Theorem List for Intuitionistic Logic Explorer - 8201-8300 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | 2addsub 8201 |
Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
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Theorem | addsubeq4 8202 |
Relation between sums and differences. (Contributed by Jeff Madsen,
17-Jun-2010.)
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Theorem | pncan3oi 8203 |
Subtraction and addition of equals. Almost but not exactly the same as
pncan3i 8264 and pncan 8193, 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 8299. (Contributed by
David A. Wheeler, 11-Oct-2018.)
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Theorem | mvrraddi 8204 |
Move RHS right addition to LHS. (Contributed by David A. Wheeler,
11-Oct-2018.)
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Theorem | mvlladdi 8205 |
Move LHS left addition to RHS. (Contributed by David A. Wheeler,
11-Oct-2018.)
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Theorem | subid 8206 |
Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.)
(Revised by Mario Carneiro, 27-May-2016.)
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Theorem | subid1 8207 |
Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised
by Mario Carneiro, 27-May-2016.)
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Theorem | npncan 8208 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
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Theorem | nppcan 8209 |
Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
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Theorem | nnpcan 8210 |
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.)
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Theorem | nppcan3 8211 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
14-Sep-2015.)
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Theorem | subcan2 8212 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
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Theorem | subeq0 8213 |
If the difference between two numbers is zero, they are equal.
(Contributed by NM, 16-Nov-1999.)
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Theorem | npncan2 8214 |
Cancellation law for subtraction. (Contributed by Scott Fenton,
21-Jun-2013.)
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Theorem | subsub2 8215 |
Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised
by Mario Carneiro, 27-May-2016.)
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Theorem | nncan 8216 |
Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.)
(Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | subsub 8217 |
Law for double subtraction. (Contributed by NM, 13-May-2004.)
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Theorem | nppcan2 8218 |
Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
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Theorem | subsub3 8219 |
Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
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Theorem | subsub4 8220 |
Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised
by Mario Carneiro, 27-May-2016.)
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Theorem | sub32 8221 |
Swap the second and third terms in a double subtraction. (Contributed by
NM, 19-Aug-2005.)
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Theorem | nnncan 8222 |
Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
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Theorem | nnncan1 8223 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
(Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | nnncan2 8224 |
Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
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Theorem | npncan3 8225 |
Cancellation law for subtraction. (Contributed by Scott Fenton,
23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | pnpcan 8226 |
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 8227 |
Cancellation law for mixed addition and subtraction. (Contributed by
Scott Fenton, 9-Jun-2006.)
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Theorem | pnncan 8228 |
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 8229 |
Cancellation law for mixed addition and subtraction. (Contributed by NM,
30-Jun-2005.)
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Theorem | addsub4 8230 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by NM, 4-Mar-2005.)
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Theorem | subadd4 8231 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by NM, 24-Aug-2006.)
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Theorem | sub4 8232 |
Rearrangement of 4 terms in a subtraction. (Contributed by NM,
23-Nov-2007.)
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Theorem | neg0 8233 |
Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
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Theorem | negid 8234 |
Addition of a number and its negative. (Contributed by NM,
14-Mar-2005.)
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Theorem | negsub 8235 |
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 8236 |
Relationship between subtraction and negative. (Contributed by NM,
10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | negneg 8237 |
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 8238 |
Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by
Mario Carneiro, 27-May-2016.)
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Theorem | negcon1 8239 |
Negative contraposition law. (Contributed by NM, 9-May-2004.)
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Theorem | negcon2 8240 |
Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
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Theorem | negeq0 8241 |
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 8242 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
(Revised by Mario Carneiro, 27-May-2016.)
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Theorem | negsubdi 8243 |
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 8244 |
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 8245 |
Distribution of negative over addition. (Contributed by NM,
1-Jan-2006.)
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Theorem | negsubdi2 8246 |
Distribution of negative over subtraction. (Contributed by NM,
4-Oct-1999.)
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Theorem | neg2sub 8247 |
Relationship between subtraction and negative. (Contributed by Paul
Chapman, 8-Oct-2007.)
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Theorem | renegcl 8248 |
Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
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Theorem | renegcli 8249 |
Closure law for negative of reals. (Note: this inference proof style
and the deduction theorem usage in renegcl 8248 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 8250 |
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 8251 |
Closure law for subtraction of reals. (Contributed by NM,
20-Jan-1997.)
