Theorem List for Intuitionistic Logic Explorer - 5801-5900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | mpoeq123dv 5801* |
An equality deduction for the maps-to notation. (Contributed by NM,
12-Sep-2011.)
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Theorem | mpoeq123i 5802 |
An equality inference for the maps-to notation. (Contributed by NM,
15-Jul-2013.)
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Theorem | mpoeq3dva 5803* |
Slightly more general equality inference for the maps-to notation.
(Contributed by NM, 17-Oct-2013.)
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Theorem | mpoeq3ia 5804 |
An equality inference for the maps-to notation. (Contributed by Mario
Carneiro, 16-Dec-2013.)
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Theorem | mpoeq3dv 5805* |
An equality deduction for the maps-to notation restricted to the value
of the operation. (Contributed by SO, 16-Jul-2018.)
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Theorem | nfmpo1 5806 |
Bound-variable hypothesis builder for an operation in maps-to notation.
(Contributed by NM, 27-Aug-2013.)
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Theorem | nfmpo2 5807 |
Bound-variable hypothesis builder for an operation in maps-to notation.
(Contributed by NM, 27-Aug-2013.)
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Theorem | nfmpo 5808* |
Bound-variable hypothesis builder for the maps-to notation.
(Contributed by NM, 20-Feb-2013.)
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Theorem | mpo0 5809 |
A mapping operation with empty domain. (Contributed by Stefan O'Rear,
29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
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Theorem | oprab4 5810* |
Two ways to state the domain of an operation. (Contributed by FL,
24-Jan-2010.)
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Theorem | cbvoprab1 5811* |
Rule used to change first bound variable in an operation abstraction,
using implicit substitution. (Contributed by NM, 20-Dec-2008.)
(Revised by Mario Carneiro, 5-Dec-2016.)
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Theorem | cbvoprab2 5812* |
Change the second bound variable in an operation abstraction.
(Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro,
11-Dec-2016.)
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Theorem | cbvoprab12 5813* |
Rule used to change first two bound variables in an operation
abstraction, using implicit substitution. (Contributed by NM,
21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
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Theorem | cbvoprab12v 5814* |
Rule used to change first two bound variables in an operation
abstraction, using implicit substitution. (Contributed by NM,
8-Oct-2004.)
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Theorem | cbvoprab3 5815* |
Rule used to change the third bound variable in an operation
abstraction, using implicit substitution. (Contributed by NM,
22-Aug-2013.)
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Theorem | cbvoprab3v 5816* |
Rule used to change the third bound variable in an operation
abstraction, using implicit substitution. (Contributed by NM,
8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)
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Theorem | cbvmpox 5817* |
Rule to change the bound variable in a maps-to function, using implicit
substitution. This version of cbvmpo 5818 allows to be a function of
. (Contributed
by NM, 29-Dec-2014.)
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Theorem | cbvmpo 5818* |
Rule to change the bound variable in a maps-to function, using implicit
substitution. (Contributed by NM, 17-Dec-2013.)
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Theorem | cbvmpov 5819* |
Rule to change the bound variable in a maps-to function, using implicit
substitution. With a longer proof analogous to cbvmpt 3993, some distinct
variable requirements could be eliminated. (Contributed by NM,
11-Jun-2013.)
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Theorem | dmoprab 5820* |
The domain of an operation class abstraction. (Contributed by NM,
17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
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Theorem | dmoprabss 5821* |
The domain of an operation class abstraction. (Contributed by NM,
24-Aug-1995.)
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Theorem | rnoprab 5822* |
The range of an operation class abstraction. (Contributed by NM,
30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
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Theorem | rnoprab2 5823* |
The range of a restricted operation class abstraction. (Contributed by
Scott Fenton, 21-Mar-2012.)
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Theorem | reldmoprab 5824* |
The domain of an operation class abstraction is a relation.
(Contributed by NM, 17-Mar-1995.)
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Theorem | oprabss 5825* |
Structure of an operation class abstraction. (Contributed by NM,
28-Nov-2006.)
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Theorem | eloprabga 5826* |
The law of concretion for operation class abstraction. Compare
elopab 4150. (Contributed by NM, 14-Sep-1999.)
(Unnecessary distinct
variable restrictions were removed by David Abernethy, 19-Jun-2012.)
(Revised by Mario Carneiro, 19-Dec-2013.)
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Theorem | eloprabg 5827* |
The law of concretion for operation class abstraction. Compare
elopab 4150. (Contributed by NM, 14-Sep-1999.) (Revised
by David
Abernethy, 19-Jun-2012.)
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Theorem | ssoprab2i 5828* |
Inference of operation class abstraction subclass from implication.
(Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy,
19-Jun-2012.)
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Theorem | mpov 5829* |
Operation with universal domain in maps-to notation. (Contributed by
NM, 16-Aug-2013.)
