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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsub32 7001 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
((A B 𝐶 ℂ) → ((AB) − 𝐶) = ((A𝐶) − B))
 
Theoremnnncan 7002 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
((A B 𝐶 ℂ) → ((A − (B𝐶)) − 𝐶) = (AB))
 
Theoremnnncan1 7003 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B 𝐶 ℂ) → ((AB) − (A𝐶)) = (𝐶B))
 
Theoremnnncan2 7004 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
((A B 𝐶 ℂ) → ((A𝐶) − (B𝐶)) = (AB))
 
Theoremnpncan3 7005 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → ((AB) + (𝐶A)) = (𝐶B))
 
Theorempnpcan 7006 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → ((A + B) − (A + 𝐶)) = (B𝐶))
 
Theorempnpcan2 7007 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
((A B 𝐶 ℂ) → ((A + 𝐶) − (B + 𝐶)) = (AB))
 
Theorempnncan 7008 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → ((A + B) − (A𝐶)) = (B + 𝐶))
 
Theoremppncan 7009 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
((A B 𝐶 ℂ) → ((A + B) + (𝐶B)) = (A + 𝐶))
 
Theoremaddsub4 7010 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) − (𝐶 + 𝐷)) = ((A𝐶) + (B𝐷)))
 
Theoremsubadd4 7011 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) − (𝐶𝐷)) = ((A + 𝐷) − (B + 𝐶)))
 
Theoremsub4 7012 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) − (𝐶𝐷)) = ((A𝐶) − (B𝐷)))
 
Theoremneg0 7013 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
-0 = 0
 
Theoremnegid 7014 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
(A ℂ → (A + -A) = 0)
 
Theoremnegsub 7015 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 B ℂ) → (A + -B) = (AB))
 
Theoremsubneg 7016 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((A B ℂ) → (A − -B) = (A + B))
 
Theoremnegneg 7017 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 7018 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B ℂ) → (-A = -BA = B))
 
Theoremnegcon1 7019 Negative contraposition law. (Contributed by NM, 9-May-2004.)
((A B ℂ) → (-A = B ↔ -B = A))
 
Theoremnegcon2 7020 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
((A B ℂ) → (A = -BB = -A))
 
Theoremnegeq0 7021 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 7022 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → ((AB) = (A𝐶) ↔ B = 𝐶))
 
Theoremnegsubdi 7023 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℂ) → -(AB) = (-A + B))
 
Theoremnegdi 7024 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℂ) → -(A + B) = (-A + -B))
 
Theoremnegdi2 7025 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
((A B ℂ) → -(A + B) = (-AB))
 
Theoremnegsubdi2 7026 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
((A B ℂ) → -(AB) = (BA))
 
Theoremneg2sub 7027 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
((A B ℂ) → (-A − -B) = (BA))
 
Theoremrenegcl 7028 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
(A ℝ → -A ℝ)
 
Theoremrenegcli 7029 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7028 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
A        -A
 
Theoremresubcli 7030 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        (AB)
 
Theoremresubcl 7031 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
((A B ℝ) → (AB) ℝ)
 
Theoremnegreb 7032 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 7033 "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 7034 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
(𝑁 ℝ → (𝑁 − 1) ℝ)
 
Theoremnegcli 7035 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
A        -A
 
Theoremnegidi 7036 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
A        (A + -A) = 0
 
Theoremnegnegi 7037 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        --A = A
 
Theoremsubidi 7038 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
A        (AA) = 0
 
Theoremsubid1i 7039 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
A        (A − 0) = A
 
Theoremnegne0bi 7040 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
A        (A ≠ 0 ↔ -A ≠ 0)
 
Theoremnegrebi 7041 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
A        (-A ℝ ↔ A ℝ)
 
Theoremnegne0i 7042 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
A     &   A ≠ 0       -A ≠ 0
 
Theoremsubcli 7043 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
A     &   B        (AB)
 
Theorempncan3i 7044 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
A     &   B        (A + (BA)) = B
 
Theoremnegsubi 7045 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     &   B        (A + -B) = (AB)
 
Theoremsubnegi 7046 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
A     &   B        (A − -B) = (A + B)
 
Theoremsubeq0i 7047 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
A     &   B        ((AB) = 0 ↔ A = B)
 
Theoremneg11i 7048 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
A     &   B        (-A = -BA = B)
 
Theoremnegcon1i 7049 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
A     &   B        (-A = B ↔ -B = A)
 
