P. 87 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8601-8700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ltsub1dd 8601	Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) < (𝐵 − 𝐶))
Theorem	ltsub2dd 8602	Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 − 𝐵) < (𝐶 − 𝐴))
Theorem	leadd1dd 8603	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))
Theorem	leadd2dd 8604	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))
Theorem	lesub1dd 8605	Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶))
Theorem	lesub2dd 8606	Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 − 𝐵) ≤ (𝐶 − 𝐴))
Theorem	le2addd 8607	Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))
Theorem	le2subd 8608	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐷) ≤ (𝐶 − 𝐵))
Theorem	ltleaddd 8609	Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
Theorem	leltaddd 8610	Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐵 < 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
Theorem	lt2addd 8611	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐵 < 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
Theorem	lt2subd 8612	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐶)    &   ⊢ (𝜑 → 𝐵 < 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐷) < (𝐶 − 𝐵))
Theorem	possumd 8613	Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴))
Theorem	sublt0d 8614	When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) < 0 ↔ 𝐴 < 𝐵))
Theorem	ltaddsublt 8615	Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) < 𝐴))
Theorem	1le1 8616	1 ≤ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
⊢ 1 ≤ 1
Theorem	gt0add 8617	A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < (𝐴 + 𝐵)) → (0 < 𝐴 ∨ 0 < 𝐵))
4.3.5  Real Apartness
Syntax	creap 8618	Class of real apartness relation.
class #ℝ
Definition	df-reap 8619*	Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #ℝ is an apartness relation on the reals (see df-ap 8626 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #ℝ and # agree (apreap 8631). (Contributed by Jim Kingdon, 26-Jan-2020.)
⊢ #ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))}
Theorem	reapval 8620	Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8632 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
Theorem	reapirr 8621	Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8649 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴)
Theorem	recexre 8622*	Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 #ℝ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Theorem	reapti 8623	Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8666. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 #ℝ 𝐵))
Theorem	recexgt0 8624*	Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
4.3.6  Complex Apartness
Syntax	cap 8625	Class of complex apartness relation.
class #
Definition	df-ap 8626*	Define complex apartness. Definition 6.1 of [Geuvers], p. 17. Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 8723 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8649), symmetry (apsym 8650), and cotransitivity (apcotr 8651). Apartness implies negated equality, as seen at apne 8667, and the converse would also follow if we assumed excluded middle. In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 8666). (Contributed by Jim Kingdon, 26-Jan-2020.)
⊢ # = {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))}
Theorem	ixi 8627	i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (i · i) = -1
Theorem	inelr 8628	The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
⊢ ¬ i ∈ ℝ
Theorem	rimul 8629	A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0)
Theorem	rereim 8630	Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 = (𝐵 + (i · 𝐶)))) → (𝐵 = 𝐴 ∧ 𝐶 = 0))
Theorem	apreap 8631	Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ 𝐴 #ℝ 𝐵))
Theorem	reaplt 8632	Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
Theorem	reapltxor 8633	Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ⊻ 𝐵 < 𝐴)))
Theorem	1ap0 8634	One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
⊢ 1 # 0
Theorem	ltmul1a 8635	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) ∧ 𝐴 < 𝐵) → (𝐴 · 𝐶) < (𝐵 · 𝐶))
Theorem	ltmul1 8636	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
Theorem	lemul1 8637	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
Theorem	reapmul1lem 8638	Lemma for reapmul1 8639. (Contributed by Jim Kingdon, 8-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
Theorem	reapmul1 8639	Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8832. (Contributed by Jim Kingdon, 8-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
Theorem	reapadd1 8640	Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
Theorem	reapneg 8641	Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
Theorem	reapcotr 8642	Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶)))
Theorem	remulext1 8643	Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
Theorem	remulext2 8644	Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵))
Theorem	apsqgt0 8645	The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴 · 𝐴))
Theorem	cru 8646	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) = (𝐶 + (i · 𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	apreim 8647	Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) # (𝐶 + (i · 𝐷)) ↔ (𝐴 # 𝐶 ∨ 𝐵 # 𝐷)))
Theorem	mulreim 8648	Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) · (𝐶 + (i · 𝐷))) = (((𝐴 · 𝐶) + -(𝐵 · 𝐷)) + (i · ((𝐶 · 𝐵) + (𝐷 · 𝐴)))))
Theorem	apirr 8649	Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
⊢ (𝐴 ∈ ℂ → ¬ 𝐴 # 𝐴)
Theorem	apsym 8650	Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ 𝐵 # 𝐴))
Theorem	apcotr 8651	Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶)))
Theorem	apadd1 8652	Addition respects apartness. Analogue of addcan 8223 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
Theorem	apadd2 8653	Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐶 + 𝐴) # (𝐶 + 𝐵)))
Theorem	addext 8654	Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5934. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷)))
Theorem	apneg 8655	Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
Theorem	mulext1 8656	Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
Theorem	mulext2 8657	Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵))
Theorem	mulext 8658	Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5934. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) # (𝐶 · 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷)))
Theorem	mulap0r 8659	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 8660	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 8661	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 8662	A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐴))
Theorem	mulge0 8663	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 8664	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 8665	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 8666	Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
Theorem	apne 8667	Apartness implies negated equality. We cannot in general prove the converse (as shown at neapmkv 15799), 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 8668	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 8669	≤ implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴))
Theorem	gt0ap0 8670	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0)
Theorem	gt0ap0i 8671	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 8672	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 𝐴 # 0
Theorem	gt0ap0d 8673	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 8674	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 8675	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 8676	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 8677	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴)
Theorem	gtapii 8678	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ 𝐵 # 𝐴
Theorem	ltapii 8679	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    ⇒   ⊢ 𝐴 # 𝐵
Theorem	ltapi 8680	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 < 𝐵 → 𝐵 # 𝐴)
Theorem	gtapd 8681	'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 8682	'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 8683	≤ 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 8684	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 8685	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 8686	Subtraction respects apartness. Analogue of subcan2 8268 for apartness. (Contributed by Jim Kingdon, 6-Jan-2022.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 − 𝐶) # (𝐵 − 𝐶)))
Theorem	subap0 8687	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 8688	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 8689	Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵 → 𝐴 = 𝐵 as defined at df-stab 832. 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 8690	Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
⊢ (𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
Theorem	apsscn 8691*	The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ⊆ ℂ
Theorem	lt0ap0 8692	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 8693	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 8694	Complex apartness (as defined at df-ap 8626) is a tight apartness (as defined at df-tap 7333). (Contributed by Jim Kingdon, 16-Feb-2025.)
⊢ # TAp ℂ
4.3.7  Reciprocals
Theorem	recextlem1 8695	Lemma for recexap 8697. (Contributed by Eric Schmidt, 23-May-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵)))
Theorem	recexaplem2 8696	Lemma for recexap 8697. (Contributed by Jim Kingdon, 20-Feb-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0)
Theorem	recexap 8697*	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 8698	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 8699	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 8700	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