927. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) # 0) ⇒ ⊢ (𝜑 → 𝐵 # 0) | ||
| Theorem | mulcanapd 8931 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) | ||
| Theorem | mulcanap2d 8932 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) | ||
| Theorem | mulcanapad 8933 | Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 8931. (Contributed by Jim Kingdon, 21-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | mulcanap2ad 8934 | Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 8932. (Contributed by Jim Kingdon, 21-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | mulcanap 8935 | 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)) | ||
| Theorem | mulcanap2 8936 | 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)) | ||
| Theorem | mulcanapi 8937 | 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) | ||
| Theorem | muleqadd 8938 | 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)) | ||
| Theorem | receuap 8939* | Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) | ||
| Theorem | mul0eqap 8940 | If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 𝐵) & ⊢ (𝜑 → (𝐴 · 𝐵) = 0) ⇒ ⊢ (𝜑 → (𝐴 = 0 ∨ 𝐵 = 0)) | ||
| Theorem | recapb 8941* | A complex number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.) |
| ⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1)) | ||
| Syntax | cdiv 8942 | Extend class notation to include division. |
| class / | ||
| Definition | df-div 8943* | Define division. Theorem divmulap 8945 relates it to multiplication, and divclap 8948 and redivclap 9001 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divvalap 8944 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.) |
| ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | ||
| Theorem | divvalap 8944* | 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)) | ||
| Theorem | divmulap 8945 | 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)) | ||
| Theorem | divmulap2 8946 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵))) | ||
| Theorem | divmulap3 8947 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶))) | ||
| Theorem | divclap 8948 | Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ) | ||
| Theorem | recclap 8949 | Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ) | ||
| Theorem | divcanap2 8950 | A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
| Theorem | divcanap1 8951 | A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
| Theorem | diveqap0 8952 | 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)) | ||
| Theorem | divap0b 8953 | 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)) | ||
| Theorem | divap0 8954 | 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) | ||
| Theorem | recap0 8955 | 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) | ||
| Theorem | recidap 8956 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1) | ||
| Theorem | recidap2 8957 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1) | ||
| Theorem | divrecap 8958 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) | ||
| Theorem | divrecap2 8959 | Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | ||
| Theorem | divassap 8960 | 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))) | ||
| Theorem | div23ap 8961 | 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)) | ||
| Theorem | div32ap 8962 | 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))) | ||
| Theorem | div13ap 8963 | 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)) | ||
| Theorem | div12ap 8964 | 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))) | ||
| Theorem | divmulassap 8965 | An associative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = ((𝐴 · 𝐵) · (𝐶 / 𝐷))) | ||
| Theorem | divmulasscomap 8966 | An associative/commutative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0)) → ((𝐴 · (𝐵 / 𝐷)) · 𝐶) = (𝐵 · ((𝐴 · 𝐶) / 𝐷))) | ||
| Theorem | divdirap 8967 | 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))) | ||
| Theorem | divcanap3 8968 | A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
| Theorem | divcanap4 8969 | A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
| Theorem | div11ap 8970 | 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)) | ||
| Theorem | dividap 8971 | A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1) | ||
| Theorem | div0ap 8972 | Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0) | ||
| Theorem | div1 8973 | 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) | ||
| Theorem | 1div1e1 8974 | 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.) |
| ⊢ (1 / 1) = 1 | ||
| Theorem | diveqap1 8975 | Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵)) | ||
| Theorem | divnegap 8976 | Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵)) | ||
| Theorem | muldivdirap 8977 | Distribution of division over addition with a multiplication. (Contributed by Jim Kingdon, 11-Nov-2021.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (((𝐶 · 𝐴) + 𝐵) / 𝐶) = (𝐴 + (𝐵 / 𝐶))) | ||
| Theorem | divsubdirap 8978 | Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 − 𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶))) | ||
| Theorem | recrecap 8979 | 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) | ||
| Theorem | rec11ap 8980 | 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)) | ||
| Theorem | rec11rap 8981 | Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴)) | ||
| Theorem | divmuldivap 8982 | 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))) | ||
| Theorem | divdivdivap 8983 | 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))) | ||
| Theorem | divcanap5 8984 | 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)) | ||
| Theorem | divmul13ap 8985 | 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))) | ||
| Theorem | divmul24ap 8986 | 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))) | ||
| Theorem | divmuleqap 8987 | 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))) | ||
| Theorem | recdivap 8988 | 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)) | ||
| Theorem | divcanap6 8989 | Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1) | ||
| Theorem | divdiv32ap 8990 | 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)) | ||
| Theorem | divcanap7 8991 | 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)) | ||
| Theorem | dmdcanap 8992 | 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)) | ||
| Theorem | divdivap1 8993 | 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))) | ||
| Theorem | divdivap2 8994 | 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)) | ||
| Theorem | recdivap2 8995 | 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))) | ||
| Theorem | ddcanap 8996 | Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵) | ||
| Theorem | divadddivap 8997 | 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))) | ||
| Theorem | divsubdivap 8998 | 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))) | ||
| Theorem | conjmulap 8999 | 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)) | ||
| Theorem | rerecclap 9000 | Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) | ||
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