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Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreapval 8301 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8313 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theoremreapirr 8302 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8330 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 # 𝐴)
 
Theoremrecexre 8303* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
Theoremreapti 8304 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8347. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
 
Theoremrecexgt0 8305* 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
 
Syntaxcap 8306 Class of complex apartness relation.
class #
 
Definitiondf-ap 8307* 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 8402 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8330), symmetry (apsym 8331), and cotransitivity (apcotr 8332). Apartness implies negated equality, as seen at apne 8348, 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 8347).

(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 # 𝑡𝑠 # 𝑢))}
 
Theoremixi 8308 i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(i · i) = -1
 
Theoreminelr 8309 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i ∈ ℝ
 
Theoremrimul 8310 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)
 
Theoremrereim 8311 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 = 𝐴𝐶 = 0))
 
Theoremapreap 8312 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵𝐴 # 𝐵))
 
Theoremreaplt 8313 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 < 𝐵𝐵 < 𝐴)))
 
Theoremreapltxor 8314 Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theorem1ap0 8315 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
1 # 0
 
Theoremltmul1a 8316 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))
 
Theoremltmul1 8317 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)))
 
Theoremlemul1 8318 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)) → (𝐴𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
 
Theoremreapmul1lem 8319 Lemma for reapmul1 8320. (Contributed by Jim Kingdon, 8-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
Theoremreapmul1 8320 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8511. (Contributed by Jim Kingdon, 8-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
Theoremreapadd1 8321 Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
 
Theoremreapneg 8322 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
 
Theoremreapcotr 8323 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 # 𝐶𝐵 # 𝐶)))
 
Theoremremulext1 8324 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
 
Theoremremulext2 8325 Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵))
 
Theoremapsqgt0 8326 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))
 
Theoremcru 8327 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 = 𝐶𝐵 = 𝐷)))
 
Theoremapreim 8328 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 # 𝐶𝐵 # 𝐷)))
 
Theoremmulreim 8329 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)))))
 
Theoremapirr 8330 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
(𝐴 ∈ ℂ → ¬ 𝐴 # 𝐴)
 
Theoremapsym 8331 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 # 𝐵𝐵 # 𝐴))
 
Theoremapcotr 8332 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 → (𝐴 # 𝐶𝐵 # 𝐶)))
 
Theoremapadd1 8333 Addition respects apartness. Analogue of addcan 7906 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
 
Theoremapadd2 8334 Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐶 + 𝐴) # (𝐶 + 𝐵)))
 
Theoremaddext 8335 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5749. 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 # 𝐶𝐵 # 𝐷)))
 
Theoremapneg 8336 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
 
Theoremmulext1 8337 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
 
Theoremmulext2 8338 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵))
 
Theoremmulext 8339 Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5749. 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 # 𝐶𝐵 # 𝐷)))
 
Theoremmulap0r 8340 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))
 
Theoremmsqge0 8341 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))
 
Theoremmsqge0i 8342 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ       0 ≤ (𝐴 · 𝐴)
 
Theoremmsqge0d 8343 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴 · 𝐴))
 
Theoremmulge0 8344 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))
 
Theoremmulge0i 8345 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))
 
Theoremmulge0d 8346 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → 0 ≤ (𝐴 · 𝐵))
 
Theoremapti 8347 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
 
Theoremapne 8348 Apartness implies negated equality. We cannot in general prove the converse, 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 # 𝐵𝐴𝐵))
 
Theoremapcon4bid 8349 Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑 → (𝐴 # 𝐵𝐶 # 𝐷))       (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
 
Theoremleltap 8350 implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (𝐴 < 𝐵𝐵 # 𝐴))
 
Theoremgt0ap0 8351 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0)
 
Theoremgt0ap0i 8352 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℝ       (0 < 𝐴𝐴 # 0)
 
Theoremgt0ap0ii 8353 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 # 0
 
Theoremgt0ap0d 8354 Positive implies apart from zero. Because of the way we define #, 𝐴 must be an element of , not just *. (Contributed by Jim Kingdon, 27-Feb-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)       (𝜑𝐴 # 0)
 
Theoremnegap0 8355 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))
 
Theoremnegap0d 8356 The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)       (𝜑 → -𝐴 # 0)
 
Theoremltleap 8357 Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐴 # 𝐵)))
 
Theoremltap 8358 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴)
 
Theoremgtapii 8359 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐵 # 𝐴
 
Theoremltapii 8360 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵       𝐴 # 𝐵
 
Theoremltapi 8361 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵𝐵 # 𝐴)
 
Theoremgtapd 8362 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵 # 𝐴)
 
Theoremltapd 8363 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴 # 𝐵)
 
Theoremleltapd 8364 implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 < 𝐵𝐵 # 𝐴))
 
Theoremap0gt0 8365 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 8366 A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 # 0)       (𝜑 → 0 < 𝐴)
 
Theoremapsub1 8367 Subtraction respects apartness. Analogue of subcan2 7951 for apartness. (Contributed by Jim Kingdon, 6-Jan-2022.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴𝐶) # (𝐵𝐶)))
 
Theoremsubap0 8368 Two numbers being apart is equivalent to their difference being apart from zero. (Contributed by Jim Kingdon, 25-Dec-2022.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) # 0 ↔ 𝐴 # 𝐵))
 
Theoremsubap0d 8369 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)
 
Theoremcnstab 8370 Equality of complex numbers is stable. Stability here means ¬ ¬ 𝐴 = 𝐵𝐴 = 𝐵 as defined at df-stab 799. (Contributed by Jim Kingdon, 1-Aug-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → STAB 𝐴 = 𝐵)
 
Theoremaprcl 8371 Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
(𝐴 # 𝐵 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
 
Theoremapsscn 8372* The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
{𝑥𝐴𝑥 # 𝐵} ⊆ ℂ
 
Theoremlt0ap0 8373 A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 # 0)
 
Theoremlt0ap0d 8374 A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 0)       (𝜑𝐴 # 0)
 
4.3.7  Reciprocals
 
Theoremrecextlem1 8375 Lemma for recexap 8377. (Contributed by Eric Schmidt, 23-May-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵)))
 
Theoremrecexaplem2 8376 Lemma for recexap 8377. (Contributed by Jim Kingdon, 20-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0)
 
Theoremrecexap 8377* Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1)
 
Theoremmulap0 8378 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 8379 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 8380 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 8381 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0))
 
Theoremmulap0d 8382 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 8383 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8382 and consequence of mulap0bd 8381. (Contributed by Jim Kingdon, 24-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) # 0)       (𝜑𝐴 # 0)
 
Theoremmulap0bbd 8384 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 8382 and consequence of mulap0bd 8381. (Contributed by Jim Kingdon, 24-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) # 0)       (𝜑𝐵 # 0)
 
Theoremmulcanapd 8385 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 8386 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 8387 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 8385. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)    &   (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremmulcanap2ad 8388 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 8386. (Contributed by Jim Kingdon, 21-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)    &   (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremmulcanap 8389 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 8390 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 8391 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 8392 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 8393* Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)
 
Theoremmul0eqap 8394 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 𝐵)    &   (𝜑 → (𝐴 · 𝐵) = 0)       (𝜑 → (𝐴 = 0 ∨ 𝐵 = 0))
 
4.3.8  Division
 
Syntaxcdiv 8395 Extend class notation to include division.
class /
 
Definitiondf-div 8396* Define division. Theorem divmulap 8398 relates it to multiplication, and divclap 8401 and redivclap 8454 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divvalap 8397 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
/ = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
 
Theoremdivvalap 8397* 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 8398 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 8399 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 8400 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)))
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