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Theorem List for Metamath Proof Explorer - 27801-27900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrgrawopreg 27801* In a friendship graph there are either no vertices or exactly one vertex having degree K, or all or all except one vertices have degree K. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V  \  A )   =>    |-  ( V FriendGrph  E  ->  (
 ( ( # `  A )  =  1  \/  A  =  (/) )  \/  ( ( # `  B )  =  1  \/  B  =  (/) ) ) )
 
Theoremfrgrawopreg1 27802* According to the proof of the friendship theorem in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V  \  A )   =>    |-  ( ( V FriendGrph  E  /\  ( # `  A )  =  1 )  ->  E. v  e.  V  A. w  e.  ( V 
 \  { v }
 ) { v ,  w }  e.  ran  E )
 
Theoremfrgrawopreg2 27803* According to the proof of the friendship theorem in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V  \  A )   =>    |-  ( ( V FriendGrph  E  /\  ( # `  B )  =  1 )  ->  E. v  e.  V  A. w  e.  ( V 
 \  { v }
 ) { v ,  w }  e.  ran  E )
 
Theoremfrgraregorufr0 27804* For each nonnegative integer K there are either no edges having degree K, or all vertices have degree K in a friendship graph, unless there is a universal friend. This corresponds to the second claim in the proof of the friendship theorem in [Huneke] p. 2: "... all vertices have degree k, unless there is a universal friend." (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  ( V FriendGrph  E  ->  ( A. v  e.  V  (
 ( V VDeg  E ) `  v )  =  K  \/  A. v  e.  V  ( ( V VDeg  E ) `  v )  =/= 
 K  \/  E. v  e.  V  A. w  e.  ( V  \  {
 v } ) {
 v ,  w }  e.  ran  E ) )
 
Theoremfrgraregorufr 27805* For each nonnegative integer K there are either no edges having degree K, or all vertices have degree K in a friendship graph, unless there is a universal friend. This corresponds to the second claim in the proof of the friendship theorem in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  ( V FriendGrph  E  ->  ( E. a  e.  V  (
 ( V VDeg  E ) `  a )  =  K  ->  ( A. v  e.  V  ( ( V VDeg 
 E ) `  v
 )  =  K  \/  E. v  e.  V  A. w  e.  ( V  \  { v } ) { v ,  w }  e.  ran  E ) ) )
 
19.23  Mathbox for David A. Wheeler

This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/.

 
19.23.1  Natural deduction
 
Theorem19.8ad 27806 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1754. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbidd 27807 An identity theorem for substitution. See sbid 1936. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
 |-  ( ph  ->  [ x  /  x ] ps )   =>    |-  ( ph  ->  ps )
 
Theoremsbidd-misc 27808 An identity theorem for substitution. See sbid 1936. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
 |-  (
 ( ph  ->  [ x  /  x ] ps )  <->  (
 ph  ->  ps ) )
 
19.23.2  Greater than, greater than or equal to.

As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems.

 
Syntaxcge-real 27809 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 27811.
 class  >_
 
Syntaxcgt 27810 Extend wff notation to include the 'greater than' relation, see df-gt 27812.
 class  >
 
Definitiondf-gte 27811 Define the 'greater than or equal' predicate over the reals. Defined in ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. This relation is merely the converse of the 'less than or equal to' relation defined by df-le 9059.

We do not write this as  ( x  >_  y  <->  y  <_  x ), and similarly we do not write ` > ` as  ( x  >  y  <->  y  <  x ), because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way:  |-  >  =  { <. x ,  y
>.  |  ( (
x  e.  RR*  /\  y  e.  RR* )  /\  y  <  x ) } and  |-  >_  =  { <. x ,  y
>.  |  ( (
x  e.  RR*  /\  y  e.  RR* )  /\  y  <_  x ) } but these are very complicated. This definition of  >_, and the similar one for  > (df-gt 27812), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 27813 for a more conventional expression of the relationship between  < and  >. As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

 |-  >_  =  `'  <_
 
Definitiondf-gt 27812 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 8936. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 27811 for a discussion on why this approach is used for the definition. See gt-lt 27814 and gt-lth 27816 for more conventional expression of the relationship between  < and  >.

