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Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremadd12i 10901 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 10902 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 10903 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 10904 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 10905 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 10906 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 10907 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 10908 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 10909 Extend class notation to include subtraction.
class

Syntaxcneg 10910 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 10909 () to prevent syntax ambiguity. For example, looking at the syntax definition co 7151, 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 10911* Define subtraction. Theorem subval 10916 shows its value (and describes how this definition works), Theorem subaddi 11012 relates it to addition, and Theorems subcli 11001 and resubcli 10987 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
− = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

Definitiondf-neg 10912 Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction () to prevent syntax ambiguity. See cneg 10910 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
-𝐴 = (0 − 𝐴)

Theorem0cnALT 10913 Alternate proof of 0cn 10672 which does not reference ax-1cn 10634. (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 10914 Alternate proof of 0cnALT 10913 which is shorter, but depends on ax-8 2114, ax-13 2380, ax-sep 5170, ax-nul 5177, ax-pow 5235, ax-pr 5299, ax-un 7460, and every complex number axiom except ax-pre-mulgt0 10653 and ax-pre-sup 10654. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ

Theoremnegeu 10915* 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 10916* 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 10917 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
(𝐴 = 𝐵 → -𝐴 = -𝐵)

Theoremnegeqi 10918 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
𝐴 = 𝐵       -𝐴 = -𝐵

Theoremnegeqd 10919 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → -𝐴 = -𝐵)

Theoremnfnegd 10920 Deduction version of nfneg 10921. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑𝑥-𝐴)

Theoremnfneg 10921 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 10922 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 10923 A negative is a set. (Contributed by NM, 4-Apr-2005.)
-𝐴 ∈ V

Theoremsubcl 10924 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) ∈ ℂ)

Theoremnegcl 10925 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
(𝐴 ∈ ℂ → -𝐴 ∈ ℂ)

Theoremnegicn 10926 -i is a complex number. (Contributed by David A. Wheeler, 7-Dec-2018.)
-i ∈ ℂ

Theoremsubf 10927 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 10928 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 10929 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 10930 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵))

Theorempncan 10931 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 10932 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

Theorempncan3 10933 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 10934 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 10935 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 10936 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 10937 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + 𝐶) = (𝐴 + (𝐶𝐵)))

Theoremaddsub12 10938 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 10940 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 10941 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 11002 and pncan 10931, 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 11037. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 + 𝐵) − 𝐵) = 𝐴

Theoremmvrraddi 10942 Move the right term in a sum on the RHS to the LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐴 = (𝐵 + 𝐶)       (𝐴𝐶) = 𝐵

Theoremmvlladdi 10943 Move the left term in a sum on the LHS to the RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   (𝐴 + 𝐵) = 𝐶       𝐵 = (𝐶𝐴)

Theoremsubid 10944 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴𝐴) = 0)

Theoremsubid1 10945 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴)

Theoremnpncan 10946 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐵𝐶)) = (𝐴𝐶))

Theoremnppcan 10947 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶))

Theoremnnpcan 10948 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 10949 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶))

Theoremsubcan2 10950 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵))

Theoremsubeq0 10951 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵))

Theoremnpncan2 10952 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) + (𝐵𝐴)) = 0)

Theoremsubsub2 10953 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = (𝐴 + (𝐶𝐵)))

Theoremnncan 10954 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴𝐵)) = 𝐵)

Theoremsubsub 10955 Law for double subtraction. (Contributed by NM, 13-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = ((𝐴𝐵) + 𝐶))

Theoremnppcan2 10956 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴𝐵))

Theoremsubsub3 10957 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = ((𝐴 + 𝐶) − 𝐵))

Theoremsubsub4 10958 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) − 𝐶) = (𝐴 − (𝐵 + 𝐶)))

Theoremsub32 10959 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) − 𝐶) = ((𝐴𝐶) − 𝐵))

Theoremnnncan 10960 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵𝐶)) − 𝐶) = (𝐴𝐵))

Theoremnnncan1 10961 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) − (𝐴𝐶)) = (𝐶𝐵))

Theoremnnncan2 10962 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) − (𝐵𝐶)) = (𝐴𝐵))

Theoremnpncan3 10963 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐶𝐴)) = (𝐶𝐵))

Theorempnpcan 10964 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by SN, 13-Nov-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵𝐶))

Theorempnpcan2 10965 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) − (𝐵 + 𝐶)) = (𝐴𝐵))

Theorempnncan 10966 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴𝐶)) = (𝐵 + 𝐶))

Theoremppncan 10967 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + (𝐶𝐵)) = (𝐴 + 𝐶))

Theoremaddsub4 10968 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴𝐶) + (𝐵𝐷)))

Theoremsubadd4 10969 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) − (𝐶𝐷)) = ((𝐴 + 𝐷) − (𝐵 + 𝐶)))

Theoremsub4 10970 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴𝐵) − (𝐶𝐷)) = ((𝐴𝐶) − (𝐵𝐷)))

Theoremneg0 10971 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
-0 = 0

Theoremnegid 10972 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
(𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0)

Theoremnegsub 10973 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴𝐵))

Theoremsubneg 10974 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵))

Theoremnegneg 10975 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → --𝐴 = 𝐴)

Theoremneg11 10976 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = -𝐵𝐴 = 𝐵))

Theoremnegcon1 10977 Negative contraposition law. (Contributed by NM, 9-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))

Theoremnegcon2 10978 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵𝐵 = -𝐴))

Theoremnegeq0 10979 A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℂ → (𝐴 = 0 ↔ -𝐴 = 0))

Theoremsubcan 10980 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶))

Theoremnegsubdi 10981 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (-𝐴 + 𝐵))

Theoremnegdi 10982 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))

Theoremnegdi2 10983 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴𝐵))

Theoremnegsubdi2 10984 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (𝐵𝐴))

Theoremneg2sub 10985 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵𝐴))

Theoremrenegcli 10986 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 10988 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ ℝ       -𝐴 ∈ ℝ

Theoremresubcli 10987 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵) ∈ ℝ

Theoremrenegcl 10988 Closure law for negative of reals. The weak deduction theorem dedth 4479 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 10986, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)

Theoremresubcl 10989 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵) ∈ ℝ)

Theoremnegreb 10990 The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
(𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ))

Theorempeano2cnm 10991 "Reverse" second Peano postulate analogue for complex numbers: A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
(𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ)

Theorempeano2rem 10992 "Reverse" second Peano postulate analogue for reals. (Contributed by NM, 6-Feb-2007.)
(𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)

Theoremnegcli 10993 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       -𝐴 ∈ ℂ

Theoremnegidi 10994 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴 + -𝐴) = 0

Theoremnegnegi 10995 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 8-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ ℂ       --𝐴 = 𝐴

Theoremsubidi 10996 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴𝐴) = 0

Theoremsubid1i 10997 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 − 0) = 𝐴

Theoremnegne0bi 10998 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)

Theoremnegrebi 10999 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℂ       (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)

Theoremnegne0i 11000 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       -𝐴 ≠ 0

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