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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubcan2i 7101 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
A     &   B     &   𝐶        ((A𝐶) = (B𝐶) ↔ A = B)

Theorempnncani 7102 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
A     &   B     &   𝐶        ((A + B) − (A𝐶)) = (B + 𝐶)

Theoremaddsub4i 7103 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)
A     &   B     &   𝐶     &   𝐷        ((A + B) − (𝐶 + 𝐷)) = ((A𝐶) + (B𝐷))

Theorem0reALT 7104 Alternate proof of 0re 6825. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
0

Theoremnegcld 7105 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → -A ℂ)

Theoremsubidd 7106 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (AA) = 0)

Theoremsubid1d 7107 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A − 0) = A)

Theoremnegidd 7108 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A + -A) = 0)

Theoremnegnegd 7109 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → --A = A)

Theoremnegeq0d 7110 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A = 0 ↔ -A = 0))

Theoremnegne0bd 7111 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A ≠ 0 ↔ -A ≠ 0))

Theoremnegcon1d 7112 Contraposition law for unary minus. Deduction form of negcon1 7059. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)       (φ → (-A = B ↔ -B = A))

Theoremnegcon1ad 7113 Contraposition law for unary minus. One-way deduction form of negcon1 7059. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φ → -A = B)       (φ → -B = A)

Theoremneg11ad 7114 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7058. Generalization of neg11d 7130. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)       (φ → (-A = -BA = B))

Theoremnegned 7115 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7130. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φAB)       (φ → -A ≠ -B)

Theoremnegne0d 7116 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φA ≠ 0)       (φ → -A ≠ 0)

Theoremnegrebd 7117 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φ → -A ℝ)       (φA ℝ)

Theoremsubcld 7118 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (AB) ℂ)

Theorempncand 7119 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → ((A + B) − B) = A)

Theorempncan2d 7120 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → ((A + B) − A) = B)

Theorempncan3d 7121 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A + (BA)) = B)

Theoremnpcand 7122 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → ((AB) + B) = A)

Theoremnncand 7123 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A − (AB)) = B)

Theoremnegsubd 7124 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A + -B) = (AB))

Theoremsubnegd 7125 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A − -B) = (A + B))

Theoremsubeq0d 7126 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (AB) = 0)       (φA = B)

Theoremsubne0d 7127 Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φAB)       (φ → (AB) ≠ 0)

Theoremsubeq0ad 7128 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 7033. Generalization of subeq0d 7126. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)       (φ → ((AB) = 0 ↔ A = B))

Theoremsubne0ad 7129 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 7127. Contrapositive of subeq0bd 7173. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (AB) ≠ 0)       (φAB)

Theoremneg11d 7130 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ → -A = -B)       (φA = B)

Theoremnegdid 7131 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → -(A + B) = (-A + -B))

Theoremnegdi2d 7132 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → -(A + B) = (-AB))

Theoremnegsubdid 7133 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → -(AB) = (-A + B))

Theoremnegsubdi2d 7134 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → -(AB) = (BA))

Theoremneg2subd 7135 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (-A − -B) = (BA))

Theoremsubaddd 7136 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) = 𝐶 ↔ (B + 𝐶) = A))

Theoremsubadd2d 7137 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) = 𝐶 ↔ (𝐶 + B) = A))

Theoremaddsubassd 7138 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) − 𝐶) = (A + (B𝐶)))

Theoremaddsubd 7139 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) − 𝐶) = ((A𝐶) + B))

Theoremsubadd23d 7140 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) + 𝐶) = (A + (𝐶B)))

Theoremaddsub12d 7141 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A + (B𝐶)) = (B + (A𝐶)))

Theoremnpncand 7142 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) + (B𝐶)) = (A𝐶))

Theoremnppcand 7143 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (((AB) + 𝐶) + B) = (A + 𝐶))

Theoremnppcan2d 7144 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A − (B + 𝐶)) + 𝐶) = (AB))

Theoremnppcan3d 7145 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) + (𝐶 + B)) = (A + 𝐶))

Theoremsubsubd 7146 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A − (B𝐶)) = ((AB) + 𝐶))

Theoremsubsub2d 7147 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A − (B𝐶)) = (A + (𝐶B)))

Theoremsubsub3d 7148 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A − (B𝐶)) = ((A + 𝐶) − B))

Theoremsubsub4d 7149 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) − 𝐶) = (A − (B + 𝐶)))

Theoremsub32d 7150 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) − 𝐶) = ((A𝐶) − B))

Theoremnnncand 7151 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A − (B𝐶)) − 𝐶) = (AB))

Theoremnnncan1d 7152 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) − (A𝐶)) = (𝐶B))

Theoremnnncan2d 7153 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A𝐶) − (B𝐶)) = (AB))

Theoremnpncan3d 7154 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) + (𝐶A)) = (𝐶B))

Theorempnpcand 7155 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) − (A + 𝐶)) = (B𝐶))

Theorempnpcan2d 7156 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + 𝐶) − (B + 𝐶)) = (AB))