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Theorem | negreb 8252 |
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 8253 |
"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 8254 |
"Reverse" second Peano postulate analog for reals. (Contributed by
NM,
6-Feb-2007.)
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Theorem | negcli 8255 |
Closure law for negative. (Contributed by NM, 26-Nov-1994.)
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Theorem | negidi 8256 |
Addition of a number and its negative. (Contributed by NM,
26-Nov-1994.)
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Theorem | negnegi 8257 |
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 8258 |
Subtraction of a number from itself. (Contributed by NM,
26-Nov-1994.)
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Theorem | subid1i 8259 |
Identity law for subtraction. (Contributed by NM, 29-May-1999.)
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Theorem | negne0bi 8260 |
A number is nonzero iff its negative is nonzero. (Contributed by NM,
10-Aug-1999.)
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Theorem | negrebi 8261 |
The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
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Theorem | negne0i 8262 |
The negative of a nonzero number is nonzero. (Contributed by NM,
30-Jul-2004.)
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Theorem | subcli 8263 |
Closure law for subtraction. (Contributed by NM, 26-Nov-1994.)
(Revised by Mario Carneiro, 21-Dec-2013.)
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Theorem | pncan3i 8264 |
Subtraction and addition of equals. (Contributed by NM,
26-Nov-1994.)
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Theorem | negsubi 8265 |
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 8266 |
Relationship between subtraction and negative. (Contributed by NM,
1-Dec-2005.)
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Theorem | subeq0i 8267 |
If the difference between two numbers is zero, they are equal.
(Contributed by NM, 8-May-1999.)
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Theorem | neg11i 8268 |
Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
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Theorem | negcon1i 8269 |
Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
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Theorem | negcon2i 8270 |
Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
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Theorem | negdii 8271 |
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 8272 |
Distribution of negative over subtraction. (Contributed by NM,
6-Aug-1999.)
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Theorem | negsubdi2i 8273 |
Distribution of negative over subtraction. (Contributed by NM,
1-Oct-1999.)
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Theorem | subaddi 8274 |
Relationship between subtraction and addition. (Contributed by NM,
26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
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Theorem | subadd2i 8275 |
Relationship between subtraction and addition. (Contributed by NM,
15-Dec-2006.)
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Theorem | subaddrii 8276 |
Relationship between subtraction and addition. (Contributed by NM,
16-Dec-2006.)
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Theorem | subsub23i 8277 |
Swap subtrahend and result of subtraction. (Contributed by NM,
7-Oct-1999.)
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Theorem | addsubassi 8278 |
Associative-type law for subtraction and addition. (Contributed by NM,
16-Sep-1999.)
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Theorem | addsubi 8279 |
Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
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Theorem | subcani 8280 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
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Theorem | subcan2i 8281 |
Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
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Theorem | pnncani 8282 |
Cancellation law for mixed addition and subtraction. (Contributed by
NM, 14-Jan-2006.)
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Theorem | addsub4i 8283 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by NM, 17-Oct-1999.)
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Theorem | 0reALT 8284 |
Alternate proof of 0re 7987. (Contributed by NM, 19-Feb-2005.)
(Proof modification is discouraged.) (New usage is discouraged.)
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Theorem | negcld 8285 |
Closure law for negative. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | subidd 8286 |
Subtraction of a number from itself. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | subid1d 8287 |
Identity law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | negidd 8288 |
Addition of a number and its negative. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | negnegd 8289 |
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 8290 |
A number is zero iff its negative is zero. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | negne0bd 8291 |
A number is nonzero iff its negative is nonzero. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | negcon1d 8292 |
Contraposition law for unary minus. Deduction form of negcon1 8239.
(Contributed by David Moews, 28-Feb-2017.)
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Theorem | negcon1ad 8293 |
Contraposition law for unary minus. One-way deduction form of
negcon1 8239. (Contributed by David Moews, 28-Feb-2017.)
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Theorem | neg11ad 8294 |
The negatives of two complex numbers are equal iff they are equal.
Deduction form of neg11 8238. Generalization of neg11d 8310.
(Contributed by David Moews, 28-Feb-2017.)
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Theorem | negned 8295 |
If two complex numbers are unequal, so are their negatives.
Contrapositive of neg11d 8310. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | negne0d 8296 |
The negative of a nonzero number is nonzero. See also negap0d 8618 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 8297 |
The negative of a real is real. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | subcld 8298 |
Closure law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | pncand 8299 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | pncan2d 8300 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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