As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

 |-  >  =  `'  <
 
Theoremgte-lte 27813 Simple relationship between  <_ and  >_. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >_  B  <->  B 
 <_  A ) )
 
Theoremgt-lt 27814 Simple relationship between  < and  >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >  B  <->  B  <  A ) )
 
Theoremgte-lteh 27815 Relationship between  <_ and  >_ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  >_  B  <->  B  <_  A )
 
Theoremgt-lth 27816 Relationship between  < and  > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  >  B  <->  B  <  A )
 
Theoremex-gt 27817 Simple example of  >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
 |-  -.  0  >  0
 
Theoremex-gte 27818 Simple example of  >_, in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
 |-  0  >_  0
 
19.23.3  Hyperbolic trig functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as  ( cos `  ( _i  x.  x ) ). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved.

 
Syntaxcsinh 27819 Extend class notation to include the hyperbolic sine function, see df-sinh 27822.
 class sinh
 
Syntaxccosh 27820 Extend class notation to include the hyperbolic cosine function. see df-cosh 27823.
 class cosh
 
Syntaxctanh 27821 Extend class notation to include the hyperbolic tangent function, see df-tanh 27824.
 class tanh
 
Definitiondf-sinh 27822 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4207, which requires the form  (
x  e.  A  |->  B ). See sinhval-named 27825 for a simple way to evaluate it. We define this function by dividing by  _i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in metamath). See sinh-conventional 27828 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
 |- sinh  =  ( x  e.  CC  |->  ( ( sin `  ( _i  x.  x ) ) 
 /  _i ) )
 
Definitiondf-cosh 27823 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4207, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- cosh  =  ( x  e.  CC  |->  ( cos `  ( _i  x.  x ) ) )
 
Definitiondf-tanh 27824 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4207, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- tanh  =  ( x  e.  ( `'cosh " ( CC  \  { 0 } )
 )  |->  ( ( tan `  ( _i  x.  x ) )  /  _i ) )
 
Theoremsinhval-named 27825 Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 27822. See sinhval 12682 for a theorem to convert this further. See sinh-conventional 27828 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
 |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremcoshval-named 27826 Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 27823. See coshval 12683 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
 
Theoremtanhval-named 27827 Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 27824. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  ( `'cosh " ( CC  \  {
 0 } ) ) 
 ->  (tanh `  A )  =  ( ( tan `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremsinh-conventional 27828 Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using metamath. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (sinh `  A )  =  (
 -u _i  x.  ( sin `  ( _i  x.  A ) ) ) )
 
Theoremsinhpcosh 27829 Prove that  (sinh `  A
)  +  (cosh `  A )  =  ( exp `  A ) using the conventional hyperbolic trig functions. (Contributed by David A. Wheeler, 27-May-2015.)
 |-  ( A  e.  CC  ->  ( (sinh `  A )  +  (cosh `  A )
 )  =  ( exp `  A ) )
 
19.23.4  Reciprocal trig functions (sec, csc, cot)

Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.

 
Syntaxcsec 27830 Extend class notation to include the secant function, see df-sec 27833.
 class  sec
 
Syntaxccsc 27831 Extend class notation to include the cosecant function, see df-csc 27834.
 class  csc
 
Syntaxccot 27832 Extend class notation to include the cotangent function, see df-cot 27835.
 class  cot
 
Definitiondf-sec 27833* Define the secant function. We define it this way for cmpt 4207, which requires the form  ( x  e.  A  |->  B ). The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  sec  =  ( x  e.  {
 y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1 
 /  ( cos `  x ) ) )
 
Definitiondf-csc 27834* Define the cosecant function. We define it this way for cmpt 4207, which requires the form  ( x  e.  A  |->  B ). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  csc  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1 
 /  ( sin `  x ) ) )
 
Definitiondf-cot 27835* Define the cotangent function. We define it this way for cmpt 4207, which requires the form  ( x  e.  A  |->  B ). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  cot  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x )  /  ( sin `  x ) ) )
 
Theoremsecval 27836 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  =  ( 1  /  ( cos `  A ) ) )
 
Theoremcscval 27837 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  =  ( 1  /  ( sin `  A ) ) )
 
Theoremcotval 27838 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  =  ( ( cos `  A )  /  ( sin `  A ) ) )
 
Theoremseccl 27839 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  CC )
 
Theoremcsccl 27840 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  CC )
 
Theoremcotcl 27841 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  CC )
 
Theoremreseccl 27842 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  RR )
 
Theoremrecsccl 27843 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  RR )
 
Theoremrecotcl 27844 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  RR )
 
Theoremrecsec 27845 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( cos `  A )  =  ( 1  /  ( sec `  A ) ) )
 
Theoremreccsc 27846 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( sin `  A )  =  ( 1  /  ( csc `  A ) ) )
 
Theoremreccot 27847 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( 1  /  ( cot `  A ) ) )
 
Theoremrectan 27848 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( cot `  A )  =  ( 1  /  ( tan `  A ) ) )
 