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Theorem | mpomptx 5830* |
Express a two-argument function as a one-argument function, or
vice-versa. In this version is not assumed to be constant
w.r.t .
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | mpompt 5831* |
Express a two-argument function as a one-argument function, or
vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by
Mario Carneiro, 29-Dec-2014.)
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Theorem | mpodifsnif 5832 |
A mapping with two arguments with the first argument from a difference set
with a singleton and a conditional as result. (Contributed by AV,
13-Feb-2019.)
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Theorem | mposnif 5833 |
A mapping with two arguments with the first argument from a singleton and
a conditional as result. (Contributed by AV, 14-Feb-2019.)
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Theorem | fconstmpo 5834* |
Representation of a constant operation using the mapping operation.
(Contributed by SO, 11-Jul-2018.)
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Theorem | resoprab 5835* |
Restriction of an operation class abstraction. (Contributed by NM,
10-Feb-2007.)
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Theorem | resoprab2 5836* |
Restriction of an operator abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | resmpo 5837* |
Restriction of the mapping operation. (Contributed by Mario Carneiro,
17-Dec-2013.)
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Theorem | funoprabg 5838* |
"At most one" is a sufficient condition for an operation class
abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
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Theorem | funoprab 5839* |
"At most one" is a sufficient condition for an operation class
abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
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Theorem | fnoprabg 5840* |
Functionality and domain of an operation class abstraction.
(Contributed by NM, 28-Aug-2007.)
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Theorem | mpofun 5841* |
The maps-to notation for an operation is always a function.
(Contributed by Scott Fenton, 21-Mar-2012.)
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Theorem | fnoprab 5842* |
Functionality and domain of an operation class abstraction.
(Contributed by NM, 15-May-1995.)
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Theorem | ffnov 5843* |
An operation maps to a class to which all values belong. (Contributed
by NM, 7-Feb-2004.)
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Theorem | fovcl 5844 |
Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
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Theorem | eqfnov 5845* |
Equality of two operations is determined by their values. (Contributed
by NM, 1-Sep-2005.)
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Theorem | eqfnov2 5846* |
Two operators with the same domain are equal iff their values at each
point in the domain are equal. (Contributed by Jeff Madsen,
7-Jun-2010.)
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Theorem | fnovim 5847* |
Representation of a function in terms of its values. (Contributed by
Jim Kingdon, 16-Jan-2019.)
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Theorem | mpo2eqb 5848* |
Bidirectional equality theorem for a mapping abstraction. Equivalent to
eqfnov2 5846. (Contributed by Mario Carneiro,
4-Jan-2017.)
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Theorem | rnmpo 5849* |
The range of an operation given by the maps-to notation. (Contributed
by FL, 20-Jun-2011.)
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Theorem | reldmmpo 5850* |
The domain of an operation defined by maps-to notation is a relation.
(Contributed by Stefan O'Rear, 27-Nov-2014.)
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Theorem | elrnmpog 5851* |
Membership in the range of an operation class abstraction. (Contributed
by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
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Theorem | elrnmpo 5852* |
Membership in the range of an operation class abstraction.
(Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | ralrnmpo 5853* |
A restricted quantifier over an image set. (Contributed by Mario
Carneiro, 1-Sep-2015.)
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Theorem | rexrnmpo 5854* |
A restricted quantifier over an image set. (Contributed by Mario
Carneiro, 1-Sep-2015.)
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Theorem | ovid 5855* |
The value of an operation class abstraction. (Contributed by NM,
16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
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Theorem | ovidig 5856* |
The value of an operation class abstraction. Compare ovidi 5857. The
condition is been
removed. (Contributed by
Mario Carneiro, 29-Dec-2014.)
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Theorem | ovidi 5857* |
The value of an operation class abstraction (weak version).
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | ov 5858* |
The value of an operation class abstraction. (Contributed by NM,
16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
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Theorem | ovigg 5859* |
The value of an operation class abstraction. Compare ovig 5860.
The
condition is been
removed. (Contributed by FL,
24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
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Theorem | ovig 5860* |
The value of an operation class abstraction (weak version).
(Unnecessary distinct variable restrictions were removed by David
Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by
Mario Carneiro, 19-Dec-2013.)
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Theorem | ovmpt4g 5861* |
Value of a function given by the maps-to notation. (This is the
operation analog of fvmpt2 5472.) (Contributed by NM, 21-Feb-2004.)
(Revised by Mario Carneiro, 1-Sep-2015.)
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Theorem | ovmpos 5862* |
Value of a function given by the maps-to notation, expressed using
explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
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Theorem | ov2gf 5863* |
The value of an operation class abstraction. A version of ovmpog 5873
using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.)
(Revised by Mario Carneiro, 19-Dec-2013.)
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Theorem | ovmpodxf 5864* |
Value of an operation given by a maps-to rule, deduction form.
