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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddgt0d 8201 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 < 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremaddge0d 8202 Addition of 2 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))
 
Theoremltnegd 8203 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
 
Theoremlenegd 8204 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ -𝐵 ≤ -𝐴))
 
Theoremltnegcon1d 8205 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴 < 𝐵)       (𝜑 → -𝐵 < 𝐴)
 
Theoremltnegcon2d 8206 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < -𝐵)       (𝜑𝐵 < -𝐴)
 
Theoremlenegcon1d 8207 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴𝐵)       (𝜑 → -𝐵𝐴)
 
Theoremlenegcon2d 8208 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 ≤ -𝐵)       (𝜑𝐵 ≤ -𝐴)
 
Theoremltaddposd 8209 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))
 
Theoremltaddpos2d 8210 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴𝐵 < (𝐴 + 𝐵)))
 
Theoremltsubposd 8211 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴 ↔ (𝐵𝐴) < 𝐵))
 
Theoremposdifd 8212 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
 
Theoremaddge01d 8213 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵𝐴 ≤ (𝐴 + 𝐵)))
 
Theoremaddge02d 8214 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵𝐴 ≤ (𝐵 + 𝐴)))
 
Theoremsubge0d 8215 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴))
 
Theoremsuble0d 8216 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 0 ↔ 𝐴𝐵))
 
Theoremsubge02d 8217 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵 ↔ (𝐴𝐵) ≤ 𝐴))
 
Theoremltadd1d 8218 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶)))
 
Theoremleadd1d 8219 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))
 
Theoremleadd2d 8220 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)))
 
Theoremltsubaddd 8221 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 𝐶𝐴 < (𝐶 + 𝐵)))
 
Theoremlesubaddd 8222 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐶 + 𝐵)))
 
Theoremltsubadd2d 8223 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))
 
Theoremlesubadd2d 8224 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐵 + 𝐶)))
 
Theoremltaddsubd 8225 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵)))
 
Theoremltaddsub2d 8226 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) < 𝐶𝐵 < (𝐶𝐴)))
 
Theoremleaddsub2d 8227 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) ≤ 𝐶𝐵 ≤ (𝐶𝐴)))
 
Theoremsubled 8228 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴𝐵) ≤ 𝐶)       (𝜑 → (𝐴𝐶) ≤ 𝐵)
 
Theoremlesubd 8229 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ≤ (𝐵𝐶))       (𝜑𝐶 ≤ (𝐵𝐴))
 
Theoremltsub23d 8230 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴𝐵) < 𝐶)       (𝜑 → (𝐴𝐶) < 𝐵)
 
Theoremltsub13d 8231 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐵𝐶))       (𝜑𝐶 < (𝐵𝐴))
 
Theoremlesub1d 8232 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝐶) ≤ (𝐵𝐶)))
 
Theoremlesub2d 8233 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐶𝐵) ≤ (𝐶𝐴)))
 
Theoremltsub1d 8234 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴𝐶) < (𝐵𝐶)))
 
Theoremltsub2d 8235 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐶𝐵) < (𝐶𝐴)))
 
Theoremltadd1dd 8236 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴 + 𝐶) < (𝐵 + 𝐶))
 
Theoremltsub1dd 8237 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴𝐶) < (𝐵𝐶))
 
Theoremltsub2dd 8238 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐶𝐵) < (𝐶𝐴))
 
Theoremleadd1dd 8239 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))
 
Theoremleadd2dd 8240 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))
 
Theoremlesub1dd 8241 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ≤ (𝐵𝐶))
 
Theoremlesub2dd 8242 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶𝐵) ≤ (𝐶𝐴))
 
Theoremle2addd 8243 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))
 
Theoremle2subd 8244 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴𝐷) ≤ (𝐶𝐵))
 
Theoremltleaddd 8245 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵𝐷)       (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
 
Theoremleltaddd 8246 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵 < 𝐷)       (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
 
Theoremlt2addd 8247 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵 < 𝐷)       (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷))
 
Theoremlt2subd 8248 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 < 𝐶)    &   (𝜑𝐵 < 𝐷)       (𝜑 → (𝐴𝐷) < (𝐶𝐵))
 
Theorempossumd 8249 Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴))
 
Theoremsublt0d 8250 When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 0 ↔ 𝐴 < 𝐵))
 
Theoremltaddsublt 8251 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))
 
Theorem1le1 8252 1 ≤ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
1 ≤ 1
 
Theoremgt0add 8253 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
 
Syntaxcreap 8254 Class of real apartness relation.
class #
 
Definitiondf-reap 8255* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # is an apartness relation on the reals (see df-ap 8262 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, # and # agree (apreap 8267). (Contributed by Jim Kingdon, 26-Jan-2020.)
# = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦𝑦 < 𝑥))}
 
Theoremreapval 8256 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8268 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theoremreapirr 8257 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8285 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
(𝐴 ∈ ℝ → ¬ 𝐴 # 𝐴)
 
Theoremrecexre 8258* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
Theoremreapti 8259 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8302. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
 
Theoremrecexgt0 8260* 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 8261 Class of complex apartness relation.
class #
 
Definitiondf-ap 8262* 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 8352 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8285), symmetry (apsym 8286), and cotransitivity (apcotr 8287). Apartness implies negated equality, as seen at apne 8303, 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 8302).

(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 8263 i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(i · i) = -1
 
Theoreminelr 8264 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i ∈ ℝ
 
Theoremrimul 8265 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 8266 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 8267 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 8268 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 8269 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 8270 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
1 # 0
 
Theoremltmul1a 8271 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 8272 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 8273 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 8274 Lemma for reapmul1 8275. (Contributed by Jim Kingdon, 8-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
Theoremreapmul1 8275 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8461. (Contributed by Jim Kingdon, 8-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
Theoremreapadd1 8276 Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
 
Theoremreapneg 8277 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
 
Theoremreapcotr 8278 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 8279 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
 
Theoremremulext2 8280 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 8281 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 8282 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 8283 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 8284 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 8285 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
(𝐴 ∈ ℂ → ¬ 𝐴 # 𝐴)
 
Theoremapsym 8286 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 8287 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 → (𝐴 # 𝐶𝐵 # 𝐶)))
 
Theoremapadd1 8288 Addition respects apartness. Analogue of addcan 7865 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
 
Theoremapadd2 8289 Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐶 + 𝐴) # (𝐶 + 𝐵)))
 
Theoremaddext 8290 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5737. 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 8291 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
 
Theoremmulext1 8292 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 8293 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 8294 Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5737. 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 8295 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 8296 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 8297 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ       0 ≤ (𝐴 · 𝐴)
 
Theoremmsqge0d 8298 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴 · 𝐴))
 
Theoremmulge0 8299 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 8300 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))
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