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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnppcan 8101 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐵) + 𝐶) + 𝐵) = (𝐴 + 𝐶))
 
Theoremnnpcan 8102 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 8103 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐵) + (𝐶 + 𝐵)) = (𝐴 + 𝐶))
 
Theoremsubcan2 8104 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵))
 
Theoremsubeq0 8105 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 8106 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵) + (𝐵𝐴)) = 0)
 
Theoremsubsub2 8107 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 8108 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 8109 Law for double subtraction. (Contributed by NM, 13-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = ((𝐴𝐵) + 𝐶))
 
Theoremnppcan2 8110 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵 + 𝐶)) + 𝐶) = (𝐴𝐵))
 
Theoremsubsub3 8111 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = ((𝐴 + 𝐶) − 𝐵))
 
Theoremsubsub4 8112 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 8113 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 8114 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵𝐶)) − 𝐶) = (𝐴𝐵))
 
Theoremnnncan1 8115 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 8116 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) − (𝐵𝐶)) = (𝐴𝐵))
 
Theoremnpncan3 8117 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 8118 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵𝐶))
 
Theorempnpcan2 8119 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 8120 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 8121 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 8122 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 8123 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 8124 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 8125 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
-0 = 0
 
Theoremnegid 8126 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
(𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0)
 
Theoremnegsub 8127 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 8128 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 8129 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 8130 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 8131 Negative contraposition law. (Contributed by NM, 9-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
 
Theoremnegcon2 8132 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵𝐵 = -𝐴))
 
Theoremnegeq0 8133 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 8134 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 8135 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 8136 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 8137 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴𝐵))
 
Theoremnegsubdi2 8138 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (𝐵𝐴))
 
Theoremneg2sub 8139 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵𝐴))
 
Theoremrenegcl 8140 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
 
Theoremrenegcli 8141 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 8140 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 8142 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
 
Theoremresubcl 8143 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵) ∈ ℝ)
 
Theoremnegreb 8144 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 8145 "Reverse" second Peano postulate analog 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 8146 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
(𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
 
Theoremnegcli 8147 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       -𝐴 ∈ ℂ
 
Theoremnegidi 8148 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴 + -𝐴) = 0
 
Theoremnegnegi 8149 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 8150 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴𝐴) = 0
 
Theoremsubid1i 8151 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 − 0) = 𝐴
 
Theoremnegne0bi 8152 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)
 
Theoremnegrebi 8153 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℂ       (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)
 
Theoremnegne0i 8154 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       -𝐴 ≠ 0
 
Theoremsubcli 8155 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴𝐵) ∈ ℂ
 
Theorempncan3i 8156 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + (𝐵𝐴)) = 𝐵
 
Theoremnegsubi 8157 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + -𝐵) = (𝐴𝐵)
 
Theoremsubnegi 8158 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 − -𝐵) = (𝐴 + 𝐵)
 
Theoremsubeq0i 8159 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵)
 
Theoremneg11i 8160 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 = -𝐵𝐴 = 𝐵)
 
Theoremnegcon1i 8161 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)
 
Theoremnegcon2i 8162 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 = -𝐵𝐵 = -𝐴)
 
Theoremnegdii 8163 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴 + 𝐵) = (-𝐴 + -𝐵)
 
Theoremnegsubdii 8164 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴𝐵) = (-𝐴 + 𝐵)
 
Theoremnegsubdi2i 8165 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴𝐵) = (𝐵𝐴)
 
Theoremsubaddi 8166 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴)
 
Theoremsubadd2i 8167 Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)
 
Theoremsubaddrii 8168 Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   (𝐵 + 𝐶) = 𝐴       (𝐴𝐵) = 𝐶
 
Theoremsubsub23i 8169 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵)
 
Theoremaddsubassi 8170 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵𝐶))
 
Theoremaddsubi 8171 Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − 𝐶) = ((𝐴𝐶) + 𝐵)
 
Theoremsubcani 8172 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶)
 
Theoremsubcan2i 8173 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵)
 
Theorempnncani 8174 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − (𝐴𝐶)) = (𝐵 + 𝐶)
 
Theoremaddsub4i 8175 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((𝐴 + 𝐵) − (𝐶 + 𝐷)) = ((𝐴𝐶) + (𝐵𝐷))
 
Theorem0reALT 8176 Alternate proof of 0re 7880. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℝ
 
Theoremnegcld 8177 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → -𝐴 ∈ ℂ)
 
Theoremsubidd 8178 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝐴) = 0)
 
Theoremsubid1d 8179 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 − 0) = 𝐴)
 
Theoremnegidd 8180 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + -𝐴) = 0)
 
Theoremnegnegd 8181 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 8182 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = 0 ↔ -𝐴 = 0))
 
Theoremnegne0bd 8183 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0))
 
Theoremnegcon1d 8184 Contraposition law for unary minus. Deduction form of negcon1 8131. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
 
Theoremnegcon1ad 8185 Contraposition law for unary minus. One-way deduction form of negcon1 8131. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 = 𝐵)       (𝜑 → -𝐵 = 𝐴)
 
Theoremneg11ad 8186 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 8130. Generalization of neg11d 8202. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = -𝐵𝐴 = 𝐵))
 
Theoremnegned 8187 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 8202. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → -𝐴 ≠ -𝐵)
 
Theoremnegne0d 8188 The negative of a nonzero number is nonzero. See also negap0d 8510 which is similar but for apart from zero rather than not equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → -𝐴 ≠ 0)
 
Theoremnegrebd 8189 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ)
 
Theoremsubcld 8190 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝐵) ∈ ℂ)
 
Theorempncand 8191 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴)
 
Theorempncan2d 8192 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
 
Theorempncan3d 8193 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + (𝐵𝐴)) = 𝐵)
 
Theoremnpcand 8194 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + 𝐵) = 𝐴)
 
Theoremnncand 8195 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 − (𝐴𝐵)) = 𝐵)
 
Theoremnegsubd 8196 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 8197 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵))
 
Theoremsubeq0d 8198 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 8199 Two unequal numbers have nonzero difference. See also subap0d 8523 which is the same thing for apartness rather than negated equality. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝐵) ≠ 0)
 
Theoremsubeq0ad 8200 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 8105. Generalization of subeq0d 8198. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵))
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