Theorem List for Intuitionistic Logic Explorer - 8001-8100 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | pnpcan 8001 |
Cancellation law for mixed addition and subtraction. (Contributed by NM,
4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | pnpcan2 8002 |
Cancellation law for mixed addition and subtraction. (Contributed by
Scott Fenton, 9-Jun-2006.)
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Theorem | pnncan 8003 |
Cancellation law for mixed addition and subtraction. (Contributed by NM,
30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | ppncan 8004 |
Cancellation law for mixed addition and subtraction. (Contributed by NM,
30-Jun-2005.)
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Theorem | addsub4 8005 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by NM, 4-Mar-2005.)
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Theorem | subadd4 8006 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by NM, 24-Aug-2006.)
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Theorem | sub4 8007 |
Rearrangement of 4 terms in a subtraction. (Contributed by NM,
23-Nov-2007.)
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Theorem | neg0 8008 |
Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
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Theorem | negid 8009 |
Addition of a number and its negative. (Contributed by NM,
14-Mar-2005.)
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Theorem | negsub 8010 |
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.)
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Theorem | subneg 8011 |
Relationship between subtraction and negative. (Contributed by NM,
10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | negneg 8012 |
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.)
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Theorem | neg11 8013 |
Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by
Mario Carneiro, 27-May-2016.)
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Theorem | negcon1 8014 |
Negative contraposition law. (Contributed by NM, 9-May-2004.)
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Theorem | negcon2 8015 |
Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
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Theorem | negeq0 8016 |
A number is zero iff its negative is zero. (Contributed by NM,
12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | subcan 8017 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
(Revised by Mario Carneiro, 27-May-2016.)
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Theorem | negsubdi 8018 |
Distribution of negative over subtraction. (Contributed by NM,
15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | negdi 8019 |
Distribution of negative over addition. (Contributed by NM, 10-May-2004.)
(Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | negdi2 8020 |
Distribution of negative over addition. (Contributed by NM,
1-Jan-2006.)
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Theorem | negsubdi2 8021 |
Distribution of negative over subtraction. (Contributed by NM,
4-Oct-1999.)
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Theorem | neg2sub 8022 |
Relationship between subtraction and negative. (Contributed by Paul
Chapman, 8-Oct-2007.)
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Theorem | renegcl 8023 |
Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
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Theorem | renegcli 8024 |
Closure law for negative of reals. (Note: this inference proof style
and the deduction theorem usage in renegcl 8023 is deprecated, but is
retained for its demonstration value.) (Contributed by NM,
17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
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Theorem | resubcli 8025 |
Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.)
(Revised by Mario Carneiro, 27-May-2016.)
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Theorem | resubcl 8026 |
Closure law for subtraction of reals. (Contributed by NM,
20-Jan-1997.)
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Theorem | negreb 8027 |
The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
(Revised by Mario Carneiro, 14-Jul-2014.)
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Theorem | peano2cnm 8028 |
"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.)
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Theorem | peano2rem 8029 |
"Reverse" second Peano postulate analog for reals. (Contributed by
NM,
6-Feb-2007.)
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Theorem | negcli 8030 |
Closure law for negative. (Contributed by NM, 26-Nov-1994.)
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Theorem | negidi 8031 |
Addition of a number and its negative. (Contributed by NM,
26-Nov-1994.)
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Theorem | negnegi 8032 |
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.)
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Theorem | subidi 8033 |
Subtraction of a number from itself. (Contributed by NM,
26-Nov-1994.)
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Theorem | subid1i 8034 |
Identity law for subtraction. (Contributed by NM, 29-May-1999.)
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Theorem | negne0bi 8035 |
A number is nonzero iff its negative is nonzero. (Contributed by NM,
10-Aug-1999.)
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Theorem | negrebi 8036 |
The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
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Theorem | negne0i 8037 |
The negative of a nonzero number is nonzero. (Contributed by NM,
30-Jul-2004.)
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Theorem | subcli 8038 |
Closure law for subtraction. (Contributed by NM, 26-Nov-1994.)
(Revised by Mario Carneiro, 21-Dec-2013.)
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Theorem | pncan3i 8039 |
Subtraction and addition of equals. (Contributed by NM,
26-Nov-1994.)
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Theorem | negsubi 8040 |
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.)
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Theorem | subnegi 8041 |
Relationship between subtraction and negative. (Contributed by NM,
1-Dec-2005.)
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Theorem | subeq0i 8042 |
If the difference between two numbers is zero, they are equal.
(Contributed by NM, 8-May-1999.)
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Theorem | neg11i 8043 |
Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
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Theorem | negcon1i 8044 |
Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
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Theorem | negcon2i 8045 |
Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
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Theorem | negdii 8046 |
Distribution of negative over addition. (Contributed by NM,
28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | negsubdii 8047 |
Distribution of negative over subtraction. (Contributed by NM,
6-Aug-1999.)
