HomeTheorem List (p. 77 of 100)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ltapi 7601 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 # 𝐴) | ||
| Theorem | gtapd 7602 | '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 7603 | '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 7604 | '<_' 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 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)) | ||
| Theorem | ap0gt0d 7606 | 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 | subap0d 7607 | Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐵) # 0) | ||
| Theorem | recextlem1 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))) | ||
| Theorem | recexaplem2 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) | ||
| Theorem | recexap 7610* | 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 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) | ||
| Theorem | mulap0b 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)) | ||
| Theorem | mulap0i 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 | ||
| Theorem | mulap0bd 7614 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0)) | ||
| Theorem | mulap0d 7615 | 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 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) # 0) ⇒ ⊢ (𝜑 → 𝐴 # 0) | ||
| Theorem | mulap0bbd 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) # 0) ⇒ ⊢ (𝜑 → 𝐵 # 0) | ||
| Theorem | mulcanapd 7618 | 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 7619 | 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 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | mulcanap2ad 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 # 0) & ⊢ (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| Theorem | mulcanap 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)) | ||
| Theorem | mulcanap2 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)) | ||
| Theorem | mulcanapi 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) | ||
| Theorem | muleqadd 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)) | ||
| Theorem | receuap 7626* | Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) | ||
| Syntax | cdiv 7627 | Extend class notation to include division. |
| class / | ||
| Definition | df-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)) | ||
| Theorem | divvalap 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)) | ||
| Theorem | divmulap 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)) | ||
| Theorem | divmulap2 7631 | 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 7632 | 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 7633 | Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ) | ||
| Theorem | recclap 7634 | Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ) | ||
| Theorem | divcanap2 7635 | A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
| Theorem | divcanap1 7636 | A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
| Theorem | diveqap0 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)) | ||
| Theorem | divap0b 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)) | ||
| Theorem | divap0 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) | ||
| Theorem | recap0 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) | ||
| Theorem | recidap 7641 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1) | ||
| Theorem | recidap2 7642 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1) | ||
| Theorem | divrecap 7643 | 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 7644 | 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 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))) | ||
| Theorem | div23ap 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)) | ||
| Theorem | div32ap 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))) | ||
| Theorem | div13ap 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)) | ||
| Theorem | div12ap 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))) | ||
| Theorem | divdirap 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))) | ||
| Theorem | divcanap3 7651 | A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴) | ||
| Theorem | divcanap4 7652 | A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴) | ||
| Theorem | div11ap 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)) | ||
| Theorem | dividap 7654 | A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1) | ||
| Theorem | div0ap 7655 | Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0) | ||
| Theorem | div1 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) | ||
| Theorem | 1div1e1 7657 | 1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.) |
| ⊢ (1 / 1) = 1 | ||
| Theorem | diveqap1 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)) | ||
| Theorem | divnegap 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)) | ||
| Theorem | divsubdirap 7660 | 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 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) | ||
| Theorem | rec11ap 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)) | ||
| Theorem | rec11rap 7663 | 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 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))) | ||
| Theorem | divdivdivap 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))) | ||
| Theorem | divcanap5 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)) | ||
| Theorem | divmul13ap 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))) | ||
| Theorem | divmul24ap 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))) | ||
| Theorem | divmuleqap 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))) | ||
| Theorem | recdivap 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)) | ||
| Theorem | divcanap6 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) | ||
| Theorem | divdiv32ap 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)) | ||
| Theorem | divcanap7 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)) | ||
| Theorem | dmdcanap 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)) | ||
| Theorem | divdivap1 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))) | ||
| Theorem | divdivap2 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)) | ||
| Theorem | recdivap2 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))) | ||
| Theorem | ddcanap 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) | ||
| Theorem | divadddivap 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))) | ||
| Theorem | divsubdivap 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))) | ||
| Theorem | conjmulap 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)) | ||
| Theorem | rerecclap 7682 | Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ) | ||
| Theorem | redivclap 7683 | Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ) | ||
| Theorem | eqneg 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 = -𝐴 ↔ 𝐴 = 0)) | ||
| Theorem | eqnegd 7685 | A complex number equals its negative iff it is zero. Deduction form of eqneg 7684. (Contributed by David Moews, 28-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴 = -𝐴 ↔ 𝐴 = 0)) | ||
| Theorem | eqnegad 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.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 = -𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
| Theorem | div2negap 7687 | Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) | ||
| Theorem | divneg2ap 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)) | ||
| Theorem | recclapzi 7689 | Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 # 0 → (1 / 𝐴) ∈ ℂ) | ||
| Theorem | recap0apzi 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) | ||
| Theorem | recidapzi 7691 | Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1) | ||
| Theorem | div1i 7692 | A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 / 1) = 𝐴 | ||
| Theorem | eqnegi 7693 | A number equal to its negative is zero. (Contributed by NM, 29-May-1999.) |
| ⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 = -𝐴 ↔ 𝐴 = 0) | ||
| Theorem | recclapi 7694 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (1 / 𝐴) ∈ ℂ | ||
| Theorem | recidapi 7695 | Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (𝐴 · (1 / 𝐴)) = 1 | ||
| Theorem | recrecapi 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 | ||
| Theorem | dividapi 7697 | A number divided by itself is one. (Contributed by NM, 9-Feb-1995.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (𝐴 / 𝐴) = 1 | ||
| Theorem | div0api 7698 | Division into zero is zero. (Contributed by NM, 12-Aug-1999.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 # 0 ⇒ ⊢ (0 / 𝐴) = 0 | ||
| Theorem | divclapzi 7699 | Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ) | ||
| Theorem | divcanap1zi 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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