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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlensymd 7801 '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 7802 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 7803 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 7804 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 7805 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 7806 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 7807 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))

3.2.5  Initial properties of the complex numbers

Theoremmul12 7808 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))

Theoremmul32 7809 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵))

Theoremmul31 7810 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))

Theoremmul4 7811 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 7812 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 7813 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 7814 A theorem for complex numbers analogous the second Peano postulate peano2 4467. (Contributed by NM, 17-Aug-2005.)
(𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)

Theorempeano2re 7815 A theorem for reals analogous the second Peano postulate peano2 4467. (Contributed by NM, 5-Jul-2005.)
(𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)

((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremaddid1 7817 0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
(𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)

Theoremaddid2 7818 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Theoremreaddcan 7819 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 7820 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
(0 + 0) = 0

Theoremaddid1i 7821 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ       (𝐴 + 0) = 𝐴

Theoremaddid2i 7822 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
𝐴 ∈ ℂ       (0 + 𝐴) = 𝐴

Theoremaddcomi 7823 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 7825 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 7826 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 7827 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 7828 0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + 0) = 𝐴)

Theoremaddid2d 7829 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (0 + 𝐴) = 𝐴)

Theoremaddcomd 7830 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 7831 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 7832 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 7833 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))

Theoremmul4d 7834 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 7835 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 7836 Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵 + 𝐶))       (𝜑𝐴 = (𝐶 + 𝐵))

3.3  Real and complex numbers - basic operations

Theoremadd12 7837 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 7838 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 7839 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 7840 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 7841 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 7842 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 7843 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 7844 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 7845 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 7846 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 7847 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 7848 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 7849 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)))

3.3.2  Subtraction

Syntaxcmin 7850 Extend class notation to include subtraction.
class

Syntaxcneg 7851 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 7850 () to prevent syntax ambiguity. For example, looking at the syntax definition co 5726, 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 7852* Define subtraction. Theorem subval 7871 shows its value (and describes how this definition works), theorem subaddi 7966 relates it to addition, and theorems subcli 7955 and resubcli 7942 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
− = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

Definitiondf-neg 7853 Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction () to prevent syntax ambiguity. See cneg 7851 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
-𝐴 = (0 − 𝐴)

Theoremcnegexlem1 7854 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7857. (Contributed by Eric Schmidt, 22-May-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))

Theoremcnegexlem2 7855 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 7857. (Contributed by Eric Schmidt, 22-May-2007.)
𝑦 ∈ ℝ (i · 𝑦) ∈ ℝ

Theoremcnegexlem3 7856* Existence of real number difference. Lemma for cnegex 7857. (Contributed by Eric Schmidt, 22-May-2007.)
((𝑏 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ∃𝑐 ∈ ℝ (𝑏 + 𝑐) = 𝑦)

Theoremcnegex 7857* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)

Theoremcnegex2 7858* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0)

Theoremaddcan 7859 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 7860 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 7861 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 7862 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 7863 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 7864 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 7865 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 7863. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = (𝐴 + 𝐶))       (𝜑𝐵 = 𝐶)

Theoremaddcan2ad 7866 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7864. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶))       (𝜑𝐴 = 𝐵)

Theoremaddneintrd 7867 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7865. Consequence of addcand 7863. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴 + 𝐵) ≠ (𝐴 + 𝐶))

Theoremaddneintr2d 7868 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7866. Consequence of addcan2d 7864. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶))

Theorem0cnALT 7869 Alternate proof of 0cn 7676. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ

Theoremnegeu 7870* 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 7871* 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 7872 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
(𝐴 = 𝐵 → -𝐴 = -𝐵)

Theoremnegeqi 7873 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
𝐴 = 𝐵       -𝐴 = -𝐵

Theoremnegeqd 7874 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → -𝐴 = -𝐵)

Theoremnfnegd 7875 Deduction version of nfneg 7876. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥-𝐴)

Theoremnfneg 7876 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 7877 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 7878 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) ∈ ℂ)

Theoremnegcl 7879 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
(𝐴 ∈ ℂ → -𝐴 ∈ ℂ)

Theoremnegicn 7880 -i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
-i ∈ ℂ

Theoremsubf 7881 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 7882 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 7883 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 7884 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵))

Theorempncan 7885 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 7886 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

Theorempncan3 7887 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵𝐴)) = 𝐵)

Theoremnpcan 7888 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 7889 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 7890 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 7891 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + 𝐶) = (𝐴 + (𝐶𝐵)))

Theoremaddsub12 7892 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵𝐶)) = (𝐵 + (𝐴𝐶)))

(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 + 𝐵) + 𝐶) − 𝐷) = (((𝐴 + 𝐶) − 𝐷) + 𝐵))

Theoremaddsubeq4 7894 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐶𝐴) = (𝐵𝐷)))

Theorempncan3oi 7895 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 7956 and pncan 7885, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 7991. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 + 𝐵) − 𝐵) = 𝐴

Theoremmvrraddi 7896 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐴 = (𝐵 + 𝐶)       (𝐴𝐶) = 𝐵

Theoremmvlladdi 7897 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       𝐵 = (𝐶𝐴)

Theoremsubid 7898 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴𝐴) = 0)

Theoremsubid1 7899 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)

Theoremnpncan 7900 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐵𝐶)) = (𝐴𝐶))

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