Theorem List for Intuitionistic Logic Explorer - 12801-12900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | limcmpted 12801* |
Express the limit operator for a function defined by a mapping, via
epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
|
lim #
|
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Theorem | limcimolemlt 12802* |
Lemma for limcimo 12803. (Contributed by Jim Kingdon, 3-Jul-2023.)
|
↾t # lim lim #
#
|
|
Theorem | limcimo 12803* |
Conditions which ensure there is at most one limit value of at
.
(Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by
Jim Kingdon, 8-Jul-2023.)
|
↾t # lim |
|
Theorem | limcresi 12804 |
Any limit of is also
a limit of the restriction of .
(Contributed by Mario Carneiro, 28-Dec-2016.)
|
lim lim |
|
Theorem | cnplimcim 12805 |
If a function is continuous at , its limit at equals the
value of the function there. (Contributed by Mario Carneiro,
28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
|
↾t
lim |
|
Theorem | cnplimclemle 12806 |
Lemma for cnplimccntop 12808. Satisfying the epsilon condition for
continuity. (Contributed by Mario Carneiro and Jim Kingdon,
17-Nov-2023.)
|
↾t lim
#
|
|
Theorem | cnplimclemr 12807 |
Lemma for cnplimccntop 12808. The reverse direction. (Contributed by
Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
|
↾t lim |
|
Theorem | cnplimccntop 12808 |
A function is continuous at iff its limit at equals the
value of the function there. (Contributed by Mario Carneiro,
28-Dec-2016.)
|
↾t
lim |
|
Theorem | cnlimcim 12809* |
If is a continuous
function, the limit of the function at each
point equals the value of the function. (Contributed by Mario Carneiro,
28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
|
lim |
|
Theorem | cnlimc 12810* |
is a continuous
function iff the limit of the function at each
point equals the value of the function. (Contributed by Mario Carneiro,
28-Dec-2016.)
|
lim |
|
Theorem | cnlimci 12811 |
If is a continuous
function, then the limit of the function at any
point equals its value. (Contributed by Mario Carneiro,
28-Dec-2016.)
|
lim |
|
Theorem | cnmptlimc 12812* |
If is a continuous
function, then the limit of the function at any
point equals its value. (Contributed by Mario Carneiro,
28-Dec-2016.)
|
lim |
|
Theorem | limccnpcntop 12813 |
If the limit of at
is and is continuous at
, then the
limit of at is .
(Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon,
18-Jun-2023.)
|
↾t lim
lim |
|
Theorem | limccnp2lem 12814* |
Lemma for limccnp2cntop 12815. This is most of the result, expressed in
epsilon-delta form, with a large number of hypotheses so that lengthy
expressions do not need to be repeated. (Contributed by Jim Kingdon,
9-Nov-2023.)
|
↾t
lim
lim
#
#
# |
|
Theorem | limccnp2cntop 12815* |
The image of a convergent sequence under a continuous map is convergent
to the image of the original point. Binary operation version.
(Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon,
14-Nov-2023.)
|
↾t
lim
lim lim |
|
Theorem | limccoap 12816* |
Composition of two limits. This theorem is only usable in the case
where # implies R(x) # so it is less general than
might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.)
(Revised by Jim Kingdon, 18-Dec-2023.)
|
#
#
#
# lim
# lim
#
lim |
|
Theorem | reldvg 12817 |
The derivative function is a relation. (Contributed by Mario Carneiro,
7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
|
|
|
Theorem | dvlemap 12818* |
Closure for a difference quotient. (Contributed by Mario Carneiro,
1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|
#
|
|
Theorem | dvfvalap 12819* |
Value and set bounds on the derivative operator. (Contributed by Mario
Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|
↾t
# lim
|
|
Theorem | eldvap 12820* |
The differentiable predicate. A function is differentiable at
with
derivative iff is defined in a
neighborhood of
and the
difference quotient has limit at .
(Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon,
27-Jun-2023.)
|
↾t #
lim |
|
Theorem | dvcl 12821 |
The derivative function takes values in the complex numbers.
(Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario
Carneiro, 9-Feb-2015.)
|
|
|
Theorem | dvbssntrcntop 12822 |
The set of differentiable points is a subset of the interior of the
domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.)
(Revised by Jim Kingdon, 27-Jun-2023.)
|
↾t |
|
Theorem | dvbss 12823 |
The set of differentiable points is a subset of the domain of the
function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by
Mario Carneiro, 9-Feb-2015.)
|
|
|
Theorem | dvbsssg 12824 |
The set of differentiable points is a subset of the ambient topology.
(Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon,
28-Jun-2023.)
|
|
|
Theorem | recnprss 12825 |
Both and are subsets of . (Contributed by Mario
Carneiro, 10-Feb-2015.)
|
|
|
Theorem | dvfgg 12826 |
Explicitly write out the functionality condition on derivative for
and . (Contributed by Mario Carneiro, 9-Feb-2015.)
(Revised by Jim Kingdon, 28-Jun-2023.)
|
|
|
Theorem | dvfpm 12827 |
The derivative is a function. (Contributed by Mario Carneiro,
8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
|
|
|
Theorem | dvfcnpm 12828 |
The derivative is a function. (Contributed by Mario Carneiro,
9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)
|
|
|
Theorem | dvidlemap 12829* |
Lemma for dvid 12831 and dvconst 12830. (Contributed by Mario Carneiro,
8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|
#
|
|
Theorem | dvconst 12830 |
Derivative of a constant function. (Contributed by Mario Carneiro,
8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|
|
|
Theorem | dvid 12831 |
Derivative of the identity function. (Contributed by Mario Carneiro,
8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|
|
|
Theorem | dvcnp2cntop 12832 |
A function is continuous at each point for which it is differentiable.
(Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario
Carneiro, 28-Dec-2016.)
|
↾t
|
|
Theorem | dvcn 12833 |
A differentiable function is continuous. (Contributed by Mario
Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)
|
|
|
Theorem | dvaddxxbr 12834 |
The sum rule for derivatives at a point. That is, if the derivative
of at is and the derivative of at is
, then the
derivative of the pointwise sum of those two
functions at
is . (Contributed by Mario Carneiro,
9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
|
|
|
Theorem | dvmulxxbr 12835 |
The product rule for derivatives at a point. For the (simpler but
more limited) function version, see dvmulxx 12837. (Contributed by Mario
Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
|
|
|
Theorem | dvaddxx 12836 |
The sum rule for derivatives at a point. For the (more general)
relation version, see dvaddxxbr 12834. (Contributed by Mario Carneiro,
9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
|
|
|
Theorem | dvmulxx 12837 |
The product rule for derivatives at a point. For the (more general)
relation version, see dvmulxxbr 12835. (Contributed by Mario Carneiro,
9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
|
|
|
Theorem | dviaddf 12838 |
The sum rule for everywhere-differentiable functions. (Contributed by
Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro,
10-Feb-2015.)
|
|
|
Theorem | dvimulf 12839 |
The product rule for everywhere-differentiable functions. (Contributed
by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro,
10-Feb-2015.)
|
|
|
Theorem | dvcoapbr 12840* |
The chain rule for derivatives at a point. The
# #
hypothesis constrains what
functions work for . (Contributed by Mario Carneiro,
9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
|
# #
|
|
Theorem | dvcjbr 12841 |
The derivative of the conjugate of a function. For the (simpler but
more limited) function version, see dvcj 12842. (Contributed by Mario
Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|
|
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Theorem | dvcj 12842 |
The derivative of the conjugate of a function. For the (more general)
relation version, see dvcjbr 12841. (Contributed by Mario Carneiro,
1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|
|
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Theorem | dvfre 12843 |
The derivative of a real function is real. (Contributed by Mario
Carneiro, 1-Sep-2014.)
|
|
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Theorem | dvexp 12844* |
Derivative of a power function. (Contributed by Mario Carneiro,
9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
|
|
|
Theorem | dvexp2 12845* |
Derivative of an exponential, possibly zero power. (Contributed by
Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro,
10-Feb-2015.)
|
|
|
Theorem | dvrecap 12846* |
Derivative of the reciprocal function. (Contributed by Mario Carneiro,
25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
|
# #
|
|
Theorem | dvmptidcn 12847 |
Function-builder for derivative: derivative of the identity.
(Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon,
30-Dec-2023.)
|
|
|
Theorem | dvmptccn 12848* |
Function-builder for derivative: derivative of a constant. (Contributed
by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon,
30-Dec-2023.)
|
|
|
Theorem | dvmptclx 12849* |
Closure lemma for dvmptmulx 12851 and other related theorems. (Contributed
by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro,
11-Feb-2015.)
|
|
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Theorem | dvmptaddx 12850* |
Function-builder for derivative, addition rule. (Contributed by Mario
Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|
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Theorem | dvmptmulx 12851* |
Function-builder for derivative, product rule. (Contributed by Mario
Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
|
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|
Theorem | dvmptcmulcn 12852* |
Function-builder for derivative, product rule for constant multiplier.
(Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon,
31-Dec-2023.)
|
|
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Theorem | dvmptnegcn 12853* |
Function-builder for derivative, product rule for negatives.
(Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon,
31-Dec-2023.)
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Theorem | dvmptsubcn 12854* |
Function-builder for derivative, subtraction rule. (Contributed by
Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon,
31-Dec-2023.)
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|
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Theorem | dveflem 12855 |
Derivative of the exponential function at 0. The key step in the proof
is eftlub 11396, to show that
.
(Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario
Carneiro, 28-Dec-2016.)
|
|
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Theorem | dvef 12856 |
Derivative of the exponential function. (Contributed by Mario Carneiro,
9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
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PART 9 BASIC REAL AND COMPLEX
FUNCTIONS
|
|
9.1 Basic trigonometry
|
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9.1.1 The exponential, sine, and cosine
functions (cont.)
|
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Theorem | efcn 12857 |
The exponential function is continuous. (Contributed by Paul Chapman,
15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
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Theorem | sincn 12858 |
Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
(Revised by Mario Carneiro, 3-Sep-2014.)
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Theorem | coscn 12859 |
Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
(Revised by Mario Carneiro, 3-Sep-2014.)
|
|
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9.1.2 Properties of pi =
3.14159...
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Theorem | pilem1 12860 |
Lemma for pire , pigt2lt4 and sinpi . (Contributed by Mario Carneiro,
9-May-2014.)
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Theorem | cosz12 12861 |
Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and
Jim Kingdon, 7-Mar-2024.)
|
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Theorem | sin0pilem1 12862* |
Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim
Kingdon, 8-Mar-2024.)
|
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Theorem | sin0pilem2 12863* |
Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim
Kingdon, 8-Mar-2024.)
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Theorem | pilem3 12864 |
Lemma for pi related theorems. (Contributed by Jim Kingdon,
9-Mar-2024.)
|
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Theorem | pigt2lt4 12865 |
is between 2 and 4.
(Contributed by Paul Chapman, 23-Jan-2008.)
(Revised by Mario Carneiro, 9-May-2014.)
|
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Theorem | sinpi 12866 |
The sine of is 0.
(Contributed by Paul Chapman, 23-Jan-2008.)
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Theorem | pire 12867 |
is a real number.
(Contributed by Paul Chapman, 23-Jan-2008.)
|
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Theorem | picn 12868 |
is a complex number.
(Contributed by David A. Wheeler,
6-Dec-2018.)
|
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Theorem | pipos 12869 |
is positive.
(Contributed by Paul Chapman, 23-Jan-2008.)
(Revised by Mario Carneiro, 9-May-2014.)
|
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Theorem | pirp 12870 |
is a positive real.
