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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlttri3d 7901 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))

Theoremletri3d 7902 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴)))

Theoremlenltd 7903 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))

Theoremltled 7904 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)

Theoremltnsymd 7905 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → ¬ 𝐵 < 𝐴)

Theoremnltled 7906 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ¬ 𝐵 < 𝐴)       (𝜑𝐴𝐵)

Theoremlensymd 7907 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → ¬ 𝐵 < 𝐴)

Theoremmulgt0d 7908 The product of two 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))

Theoremletrd 7909 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremlelttrd 7910 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑𝐴 < 𝐶)

Theoremlttrd 7911 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵 < 𝐶)       (𝜑𝐴 < 𝐶)

Theorem0lt1 7912 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)
0 < 1

Theoremltntri 7913 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, 𝐴 < 𝐵𝐴 = 𝐵𝐵 < 𝐴. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴))

4.2.5  Initial properties of the complex numbers

Theoremmul12 7914 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))

Theoremmul32 7915 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))

Theoremmul31 7916 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))

Theoremmul4 7917 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)))

Theoremmuladd11 7918 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵))))

Theorem1p1times 7919 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴))

Theorempeano2cn 7920 A theorem for complex numbers analogous the second Peano postulate peano2 4516. (Contributed by NM, 17-Aug-2005.)
(𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Theorempeano2re 7921 A theorem for reals analogous the second Peano postulate peano2 4516. (Contributed by NM, 5-Jul-2005.)
(𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)

((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremaddid1 7923 0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
(𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Theoremaddid2 7924 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Theoremreaddcan 7925 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵))

Theorem00id 7926 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
(0 + 0) = 0

Theoremaddid1i 7927 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ       (𝐴 + 0) = 𝐴

Theoremaddid2i 7928 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
𝐴 ∈ ℂ       (0 + 𝐴) = 𝐴

Theoremaddcomi 7929 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) = (𝐵 + 𝐴)

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       (𝐵 + 𝐴) = 𝐶

Theoremmul12i 7931 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))

Theoremmul32i 7932 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)

Theoremmul4i 7933 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))

Theoremaddid1d 7934 0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + 0) = 𝐴)

Theoremaddid2d 7935 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (0 + 𝐴) = 𝐴)

Theoremaddcomd 7936 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremmul12d 7937 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))

Theoremmul32d 7938 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))

Theoremmul31d 7939 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))

Theoremmul4d 7940 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)))

Theoremmuladd11r 7941 A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 1) · (𝐵 + 1)) = (((𝐴 · 𝐵) + (𝐴 + 𝐵)) + 1))

Theoremcomraddd 7942 Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵 + 𝐶))       (𝜑𝐴 = (𝐶 + 𝐵))

4.3  Real and complex numbers - basic operations

Theoremadd12 7943 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))

Theoremadd32 7944 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))

Theoremadd32r 7945 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = ((𝐴 + 𝐶) + 𝐵))

Theoremadd4 7946 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))

Theoremadd42 7947 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))

Theoremadd12i 7948 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))

Theoremadd32i 7949 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)

Theoremadd4i 7950 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))

Theoremadd42i 7951 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))

Theoremadd12d 7952 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))

Theoremadd32d 7953 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))

Theoremadd4d 7954 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))

Theoremadd42d 7955 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))

4.3.2  Subtraction

Syntaxcmin 7956 Extend class notation to include subtraction.
class

Syntaxcneg 7957 Extend class notation to include unary minus. The symbol - is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus (-) and subtraction cmin 7956 () to prevent syntax ambiguity. For example, looking at the syntax definition co 5781, if we used the same symbol then "( − 𝐴𝐵) " could mean either "𝐴 " minus "𝐵", or it could represent the (meaningless) operation of classes " " and "𝐵 " connected with "operation" "𝐴". On the other hand, "(-𝐴𝐵) " is unambiguous.
class -𝐴

Definitiondf-sub 7958* Define subtraction. Theorem subval 7977 shows its value (and describes how this definition works), theorem subaddi 8072 relates it to addition, and theorems subcli 8061 and resubcli 8048 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
− = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

Definitiondf-neg 7959 Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction () to prevent syntax ambiguity. See cneg 7957 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
-𝐴 = (0 − 𝐴)

Theoremcnegexlem1 7960 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7963. (Contributed by Eric Schmidt, 22-May-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Theoremcnegexlem2 7961 Existence of a real number which produces a real number when multiplied by i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 7963. (Contributed by Eric Schmidt, 22-May-2007.)
𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ

Theoremcnegexlem3 7962* Existence of real number difference. Lemma for cnegex 7963. (Contributed by Eric Schmidt, 22-May-2007.)
((𝑏 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ∃𝑐 ∈ ℝ (𝑏 + 𝑐) = 𝑦)

Theoremcnegex 7963* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)

Theoremcnegex2 7964* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0)

Theoremaddcan 7965 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Theoremaddcan2 7966 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))

Theoremaddcani 7967 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)

Theoremaddcan2i 7968 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)

Theoremaddcand 7969 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Theoremaddcan2d 7970 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))

Theoremaddcanad 7971 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 7969. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = (𝐴 + 𝐶))       (𝜑𝐵 = 𝐶)

Theoremaddcan2ad 7972 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7970. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶))       (𝜑𝐴 = 𝐵)

Theoremaddneintrd 7973 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7971. Consequence of addcand 7969. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴 + 𝐵) ≠ (𝐴 + 𝐶))

Theoremaddneintr2d 7974 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7972. Consequence of addcan2d 7970. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶))

Theorem0cnALT 7975 Alternate proof of 0cn 7781. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ

Theoremnegeu 7976* Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)

Theoremsubval 7977* Value of subtraction, which is the (unique) element 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))

Theoremnegeq 7978 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
(𝐴 = 𝐵 → -𝐴 = -𝐵)

Theoremnegeqi 7979 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
𝐴 = 𝐵       -𝐴 = -𝐵

Theoremnegeqd 7980 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → -𝐴 = -𝐵)

Theoremnfnegd 7981 Deduction version of nfneg 7982. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥-𝐴)

Theoremnfneg 7982 Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥-𝐴

Theoremcsbnegg 7983 Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐴𝑉𝐴 / 𝑥-𝐵 = -𝐴 / 𝑥𝐵)

Theoremsubcl 7984 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) ∈ ℂ)

Theoremnegcl 7985 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
(𝐴 ∈ ℂ → -𝐴 ∈ ℂ)

Theoremnegicn 7986 -i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
-i ∈ ℂ

Theoremsubf 7987 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
− :(ℂ × ℂ)⟶ℂ

Theoremsubadd 7988 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))

Theoremsubadd2 7989 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴))

Theoremsubsub23 7990 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵))

Theorempncan 7991 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴)

Theorempncan2 7992 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

Theorempncan3 7993 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵𝐴)) = 𝐵)

Theoremnpcan 7994 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) + 𝐵) = 𝐴)

Theoremaddsubass 7995 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵𝐶)))

Theoremaddsub 7996 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴𝐶) + 𝐵))

Theoremsubadd23 7997 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + 𝐶) = (𝐴 + (𝐶𝐵)))

Theoremaddsub12 7998 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵𝐶)) = (𝐵 + (𝐴𝐶)))