P. 85 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mulap0r 8401	A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 · 𝐵) # 0) → (𝐴 # 0 ∧ 𝐵 # 0))
Theorem	msqge0 8402	A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴))
Theorem	msqge0i 8403	A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 0 ≤ (𝐴 · 𝐴)
Theorem	msqge0d 8404	A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐴))
Theorem	mulge0 8405	The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 · 𝐵))
Theorem	mulge0i 8406	The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 · 𝐵))
Theorem	mulge0d 8407	The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵))
Theorem	apti 8408	Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
Theorem	apne 8409	Apartness implies negated equality. We cannot in general prove the converse (as shown at neapmkv 13425), which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 → 𝐴 ≠ 𝐵))
Theorem	apcon4bid 8410	Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 # 𝐵 ↔ 𝐶 # 𝐷))    ⇒   ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))
Theorem	leltap 8411	≤ implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴))
Theorem	gt0ap0 8412	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0)
Theorem	gt0ap0i 8413	Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 < 𝐴 → 𝐴 # 0)
Theorem	gt0ap0ii 8414	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 𝐴 # 0
Theorem	gt0ap0d 8415	Positive implies apart from zero. Because of the way we define #, 𝐴 must be an element of ℝ, not just ℝ*. (Contributed by Jim Kingdon, 27-Feb-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 # 0)
Theorem	negap0 8416	A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ -𝐴 # 0))
Theorem	negap0d 8417	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 8418	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 8419	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴)
Theorem	gtapii 8420	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ 𝐵 # 𝐴
Theorem	ltapii 8421	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ 𝐴 # 𝐵
Theorem	ltapi 8422	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 < 𝐵 → 𝐵 # 𝐴)
Theorem	gtapd 8423	'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 8424	'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 8425	≤ 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 8426	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 8427	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 8428	Subtraction respects apartness. Analogue of subcan2 8011 for apartness. (Contributed by Jim Kingdon, 6-Jan-2022.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 − 𝐶) # (𝐵 − 𝐶)))
Theorem	subap0 8429	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 8430	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	cnstab 8431	Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 817. This theorem for real numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim Kingdon, 1-Aug-2023.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → STAB 𝐴 = 𝐵)
Theorem	aprcl 8432	Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
⊢ (𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
Theorem	apsscn 8433*	The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ⊆ ℂ
Theorem	lt0ap0 8434	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 8435	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)
4.3.7  Reciprocals
Theorem	recextlem1 8436	Lemma for recexap 8438. (Contributed by Eric Schmidt, 23-May-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵)))
Theorem	recexaplem2 8437	Lemma for recexap 8438. (Contributed by Jim Kingdon, 20-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0)
Theorem	recexap 8438*	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 8439	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 8440	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 8441	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 8442	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 8443	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 8444	A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8443 and consequence of mulap0bd 8442. (Contributed by Jim Kingdon, 24-Feb-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) # 0)    ⇒   ⊢ (𝜑 → 𝐴 # 0)
Theorem	mulap0bbd 8445	A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8443 and consequence of mulap0bd 8442. (Contributed by Jim Kingdon, 24-Feb-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) # 0)    ⇒   ⊢ (𝜑 → 𝐵 # 0)
Theorem	mulcanapd 8446	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 8447	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 8448	Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 8446. (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 8449	Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 8447. (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 8450	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 8451	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 8452	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 8453	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 8454*	Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)
Theorem	mul0eqap 8455	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))
4.3.8  Division
Syntax	cdiv 8456	Extend class notation to include division.
class /
Definition	df-div 8457*	Define division. Theorem divmulap 8459 relates it to multiplication, and divclap 8462 and redivclap 8515 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divvalap 8458 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
Theorem	divvalap 8458*	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 8459	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 8460	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 8461	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 8462	Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ)
Theorem	recclap 8463	Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ)
Theorem	divcanap2 8464	A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
Theorem	divcanap1 8465	A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
Theorem	diveqap0 8466	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 8467	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 8468	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 8469	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 8470	Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1)
Theorem	recidap2 8471	Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1)
Theorem	divrecap 8472	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 8473	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 8474	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 8475	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 8476	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 8477	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 8478	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 8479	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 8480	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 8481	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 8482	A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
Theorem	divcanap4 8483	A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
Theorem	div11ap 8484	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 8485	A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1)
Theorem	div0ap 8486	Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0)
Theorem	div1 8487	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 8488	1 divided by 1 is 1 (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
⊢ (1 / 1) = 1
Theorem	diveqap1 8489	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 8490	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 8491	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 8492	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 8493	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 8494	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 8495	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 8496	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 8497	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 8498	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 8499	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 8500	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)))