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Theorem | negsubdi2i 8048 |
Distribution of negative over subtraction. (Contributed by NM,
1-Oct-1999.)
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Theorem | subaddi 8049 |
Relationship between subtraction and addition. (Contributed by NM,
26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
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Theorem | subadd2i 8050 |
Relationship between subtraction and addition. (Contributed by NM,
15-Dec-2006.)
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Theorem | subaddrii 8051 |
Relationship between subtraction and addition. (Contributed by NM,
16-Dec-2006.)
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Theorem | subsub23i 8052 |
Swap subtrahend and result of subtraction. (Contributed by NM,
7-Oct-1999.)
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Theorem | addsubassi 8053 |
Associative-type law for subtraction and addition. (Contributed by NM,
16-Sep-1999.)
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Theorem | addsubi 8054 |
Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
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Theorem | subcani 8055 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
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Theorem | subcan2i 8056 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
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Theorem | pnncani 8057 |
Cancellation law for mixed addition and subtraction. (Contributed by
NM, 14-Jan-2006.)
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Theorem | addsub4i 8058 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by NM, 17-Oct-1999.)
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Theorem | 0reALT 8059 |
Alternate proof of 0re 7766. (Contributed by NM, 19-Feb-2005.)
(Proof modification is discouraged.) (New usage is discouraged.)
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Theorem | negcld 8060 |
Closure law for negative. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | subidd 8061 |
Subtraction of a number from itself. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | subid1d 8062 |
Identity law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | negidd 8063 |
Addition of a number and its negative. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | negnegd 8064 |
A number is equal to the negative of its negative. Theorem I.4 of
[Apostol] p. 18. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | negeq0d 8065 |
A number is zero iff its negative is zero. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | negne0bd 8066 |
A number is nonzero iff its negative is nonzero. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | negcon1d 8067 |
Contraposition law for unary minus. Deduction form of negcon1 8014.
(Contributed by David Moews, 28-Feb-2017.)
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Theorem | negcon1ad 8068 |
Contraposition law for unary minus. One-way deduction form of
negcon1 8014. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | neg11ad 8069 |
The negatives of two complex numbers are equal iff they are equal.
Deduction form of neg11 8013. Generalization of neg11d 8085.
(Contributed by David Moews, 28-Feb-2017.)
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Theorem | negned 8070 |
If two complex numbers are unequal, so are their negatives.
Contrapositive of neg11d 8085. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | negne0d 8071 |
The negative of a nonzero number is nonzero. See also negap0d 8393 which
is similar but for apart from zero rather than not equal to zero.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | negrebd 8072 |
The negative of a real is real. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | subcld 8073 |
Closure law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | pncand 8074 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | pncan2d 8075 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | pncan3d 8076 |
Subtraction and addition of equals. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | npcand 8077 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | nncand 8078 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | negsubd 8079 |
Relationship between subtraction and negative. Theorem I.3 of [Apostol]
p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | subnegd 8080 |
Relationship between subtraction and negative. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | subeq0d 8081 |
If the difference between two numbers is zero, they are equal.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | subne0d 8082 |
Two unequal numbers have nonzero difference. See also subap0d 8406 which
is the same thing for apartness rather than negated equality.
(Contributed by Mario Carneiro, 1-Jan-2017.)
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Theorem | subeq0ad 8083 |
The difference of two complex numbers is zero iff they are equal.
Deduction form of subeq0 7988. Generalization of subeq0d 8081.
(Contributed by David Moews, 28-Feb-2017.)
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Theorem | subne0ad 8084 |
If the difference of two complex numbers is nonzero, they are unequal.
Converse of subne0d 8082. Contrapositive of subeq0bd 8141. (Contributed
by David Moews, 28-Feb-2017.)
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Theorem | neg11d 8085 |
If the difference between two numbers is zero, they are equal.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | negdid 8086 |
Distribution of negative over addition. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | negdi2d 8087 |
Distribution of negative over addition. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | negsubdid 8088 |
Distribution of negative over subtraction. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | negsubdi2d 8089 |
Distribution of negative over subtraction. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | neg2subd 8090 |
Relationship between subtraction and negative. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | subaddd 8091 |
Relationship between subtraction and addition. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | subadd2d 8092 |
Relationship between subtraction and addition. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | addsubassd 8093 |
Associative-type law for subtraction and addition. (Contributed by
Mario Carneiro, 27-May-2016.)
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Theorem | addsubd 8094 |
Law for subtraction and addition. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | subadd23d 8095 |
Commutative/associative law for addition and subtraction. (Contributed
by Mario Carneiro, 27-May-2016.)
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Theorem | addsub12d 8096 |
Commutative/associative law for addition and subtraction. (Contributed
by Mario Carneiro, 27-May-2016.)
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Theorem | npncand 8097 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | nppcand 8098 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | nppcan2d 8099 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | nppcan3d 8100 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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