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | ovmpodx 5865* |
Value of an operation given by a maps-to rule, deduction form.
(Contributed by Mario Carneiro, 29-Dec-2014.)
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Theorem | ovmpod 5866* |
Value of an operation given by a maps-to rule, deduction form.
(Contributed by Mario Carneiro, 7-Dec-2014.)
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Theorem | ovmpox 5867* |
The value of an operation class abstraction. Variant of ovmpoga 5868 which
does not require and to be
distinct. (Contributed by Jeff
Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
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Theorem | ovmpoga 5868* |
Value of an operation given by a maps-to rule. (Contributed by Mario
Carneiro, 19-Dec-2013.)
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Theorem | ovmpoa 5869* |
Value of an operation given by a maps-to rule. (Contributed by NM,
19-Dec-2013.)
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Theorem | ovmpodf 5870* |
Alternate deduction version of ovmpo 5874, suitable for iteration.
(Contributed by Mario Carneiro, 7-Jan-2017.)
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Theorem | ovmpodv 5871* |
Alternate deduction version of ovmpo 5874, suitable for iteration.
(Contributed by Mario Carneiro, 7-Jan-2017.)
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Theorem | ovmpodv2 5872* |
Alternate deduction version of ovmpo 5874, suitable for iteration.
(Contributed by Mario Carneiro, 7-Jan-2017.)
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Theorem | ovmpog 5873* |
Value of an operation given by a maps-to rule. Special case.
(Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy,
19-Jun-2012.)
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Theorem | ovmpo 5874* |
Value of an operation given by a maps-to rule. Special case.
(Contributed by NM, 16-May-1995.) (Revised by David Abernethy,
19-Jun-2012.)
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Theorem | ovi3 5875* |
The value of an operation class abstraction. Special case.
(Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro,
29-Dec-2014.)
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Theorem | ov6g 5876* |
The value of an operation class abstraction. Special case.
(Contributed by NM, 13-Nov-2006.)
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Theorem | ovg 5877* |
The value of an operation class abstraction. (Contributed by Jeff
Madsen, 10-Jun-2010.)
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Theorem | ovres 5878 |
The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
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Theorem | ovresd 5879 |
Lemma for converting metric theorems to metric space theorems.
(Contributed by Mario Carneiro, 2-Oct-2015.)
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Theorem | oprssov 5880 |
The value of a member of the domain of a subclass of an operation.
(Contributed by NM, 23-Aug-2007.)
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Theorem | fovrn 5881 |
An operation's value belongs to its codomain. (Contributed by NM,
27-Aug-2006.)
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Theorem | fovrnda 5882 |
An operation's value belongs to its codomain. (Contributed by Mario
Carneiro, 29-Dec-2016.)
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Theorem | fovrnd 5883 |
An operation's value belongs to its codomain. (Contributed by Mario
Carneiro, 29-Dec-2016.)
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Theorem | fnrnov 5884* |
The range of an operation expressed as a collection of the operation's
values. (Contributed by NM, 29-Oct-2006.)
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Theorem | foov 5885* |
An onto mapping of an operation expressed in terms of operation values.
(Contributed by NM, 29-Oct-2006.)
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Theorem | fnovrn 5886 |
An operation's value belongs to its range. (Contributed by NM,
10-Feb-2007.)
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Theorem | ovelrn 5887* |
A member of an operation's range is a value of the operation.
(Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro,
30-Jan-2014.)
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Theorem | funimassov 5888* |
Membership relation for the values of a function whose image is a
subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)
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Theorem | ovelimab 5889* |
Operation value in an image. (Contributed by Mario Carneiro,
23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
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Theorem | ovconst2 5890 |
The value of a constant operation. (Contributed by NM, 5-Nov-2006.)
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Theorem | caovclg 5891* |
Convert an operation closure law to class notation. (Contributed by
Mario Carneiro, 26-May-2014.)
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Theorem | caovcld 5892* |
Convert an operation closure law to class notation. (Contributed by
Mario Carneiro, 30-Dec-2014.)
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Theorem | caovcl 5893* |
Convert an operation closure law to class notation. (Contributed by NM,
4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
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Theorem | caovcomg 5894* |
Convert an operation commutative law to class notation. (Contributed
by Mario Carneiro, 1-Jun-2013.)
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Theorem | caovcomd 5895* |
Convert an operation commutative law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovcom 5896* |
Convert an operation commutative law to class notation. (Contributed
by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 1-Jun-2013.)
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Theorem | caovassg 5897* |
Convert an operation associative law to class notation. (Contributed
by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro,
26-May-2014.)
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Theorem | caovassd 5898* |
Convert an operation associative law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
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Theorem | caovass 5899* |
Convert an operation associative law to class notation. (Contributed
by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
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Theorem | caovcang 5900* |
Convert an operation cancellation law to class notation. (Contributed
by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
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