(Contributed by Glauco Siliprandi,
11-Dec-2019.)
|
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Theorem | negpicn 12871 |
is a real number.
(Contributed by David A. Wheeler,
8-Dec-2018.)
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Theorem | sinhalfpilem 12872 |
Lemma for sinhalfpi 12877 and coshalfpi 12878. (Contributed by Paul Chapman,
23-Jan-2008.)
|
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Theorem | halfpire 12873 |
is real. (Contributed by David Moews,
28-Feb-2017.)
|
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Theorem | neghalfpire 12874 |
is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|
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Theorem | neghalfpirx 12875 |
is an extended real. (Contributed by David A. Wheeler,
8-Dec-2018.)
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Theorem | pidiv2halves 12876 |
Adding to itself gives . See 2halves 8949.
(Contributed by David A. Wheeler, 8-Dec-2018.)
|
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Theorem | sinhalfpi 12877 |
The sine of is 1. (Contributed by Paul Chapman,
23-Jan-2008.)
|
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Theorem | coshalfpi 12878 |
The cosine of is 0. (Contributed by Paul Chapman,
23-Jan-2008.)
|
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Theorem | cosneghalfpi 12879 |
The cosine of is zero. (Contributed by David Moews,
28-Feb-2017.)
|
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Theorem | efhalfpi 12880 |
The exponential of is . (Contributed by Mario
Carneiro, 9-May-2014.)
|
|
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Theorem | cospi 12881 |
The cosine of is
. (Contributed by Paul
Chapman,
23-Jan-2008.)
|
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Theorem | efipi 12882 |
The exponential of
is . (Contributed by Paul
Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|
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Theorem | eulerid 12883 |
Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised
by Mario Carneiro, 9-May-2014.)
|
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Theorem | sin2pi 12884 |
The sine of is 0. (Contributed by
Paul Chapman,
23-Jan-2008.)
|
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Theorem | cos2pi 12885 |
The cosine of is 1. (Contributed by
Paul Chapman,
23-Jan-2008.)
|
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Theorem | ef2pi 12886 |
The exponential of is . (Contributed by Mario
Carneiro, 9-May-2014.)
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Theorem | ef2kpi 12887 |
If is an integer,
then the exponential of is .
(Contributed by Mario Carneiro, 9-May-2014.)
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Theorem | efper 12888 |
The exponential function is periodic. (Contributed by Paul Chapman,
21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)
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Theorem | sinperlem 12889 |
Lemma for sinper 12890 and cosper 12891. (Contributed by Paul Chapman,
23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|
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Theorem | sinper 12890 |
The sine function is periodic. (Contributed by Paul Chapman,
23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
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Theorem | cosper 12891 |
The cosine function is periodic. (Contributed by Paul Chapman,
23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)
|
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Theorem | sin2kpi 12892 |
If is an integer,
then the sine of is 0. (Contributed
by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro,
10-May-2014.)
|
|
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Theorem | cos2kpi 12893 |
If is an integer,
then the cosine of is 1. (Contributed
by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro,
10-May-2014.)
|
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Theorem | sin2pim 12894 |
Sine of a number subtracted from . (Contributed by Paul
Chapman, 15-Mar-2008.)
|
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Theorem | cos2pim 12895 |
Cosine of a number subtracted from . (Contributed by Paul
Chapman, 15-Mar-2008.)
|
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Theorem | sinmpi 12896 |
Sine of a number less . (Contributed by Paul Chapman,
15-Mar-2008.)
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Theorem | cosmpi 12897 |
Cosine of a number less . (Contributed by Paul Chapman,
15-Mar-2008.)
|
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Theorem | sinppi 12898 |
Sine of a number plus . (Contributed by NM, 10-Aug-2008.)
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Theorem | cosppi 12899 |
Cosine of a number plus . (Contributed by NM, 18-Aug-2008.)
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Theorem | efimpi 12900 |
The exponential function at times a real number less .
(Contributed by Paul Chapman, 15-Mar-2008.)
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