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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltletri 6901 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((A < B B𝐶) → A < 𝐶)
 
Theoremletri 6902 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((AB B𝐶) → A𝐶)
 
Theoremle2tri3i 6903 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
A     &   B     &   𝐶        ((AB B𝐶 𝐶A) ↔ (A = B B = 𝐶 𝐶 = A))
 
Theoremmulgt0i 6904 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
A     &   B        ((0 < A 0 < B) → 0 < (A · B))
 
Theoremmulgt0ii 6905 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
A     &   B     &   0 < A    &   0 < B       0 < (A · B)
 
Theoremltnrd 6906 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → ¬ A < A)
 
Theoremgtned 6907 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φA < B)       (φBA)
 
Theoremltned 6908 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φA < B)       (φAB)
 
Theoremlttri3d 6909 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A = B ↔ (¬ A < B ¬ B < A)))
 
Theoremletri3d 6910 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A = B ↔ (AB BA)))
 
Theoremlenltd 6911 '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 6912 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA < B)       (φAB)
 
Theoremltnsymd 6913 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA < B)       (φ → ¬ B < A)
 
Theoremmulgt0d 6914 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 6915 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 6916 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 6917 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 6918 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
 
3.2.5  Initial properties of the complex numbers
 
Theoremmul12 6919 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
((A B 𝐶 ℂ) → (A · (B · 𝐶)) = (B · (A · 𝐶)))
 
Theoremmul32 6920 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
((A B 𝐶 ℂ) → ((A · B) · 𝐶) = ((A · 𝐶) · B))
 
Theoremmul31 6921 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
((A B 𝐶 ℂ) → ((A · B) · 𝐶) = ((𝐶 · B) · A))
 
Theoremmul4 6922 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A · B) · (𝐶 · 𝐷)) = ((A · 𝐶) · (B · 𝐷)))
 
Theoremmuladd11 6923 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
((A B ℂ) → ((1 + A) · (1 + B)) = ((1 + A) + (B + (A · B))))
 
Theorem1p1times 6924 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 6925 A theorem for complex numbers analogous the second Peano postulate peano2 4261. (Contributed by NM, 17-Aug-2005.)
(A ℂ → (A + 1) ℂ)
 
Theorempeano2re 6926 A theorem for reals analogous the second Peano postulate peano2 4261. (Contributed by NM, 5-Jul-2005.)
(A ℝ → (A + 1) ℝ)
 
Theoremaddcom 6927 Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
((A B ℂ) → (A + B) = (B + A))
 
Theoremaddid1 6928 0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
(A ℂ → (A + 0) = A)
 
Theoremaddid2 6929 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(A ℂ → (0 + A) = A)
 
Theoremreaddcan 6930 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
((A B 𝐶 ℝ) → ((𝐶 + A) = (𝐶 + B) ↔ A = B))
 
Theorem00id 6931 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
(0 + 0) = 0
 
Theoremaddid1i 6932 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
A        (A + 0) = A
 
Theoremaddid2i 6933 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
A        (0 + A) = A
 
Theoremaddcomi 6934 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
A     &   B        (A + B) = (B + A)
 
Theoremaddcomli 6935 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
A     &   B     &   (A + B) = 𝐶       (B + A) = 𝐶
 
Theoremmul12i 6936 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     &   B     &   𝐶        (A · (B · 𝐶)) = (B · (A · 𝐶))
 
Theoremmul32i 6937 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
A     &   B     &   𝐶        ((A · B) · 𝐶) = ((A · 𝐶) · B)
 
Theoremmul4i 6938 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
A     &   B     &   𝐶     &   𝐷        ((A · B) · (𝐶 · 𝐷)) = ((A · 𝐶) · (B · 𝐷))
 
Theoremaddid1d 6939 0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A + 0) = A)
 
Theoremaddid2d 6940 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (0 + A) = A)
 
Theoremaddcomd 6941 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 6942 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 6943 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 6944 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A · B) · 𝐶) = ((𝐶 · B) · A))
 
Theoremmul4d 6945 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)))
 
3.3  Real and complex numbers - basic operations
 
3.3.1  Addition
 
Theoremadd12 6946 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
((A B 𝐶 ℂ) → (A + (B + 𝐶)) = (B + (A + 𝐶)))
 
Theoremadd32 6947 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
((A B 𝐶 ℂ) → ((A + B) + 𝐶) = ((A + 𝐶) + B))
 
Theoremadd32r 6948 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
((A B 𝐶 ℂ) → (A + (B + 𝐶)) = ((A + 𝐶) + B))
 
Theoremadd4 6949 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷)))
 
Theoremadd42 6950 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (𝐷 + B)))
 
Theoremadd12i 6951 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        (A + (B + 𝐶)) = (B + (A + 𝐶))
 
Theoremadd32i 6952 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        ((A + B) + 𝐶) = ((A + 𝐶) + B)
 
Theoremadd4i 6953 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
A     &   B     &   𝐶     &   𝐷        ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷))
 
