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Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubsub3 8001 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − (𝐵𝐶)) = ((𝐴 + 𝐶) − 𝐵))

Theoremsubsub4 8002 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 8003 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 8004 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − (𝐵𝐶)) − 𝐶) = (𝐴𝐵))

Theoremnnncan1 8005 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 8006 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐶) − (𝐵𝐶)) = (𝐴𝐵))

Theoremnpncan3 8007 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 8008 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 8009 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 8010 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 8011 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 8012 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 8013 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 8014 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 8015 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
-0 = 0

Theoremnegid 8016 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
(𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0)

Theoremnegsub 8017 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 8018 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 8019 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 8020 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 8021 Negative contraposition law. (Contributed by NM, 9-May-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))

Theoremnegcon2 8022 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = -𝐵𝐵 = -𝐴))

Theoremnegeq0 8023 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 8024 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 8025 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 8026 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 8027 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴𝐵))

Theoremnegsubdi2 8028 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴𝐵) = (𝐵𝐴))

Theoremneg2sub 8029 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 − -𝐵) = (𝐵𝐴))

Theoremrenegcl 8030 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
(𝐴 ∈ ℝ → -𝐴 ∈ ℝ)

Theoremrenegcli 8031 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 8030 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 8032 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 8033 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵) ∈ ℝ)

Theoremnegreb 8034 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 8035 "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 8036 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
(𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)

Theoremnegcli 8037 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       -𝐴 ∈ ℂ

Theoremnegidi 8038 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴 + -𝐴) = 0

Theoremnegnegi 8039 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 8040 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ       (𝐴𝐴) = 0

Theoremsubid1i 8041 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
𝐴 ∈ ℂ       (𝐴 − 0) = 𝐴

Theoremnegne0bi 8042 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
𝐴 ∈ ℂ       (𝐴 ≠ 0 ↔ -𝐴 ≠ 0)

Theoremnegrebi 8043 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℂ       (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)

Theoremnegne0i 8044 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
𝐴 ∈ ℂ    &   𝐴 ≠ 0       -𝐴 ≠ 0

Theoremsubcli 8045 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴𝐵) ∈ ℂ

Theorempncan3i 8046 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 + (𝐵𝐴)) = 𝐵

Theoremnegsubi 8047 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 8048 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 − -𝐵) = (𝐴 + 𝐵)

Theoremsubeq0i 8049 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 8050 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 = -𝐵𝐴 = 𝐵)

Theoremnegcon1i 8051 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴)

Theoremnegcon2i 8052 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐴 = -𝐵𝐵 = -𝐴)

Theoremnegdii 8053 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 8054 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴𝐵) = (-𝐴 + 𝐵)

Theoremnegsubdi2i 8055 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       -(𝐴𝐵) = (𝐵𝐴)

Theoremsubaddi 8056 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)

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐶 + 𝐵) = 𝐴)

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   (𝐵 + 𝐶) = 𝐴       (𝐴𝐵) = 𝐶

Theoremsubsub23i 8059 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = 𝐶 ↔ (𝐴𝐶) = 𝐵)

Theoremaddsubassi 8060 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − 𝐶) = (𝐴 + (𝐵𝐶))

𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴 + 𝐵) − 𝐶) = ((𝐴𝐶) + 𝐵)

Theoremsubcani 8062 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐵) = (𝐴𝐶) ↔ 𝐵 = 𝐶)

Theoremsubcan2i 8063 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ       ((𝐴𝐶) = (𝐵𝐶) ↔ 𝐴 = 𝐵)

Theorempnncani 8064 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 8065 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 8066 Alternate proof of 0re 7773. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℝ

Theoremnegcld 8067 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → -𝐴 ∈ ℂ)

Theoremsubidd 8068 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴𝐴) = 0)

Theoremsubid1d 8069 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 − 0) = 𝐴)

Theoremnegidd 8070 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 + -𝐴) = 0)

Theoremnegnegd 8071 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 8072 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 = 0 ↔ -𝐴 = 0))

Theoremnegne0bd 8073 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (𝐴 ≠ 0 ↔ -𝐴 ≠ 0))

Theoremnegcon1d 8074 Contraposition law for unary minus. Deduction form of negcon1 8021. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))

Theoremnegcon1ad 8075 Contraposition law for unary minus. One-way deduction form of negcon1 8021. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 = 𝐵)       (𝜑 → -𝐵 = 𝐴)

Theoremneg11ad 8076 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 8020. Generalization of neg11d 8092. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 = -𝐵𝐴 = 𝐵))

Theoremnegned 8077 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 8092. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴𝐵)       (𝜑 → -𝐴 ≠ -𝐵)

Theoremnegne0d 8078 The negative of a nonzero number is nonzero. See also negap0d 8400 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 8079 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → -𝐴 ∈ ℝ)       (𝜑𝐴 ∈ ℝ)

Theoremsubcld 8080 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴𝐵) ∈ ℂ)

Theorempncand 8081 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴)

Theorempncan2d 8082 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

Theorempncan3d 8083 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 + (𝐵𝐴)) = 𝐵)

Theoremnpcand 8084 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴𝐵) + 𝐵) = 𝐴)

Theoremnncand 8085 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 − (𝐴𝐵)) = 𝐵)

Theoremnegsubd 8086 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 8087 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵))

Theoremsubeq0d 8088 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 8089 Two unequal numbers have nonzero difference. See also subap0d 8413 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 8090 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 7995. Generalization of subeq0d 8088. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴𝐵) = 0 ↔ 𝐴 = 𝐵))

Theoremsubne0ad 8091 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 8089. Contrapositive of subeq0bd 8148. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) ≠ 0)       (𝜑𝐴𝐵)

Theoremneg11d 8092 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 8093 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))

Theoremnegdi2d 8094 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴 + 𝐵) = (-𝐴𝐵))

Theoremnegsubdid 8095 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (-𝐴 + 𝐵))

Theoremnegsubdi2d 8096 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → -(𝐴𝐵) = (𝐵𝐴))

Theoremneg2subd 8097 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (-𝐴 − -𝐵) = (𝐵𝐴))

Theoremsubaddd 8098 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 8099 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 8100 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)))

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