Theoremnegcon2i 7050 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
A     &   B        (A = -BB = -A)
 
Theoremnegdii 7051 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B        -(A + B) = (-A + -B)
 
Theoremnegsubdii 7052 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
A     &   B        -(AB) = (-A + B)
 
Theoremnegsubdi2i 7053 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
A     &   B        -(AB) = (BA)
 
Theoremsubaddi 7054 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
A     &   B     &   𝐶        ((AB) = 𝐶 ↔ (B + 𝐶) = A)
 
Theoremsubadd2i 7055 Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.)
A     &   B     &   𝐶        ((AB) = 𝐶 ↔ (𝐶 + B) = A)
 
Theoremsubaddrii 7056 Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.)
A     &   B     &   𝐶     &   (B + 𝐶) = A       (AB) = 𝐶
 
Theoremsubsub23i 7057 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
A     &   B     &   𝐶        ((AB) = 𝐶 ↔ (A𝐶) = B)
 
Theoremaddsubassi 7058 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
A     &   B     &   𝐶        ((A + B) − 𝐶) = (A + (B𝐶))
 
Theoremaddsubi 7059 Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
A     &   B     &   𝐶        ((A + B) − 𝐶) = ((A𝐶) + B)
 
Theoremsubcani 7060 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
A     &   B     &   𝐶        ((AB) = (A𝐶) ↔ B = 𝐶)
 
Theoremsubcan2i 7061 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
A     &   B     &   𝐶        ((A𝐶) = (B𝐶) ↔ A = B)
 
Theorempnncani 7062 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
A     &   B     &   𝐶        ((A + B) − (A𝐶)) = (B + 𝐶)
 
Theoremaddsub4i 7063 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 7064 Alternate proof of 0re 6785. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
0
 
Theoremnegcld 7065 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → -A ℂ)
 
Theoremsubidd 7066 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (AA) = 0)
 
Theoremsubid1d 7067 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A − 0) = A)
 
Theoremnegidd 7068 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A + -A) = 0)
 
Theoremnegnegd 7069 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 7070 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A = 0 ↔ -A = 0))
 
Theoremnegne0bd 7071 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A ≠ 0 ↔ -A ≠ 0))
 
Theoremnegcon1d 7072 Contraposition law for unary minus. Deduction form of negcon1 7019. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)       (φ → (-A = B ↔ -B = A))
 
Theoremnegcon1ad 7073 Contraposition law for unary minus. One-way deduction form of negcon1 7019. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φ → -A = B)       (φ → -B = A)
 
Theoremneg11ad 7074 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7018. Generalization of neg11d 7090. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)       (φ → (-A = -BA = B))
 
Theoremnegned 7075 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7090. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φAB)       (φ → -A ≠ -B)
 
Theoremnegne0d 7076 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φA ≠ 0)       (φ → -A ≠ 0)
 
Theoremnegrebd 7077 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
(φA ℂ)    &   (φ → -A ℝ)       (φA ℝ)
 
Theoremsubcld 7078 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (AB) ℂ)
 
Theorempncand 7079 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → ((A + B) − B) = A)
 
Theorempncan2d 7080 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → ((A + B) − A) = B)
 
Theorempncan3d 7081 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A + (BA)) = B)
 
Theoremnpcand 7082 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → ((AB) + B) = A)
 
Theoremnncand 7083 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A − (AB)) = B)
 
Theoremnegsubd 7084 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 7085 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A − -B) = (A + B))
 
Theoremsubeq0d 7086 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 7087 Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φAB)       (φ → (AB) ≠ 0)
 
Theoremsubeq0ad 7088 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 6993. Generalization of subeq0d 7086. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)       (φ → ((AB) = 0 ↔ A = B))
 
Theoremsubne0ad 7089 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 7087. Contrapositive of subeq0bd 7133. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ → (AB) ≠ 0)       (φAB)
 
Theoremneg11d 7090 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 7091 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → -(A + B) = (-A + -B))
 
Theoremnegdi2d 7092 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → -(A + B) = (-AB))
 
Theoremnegsubdid 7093 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → -(AB) = (-A + B))
 
Theoremnegsubdi2d 7094 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → -(AB) = (BA))
 
Theoremneg2subd 7095 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (-A − -B) = (BA))
 
Theoremsubaddd 7096 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 7097 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 7098 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 7099 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 7100 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)))
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