Theoremadd42i 6954 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
A     &   B     &   𝐶     &   𝐷        ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (𝐷 + B))
 
Theoremadd12d 6955 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 6956 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 6957 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 6958 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 6959 Extend class notation to include subtraction.
class
 
Syntaxcneg 6960 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 6959 () to prevent syntax ambiguity. For example, looking at the syntax definition co 5455, 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 6961* Define subtraction. Theorem subval 6980 shows its value (and describes how this definition works), theorem subaddi 7074 relates it to addition, and theorems subcli 7063 and resubcli 7050 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
− = (x ℂ, y ℂ ↦ (z ℂ (y + z) = x))
 
Definitiondf-neg 6962 Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction () to prevent syntax ambiguity. See cneg 6960 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
-A = (0 − A)
 
Theoremcnegexlem1 6963 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 6966. (Contributed by Eric Schmidt, 22-May-2007.)
((A B 𝐶 ℂ) → ((A + B) = (A + 𝐶) ↔ B = 𝐶))
 
Theoremcnegexlem2 6964 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 6966. (Contributed by Eric Schmidt, 22-May-2007.)
y ℝ (i · y)
 
Theoremcnegexlem3 6965* Existence of real number difference. Lemma for cnegex 6966. (Contributed by Eric Schmidt, 22-May-2007.)
((𝑏 y ℝ) → 𝑐 ℝ (𝑏 + 𝑐) = y)
 
Theoremcnegex 6966* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
(A ℂ → x ℂ (A + x) = 0)
 
Theoremcnegex2 6967* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(A ℂ → x ℂ (x + A) = 0)
 
Theoremaddcan 6968 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 B 𝐶 ℂ) → ((A + B) = (A + 𝐶) ↔ B = 𝐶))
 
Theoremaddcan2 6969 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
((A B 𝐶 ℂ) → ((A + 𝐶) = (B + 𝐶) ↔ A = B))
 
Theoremaddcani 6970 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     &   B     &   𝐶        ((A + B) = (A + 𝐶) ↔ B = 𝐶)
 
Theoremaddcan2i 6971 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     &   B     &   𝐶        ((A + 𝐶) = (B + 𝐶) ↔ A = B)
 
Theoremaddcand 6972 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 6973 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 6974 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 6972. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ → (A + B) = (A + 𝐶))       (φB = 𝐶)
 
Theoremaddcan2ad 6975 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 6973. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ → (A + 𝐶) = (B + 𝐶))       (φA = B)
 
Theoremaddneintrd 6976 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 6974. Consequence of addcand 6972. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB𝐶)       (φ → (A + B) ≠ (A + 𝐶))
 
Theoremaddneintr2d 6977 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 6975. Consequence of addcan2d 6973. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φAB)       (φ → (A + 𝐶) ≠ (B + 𝐶))
 
Theorem0cnALT 6978 Alternate proof of 0cn 6797. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
0
 
Theoremnegeu 6979* 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 B ℂ) → ∃!x ℂ (A + x) = B)
 
Theoremsubval 6980* 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 B ℂ) → (AB) = (x ℂ (B + x) = A))
 
Theoremnegeq 6981 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
(A = B → -A = -B)
 
Theoremnegeqi 6982 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
A = B       -A = -B
 
Theoremnegeqd 6983 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
(φA = B)       (φ → -A = -B)
 
Theoremnfnegd 6984 Deduction version of nfneg 6985. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
(φxA)       (φx-A)
 
Theoremnfneg 6985 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 6986 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.)
(A 𝑉A / x-B = -A / xB)
 
Theoremsubcl 6987 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
((A B ℂ) → (AB) ℂ)
 
Theoremnegcl 6988 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
(A ℂ → -A ℂ)
 
Theoremnegicn 6989 -i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
-i
 
Theoremsubf 6990 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 6991 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
((A B 𝐶 ℂ) → ((AB) = 𝐶 ↔ (B + 𝐶) = A))
 
Theoremsubadd2 6992 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → ((AB) = 𝐶 ↔ (𝐶 + B) = A))
 
Theoremsubsub23 6993 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
((A B 𝐶 ℂ) → ((AB) = 𝐶 ↔ (A𝐶) = B))
 
Theorempncan 6994 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((A B ℂ) → ((A + B) − B) = A)
 
Theorempncan2 6995 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
((A B ℂ) → ((A + B) − A) = B)
 
Theorempncan3 6996 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
((A B ℂ) → (A + (BA)) = B)
 
Theoremnpcan 6997 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((A B ℂ) → ((AB) + B) = A)
 
Theoremaddsubass 6998 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → ((A + B) − 𝐶) = (A + (B𝐶)))
 
Theoremaddsub 6999 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((A B 𝐶 ℂ) → ((A + B) − 𝐶) = ((A𝐶) + B))
 
Theoremsubadd23 7000 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
((A B 𝐶 ℂ) → ((AB) + 𝐶) = (A + (𝐶B)))
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