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Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempeano2cn 10801 A theorem for complex numbers analogous the second Peano postulate peano2nn 11639. (Contributed by NM, 17-Aug-2005.)
(𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ)
 
Theorempeano2re 10802 A theorem for reals analogous the second Peano postulate peano2nn 11639. (Contributed by NM, 5-Jul-2005.)
(𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
 
Theoremreaddcan 10803 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵))
 
Theorem00id 10804 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
(0 + 0) = 0
 
Theoremmul02lem1 10805 Lemma for mul02 10807. If any real does not produce 0 when multiplied by 0, then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.)
(((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) ∧ 𝐵 ∈ ℂ) → 𝐵 = (𝐵 + 𝐵))
 
Theoremmul02lem2 10806 Lemma for mul02 10807. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℝ → (0 · 𝐴) = 0)
 
Theoremmul02 10807 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (0 · 𝐴) = 0)
 
Theoremmul01 10808 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (𝐴 · 0) = 0)
 
Theoremaddid1 10809 0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
 
Theoremcnegex 10810* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
 
Theoremcnegex2 10811* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0)
 
Theoremaddid2 10812 0 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
 
Theoremaddcan 10813 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 10814 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))
 
Theoremaddcom 10815 Addition commutes. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremaddid1i 10816 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ       (𝐴 + 0) = 𝐴
 
Theoremaddid2i 10817 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
𝐴 ∈ ℂ       (0 + 𝐴) = 𝐴
 
Theoremmul02i 10818 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
𝐴 ∈ ℂ       (0 · 𝐴) = 0
 
Theoremmul01i 10819 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ       (𝐴 · 0) = 0
 
Theoremaddcomi 10820 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + 𝐵) = (𝐵 + 𝐴)
 
Theoremaddcomli 10821 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       (𝐵 + 𝐴) = 𝐶
 
Theoremaddcani 10822 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 10823 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)
 
Theoremmul12i 10824 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 10825 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 10826 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))
 
Theoremmul02d 10827 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (0 · 𝐴) = 0)
 
Theoremmul01d 10828 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 · 0) = 0)
 
Theoremaddid1d 10829 0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + 0) = 𝐴)
 
Theoremaddid2d 10830 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (0 + 𝐴) = 𝐴)
 
Theoremaddcomd 10831 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))
 
Theoremaddcand 10832 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 10833 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 10834 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 10832. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = (𝐴 + 𝐶))       (𝜑𝐵 = 𝐶)
 
Theoremaddcan2ad 10835 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 10833. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremaddneintrd 10836 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 10834. Consequence of addcand 10832. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴 + 𝐵) ≠ (𝐴 + 𝐶))
 
Theoremaddneintr2d 10837 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 10835. Consequence of addcan2d 10833. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶))
 
Theoremmul12d 10838 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 10839 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 10840 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴))
 
Theoremmul4d 10841 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 10842 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 10843 Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵 + 𝐶))       (𝜑𝐴 = (𝐶 + 𝐵))
 
Theoremltaddneg 10844 Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
 
Theoremltaddnegr 10845 Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐴 + 𝐵) < 𝐵))
 
5.3  Real and complex numbers - basic operations
 
5.3.1  Addition
 
Theoremadd12 10846 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 10847 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 10848 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 10849 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 10850 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 10851 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 10852 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 10853 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 10854 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))
 
Theoremadd12d 10855 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 10856 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 10857 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 10858 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)))
 
5.3.2  Subtraction
 
Syntaxcmin 10859 Extend class notation to include subtraction.
class
 
Syntaxcneg 10860 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 10859 () to prevent syntax ambiguity. For example, looking at the syntax definition co 7145, 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 10861* Define subtraction. Theorem subval 10866 shows its value (and describes how this definition works), theorem subaddi 10962 relates it to addition, and theorems subcli 10951 and resubcli 10937 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
− = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
 
Definitiondf-neg 10862 Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction () to prevent syntax ambiguity. See cneg 10860 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
-𝐴 = (0 − 𝐴)
 
Theorem0cnALT 10863 Alternate proof of 0cn 10622 which does not reference ax-1cn 10584. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ
 
Theorem0cnALT2 10864 Alternate proof of 0cnALT 10863 which is shorter, but depends on ax-8 2107, ax-13 2383, ax-sep 5195, ax-nul 5202, ax-pow 5258, ax-pr 5321, ax-un 7450, and every complex number axiom except ax-pre-mulgt0 10603 and ax-pre-sup 10604. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ
 
Theoremnegeu 10865* 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 10866* 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 10867 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
(𝐴 = 𝐵 → -𝐴 = -𝐵)
 
Theoremnegeqi 10868 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
𝐴 = 𝐵       -𝐴 = -𝐵
 
Theoremnegeqd 10869 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → -𝐴 = -𝐵)
 
Theoremnfnegd 10870 Deduction version of nfneg 10871. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥-𝐴)
 
Theoremnfneg 10871 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 10872 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)
 
Theoremnegex 10873 A negative is a set. (Contributed by NM, 4-Apr-2005.)
-𝐴 ∈ V
 
Theoremsubcl 10874 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) ∈ ℂ)
 
Theoremnegcl 10875 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
(𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
 
Theoremnegicn 10876 -i is a complex number. (Contributed by David A. Wheeler, 7-Dec-2018.)
-i ∈ ℂ
 
Theoremsubf 10877 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 10878 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 10879 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 10880 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵))
 
Theorempncan 10881 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 10882 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
 
Theorempncan3 10883 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵𝐴)) = 𝐵)
 
Theoremnpcan 10884 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 10885 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 10886 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 10887 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + 𝐶) = (𝐴 + (𝐶𝐵)))
 
Theoremaddsub12 10888 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵𝐶)) = (𝐵 + (𝐴𝐶)))
 
Theorem2addsub 10889 Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 + 𝐵) + 𝐶) − 𝐷) = (((𝐴 + 𝐶) − 𝐷) + 𝐵))
 
Theoremaddsubeq4 10890 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 10891 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 10952 and pncan 10881, 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 10987. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 + 𝐵) − 𝐵) = 𝐴
 
Theoremmvrraddi 10892 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐴 = (𝐵 + 𝐶)       (𝐴𝐶) = 𝐵
 
Theoremmvlladdi 10893 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       𝐵 = (𝐶𝐴)
 
Theoremsubid 10894 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴𝐴) = 0)
 
Theoremsubid1 10895 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)
 
Theoremnpncan 10896 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐵𝐶)) = (𝐴𝐶))
 
Theoremnppcan 10897 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶))
 
Theoremnnpcan 10898 Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐵) − 𝐶) + 𝐵) = (𝐴𝐶))
 
Theoremnppcan3 10899 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶))
 
Theoremsubcan2 10900 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵))
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