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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnegeq 7001 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
(A = B → -A = -B)

Theoremnegeqi 7002 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
A = B       -A = -B

Theoremnegeqd 7003 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
(φA = B)       (φ → -A = -B)

Theoremnfnegd 7004 Deduction version of nfneg 7005. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
(φxA)       (φx-A)

Theoremnfneg 7005 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 7006 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 7007 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
((A B ℂ) → (AB) ℂ)

Theoremnegcl 7008 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
(A ℂ → -A ℂ)

Theoremnegicn 7009 -i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
-i

Theoremsubf 7010 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 7011 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 7012 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 7013 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
((A B 𝐶 ℂ) → ((AB) = 𝐶 ↔ (A𝐶) = B))

Theorempncan 7014 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 7015 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
((A B ℂ) → ((A + B) − A) = B)

Theorempncan3 7016 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
((A B ℂ) → (A + (BA)) = B)

Theoremnpcan 7017 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
((A B ℂ) → ((AB) + B) = A)

Theoremaddsubass 7018 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 7019 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 7020 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
((A B 𝐶 ℂ) → ((AB) + 𝐶) = (A + (𝐶B)))

Theoremaddsub12 7021 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
((A B 𝐶 ℂ) → (A + (B𝐶)) = (B + (A𝐶)))

(((A B ℂ) (𝐶 𝐷 ℂ)) → (((A + B) + 𝐶) − 𝐷) = (((A + 𝐶) − 𝐷) + B))

Theoremaddsubeq4 7023 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) = (𝐶 + 𝐷) ↔ (𝐶A) = (B𝐷)))

Theorempncan3oi 7024 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 7084 and pncan 7014, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 7119. (Contributed by David A. Wheeler, 11-Oct-2018.)
A     &   B        ((A + B) − B) = A

Theoremmvlladdi 7025 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
A     &   B     &   (A + B) = 𝐶       B = (𝐶A)

Theoremsubid 7026 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(A ℂ → (AA) = 0)

Theoremsubid1 7027 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
(A ℂ → (A − 0) = A)

Theoremnpncan 7028 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((A B 𝐶 ℂ) → ((AB) + (B𝐶)) = (A𝐶))

Theoremnppcan 7029 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
((A B 𝐶 ℂ) → (((AB) + 𝐶) + B) = (A + 𝐶))

Theoremnnpcan 7030 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 B 𝐶 ℂ) → (((AB) − 𝐶) + B) = (A𝐶))

Theoremnppcan3 7031 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
((A B 𝐶 ℂ) → ((AB) + (𝐶 + B)) = (A + 𝐶))

Theoremsubcan2 7032 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
((A B 𝐶 ℂ) → ((A𝐶) = (B𝐶) ↔ A = B))

Theoremsubeq0 7033 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)
((A B ℂ) → ((AB) = 0 ↔ A = B))

Theoremnpncan2 7034 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
((A B ℂ) → ((AB) + (BA)) = 0)

Theoremsubsub2 7035 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → (A − (B𝐶)) = (A + (𝐶B)))

Theoremnncan 7036 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B ℂ) → (A − (AB)) = B)

Theoremsubsub 7037 Law for double subtraction. (Contributed by NM, 13-May-2004.)
((A B 𝐶 ℂ) → (A − (B𝐶)) = ((AB) + 𝐶))

Theoremnppcan2 7038 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
((A B 𝐶 ℂ) → ((A − (B + 𝐶)) + 𝐶) = (AB))

Theoremsubsub3 7039 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
((A B 𝐶 ℂ) → (A − (B𝐶)) = ((A + 𝐶) − B))

Theoremsubsub4 7040 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → ((AB) − 𝐶) = (A − (B + 𝐶)))

Theoremsub32 7041 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 7042 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
((A B 𝐶 ℂ) → ((A − (B𝐶)) − 𝐶) = (AB))

Theoremnnncan1 7043 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 7044 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
((A B 𝐶 ℂ) → ((A𝐶) − (B𝐶)) = (AB))

Theoremnpncan3 7045 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 7046 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 7047 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
((A B 𝐶 ℂ) → ((A + 𝐶) − (B + 𝐶)) = (AB))

Theorempnncan 7048 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 7049 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
((A B 𝐶 ℂ) → ((A + B) + (𝐶B)) = (A + 𝐶))

Theoremaddsub4 7050 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 7051 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 7052 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) − (𝐶𝐷)) = ((A𝐶) − (B𝐷)))

Theoremneg0 7053 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
-0 = 0

Theoremnegid 7054 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
(A ℂ → (A + -A) = 0)

Theoremnegsub 7055 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 7056 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 7057 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 7058 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 7059 Negative contraposition law. (Contributed by NM, 9-May-2004.)
((A B ℂ) → (-A = B ↔ -B = A))

Theoremnegcon2 7060 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
((A B ℂ) → (A = -BB = -A))

Theoremnegeq0 7061 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 7062 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 7063 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 7064 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 7065 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
((A B ℂ) → -(A + B) = (-AB))

Theoremnegsubdi2 7066 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
((A B ℂ) → -(AB) = (BA))

Theoremneg2sub 7067 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
((A B ℂ) → (-A − -B) = (BA))

Theoremrenegcl 7068 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
(A ℝ → -A ℝ)

Theoremrenegcli 7069 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7068 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 7070 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        (AB)

Theoremresubcl 7071 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
((A B ℝ) → (AB) ℝ)

Theoremnegreb 7072 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 7073 "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 7074 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
(𝑁 ℝ → (𝑁 − 1) ℝ)

Theoremnegcli 7075 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
A        -A

Theoremnegidi 7076 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
A        (A + -A) = 0

Theoremnegnegi 7077 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 7078 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
A        (AA) = 0

Theoremsubid1i 7079 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
A        (A − 0) = A

Theoremnegne0bi 7080 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
A        (A ≠ 0 ↔ -A ≠ 0)

Theoremnegrebi 7081 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
A        (-A ℝ ↔ A ℝ)

Theoremnegne0i 7082 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
A     &   A ≠ 0       -A ≠ 0

Theoremsubcli 7083 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
A     &   B        (AB)

Theorempncan3i 7084 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
A     &   B        (A + (BA)) = B

Theoremnegsubi 7085 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 7086 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
A     &   B        (A − -B) = (A + B)

Theoremsubeq0i 7087 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
A     &   B        ((AB) = 0 ↔ A = B)

Theoremneg11i 7088 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
A     &   B        (-A = -BA = B)

Theoremnegcon1i 7089 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
A     &   B        (-A = B ↔ -B = A)

Theoremnegcon2i 7090 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
A     &   B        (A = -BB = -A)

Theoremnegdii 7091 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 7092 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
A     &   B        -(AB) = (-A + B)

Theoremnegsubdi2i 7093 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
A     &   B        -(AB) = (BA)

Theoremsubaddi 7094 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)

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

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

Theoremsubsub23i 7097 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
A     &   B     &   𝐶        ((AB) = 𝐶 ↔ (A𝐶) = B)

Theoremaddsubassi 7098 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
A     &   B     &   𝐶        ((A + B) − 𝐶) = (A + (B𝐶))