Theorempnncand 7157 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) − (A𝐶)) = (B + 𝐶))

Theoremppncand 7158 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) + (𝐶B)) = (A + 𝐶))

Theoremsubcand 7159 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ → (AB) = (A𝐶))       (φB = 𝐶)

Theoremsubcan2d 7160 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ → (A𝐶) = (B𝐶))       (φA = B)

Theoremsubcanad 7161 Cancellation law for subtraction. Deduction form of subcan 7062. Generalization of subcand 7159. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) = (A𝐶) ↔ B = 𝐶))

Theoremsubneintrd 7162 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 7159. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB𝐶)       (φ → (AB) ≠ (A𝐶))

Theoremsubcan2ad 7163 Cancellation law for subtraction. Deduction form of subcan2 7032. Generalization of subcan2d 7160. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A𝐶) = (B𝐶) ↔ A = B))

Theoremsubneintr2d 7164 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 7160. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φAB)       (φ → (A𝐶) ≠ (B𝐶))

Theoremaddsub4d 7165 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((A + B) − (𝐶 + 𝐷)) = ((A𝐶) + (B𝐷)))

Theoremsubadd4d 7166 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((AB) − (𝐶𝐷)) = ((A + 𝐷) − (B + 𝐶)))

Theoremsub4d 7167 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((AB) − (𝐶𝐷)) = ((A𝐶) − (B𝐷)))

Theorem2addsubd 7168 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → (((A + B) + 𝐶) − 𝐷) = (((A + 𝐶) − 𝐷) + B))

Theoremaddsubeq4d 7169 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((A + B) = (𝐶 + 𝐷) ↔ (𝐶A) = (B𝐷)))

Theoremsubeqrev 7170 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) = (𝐶𝐷) ↔ (BA) = (𝐷𝐶)))

Theorempncan1 7171 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(A ℂ → ((A + 1) − 1) = A)

Theoremnpcan1 7172 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
(A ℂ → ((A − 1) + 1) = A)

Theoremsubeq0bd 7173 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 7128. Converse of subeq0d 7126. Contrapositive of subne0ad 7129. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φA = B)       (φ → (AB) = 0)

Theoremrenegcld 7174 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → -A ℝ)

Theoremresubcld 7175 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (AB) ℝ)

3.3.3  Multiplication

Theoremkcnktkm1cn 7176 k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
(𝐾 ℂ → (𝐾 · (𝐾 − 1)) ℂ)

Theoremmuladd 7177 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) · (𝐶 + 𝐷)) = (((A · 𝐶) + (𝐷 · B)) + ((A · 𝐷) + (𝐶 · B))))

Theoremsubdi 7178 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)
((A B 𝐶 ℂ) → (A · (B𝐶)) = ((A · B) − (A · 𝐶)))

Theoremsubdir 7179 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)
((A B 𝐶 ℂ) → ((AB) · 𝐶) = ((A · 𝐶) − (B · 𝐶)))

Theoremmul02 7180 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)
(A ℂ → (0 · A) = 0)

Theoremmul02lem2 7181 Zero times a real is zero. Although we prove it as a corollary of mul02 7180, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 7180. (Contributed by Scott Fenton, 3-Jan-2013.)
(A ℝ → (0 · A) = 0)

Theoremmul01 7182 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)

Theoremmul02i 7183 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
A        (0 · A) = 0

Theoremmul01i 7184 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
A        (A · 0) = 0

Theoremmul02d 7185 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (0 · A) = 0)

Theoremmul01d 7186 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A · 0) = 0)

Theoremine0 7187 The imaginary unit i is not zero. (Contributed by NM, 6-May-1999.)
i ≠ 0

Theoremmulneg1 7188 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℂ) → (-A · B) = -(A · B))

Theoremmulneg2 7189 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)
((A B ℂ) → (A · -B) = -(A · B))

Theoremmulneg12 7190 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
((A B ℂ) → (-A · B) = (A · -B))

Theoremmul2neg 7191 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B ℂ) → (-A · -B) = (A · B))

Theoremsubmul2 7192 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
((A B 𝐶 ℂ) → (A − (B · 𝐶)) = (A + (B · -𝐶)))

Theoremmulm1 7193 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
(A ℂ → (-1 · A) = -A)

Theoremmulsub 7194 Product of two differences. (Contributed by NM, 14-Jan-2006.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) · (𝐶𝐷)) = (((A · 𝐶) + (𝐷 · B)) − ((A · 𝐷) + (𝐶 · B))))

Theoremmulsub2 7195 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) · (𝐶𝐷)) = ((BA) · (𝐷𝐶)))

Theoremmulm1i 7196 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
A        (-1 · A) = -A

Theoremmulneg1i 7197 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        (-A · B) = -(A · B)

Theoremmulneg2i 7198 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        (A · -B) = -(A · B)

Theoremmul2negi 7199 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        (-A · -B) = (A · B)

Theoremsubdii 7200 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)
A     &   B     &   𝐶        (A · (B𝐶)) = ((A · B) − (A · 𝐶))

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