Theoremsec0 27849 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 27850 Prove the tangent squared secant squared identity  ( 1  +  ( ( tan A ) ^ 2 ) ) = ( ( sec  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 25-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( 1  +  (
 ( tan `  A ) ^ 2 ) )  =  ( ( sec `  A ) ^ 2
 ) )
 
Theoremcotsqcscsq 27851 Prove the tangent squared cosecant squared identity  ( 1  +  ( ( cot A ) ^ 2 ) ) = ( ( csc  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 27-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( 1  +  (
 ( cot `  A ) ^ 2 ) )  =  ( ( csc `  A ) ^ 2
 ) )
 
19.23.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 27852 If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3689 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
19.23.6  Not-member-of
 
TheoremAnelBC 27853 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using 
e/. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  A  =/=  B   &    |-  A  =/=  C   =>    |-  A  e/  { B ,  C }
 
19.23.7  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 27857 and df-dp2 27856 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10315.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 27854 Constant used for decimal fraction constructor. See df-dp2 27856.
 class _ A B
 
Syntaxcdp 27855 Decimal point operator. See df-dp 27857.
 class  period
 
Definitiondf-dp2 27856 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10315. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 27857* Define the  period (decimal point) operator. For example,  ( 1 period 5 )  =  ( 3  /  2 ), and  -u (; 3 2 period_ 7_ 1 8 )  = 
-u (;;;; 3 2 7 1 8  / ;;; 1 0 0 0 ) Unary minus, if applied, should normally be applied in front of the parentheses.

Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers.

The RHS is  RR, not  QQ; this should simplify some proofs. The LHS is  NN0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression  -u ( A period B ) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

 |-  period  =  ( x  e.  NN0 ,  y  e.  RR  |-> _ x y )
 
Theoremdp2cl 27858 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  -> _ A B  e.  RR )
 
Theoremdpval 27859 Define the value of the decimal point operator. See df-dp 27857. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
 
Theoremdpcl 27860 Prove that the closure of the decimal point is  RR as we have defined it. See df-dp 27857. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  e.  RR )
 
Theoremdpfrac1 27861 Prove a simple equivalence involving the decimal point. See df-dp 27857 and dpcl 27860. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  =  (; A B  /  10 ) )
 
19.23.8  Signum (sgn or sign) function
 
Syntaxcsgn 27862 Extend class notation to include the Signum function.
 class sgn
 
Definitiondf-sgn 27863 Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over  RR* (df-xr 9057) instead of  RR so that it can accept  +oo and  -oo. Note that df-psgn 27084 defines the sign of a permutation, which is different. Value shown in sgnval 27864. (Contributed by David A. Wheeler, 15-May-2015.)
 |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 , 
 0 ,  if ( x  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgnval 27864 Value of Signum function. Pronounced "signum" . See df-sgn 27863. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgn0 27865 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (sgn `  0 )  =  0
 
Theoremsgnp 27866 Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR*  /\  0  <  A ) 
 ->  (sgn `  A )  =  1 )
 
Theoremsgnrrp 27867 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
 |-  ( A  e.  RR+  ->  (sgn `  A )  =  1 )
 
Theoremsgn1 27868 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn `  1 )  =  1
 
Theoremsgnpnf 27869 Proof that the signum of  +oo is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 +oo )  =  1
 
Theoremsgnn 27870 Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR*  /\  A  <  0 ) 
 ->  (sgn `  A )  =  -u 1 )
 
Theoremsgnmnf 27871 Proof that the signum of  -oo is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` 
 -oo )  =  -u 1
 
19.23.9  Ceiling function
 
Syntaxccei 27872 Extend class notation to include the ceiling function.
 class
 
Definitiondf-ceiling 27873 The ceiling function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4.

By convention metamath users tend to expand this construct directly, instead of using the definition. However, we want to make sure this is separately and formally defined. Proof ceicl 11159 shows that the ceiling function returns an integer when provided a real. Formalized by David A. Wheeler. (Contributed by David A. Wheeler, 19-May-2015.)

 |- =  ( x  e.  RR  |->  -u ( |_ `  -u x ) )
 
Theoremceilingval 27874 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  =  -u ( |_ `  -u A ) )
 
Theoremceilingcl 27875 Closure of the ceiling function; the real work is in ceicl 11159. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  ( A  e.  RR  ->  ( `  A )  e.  ZZ )
 
19.23.10  Logarithms generalized to arbitrary base using ` logb `
 
Theoremene0 27876  _e is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  0
 
Theoremene1 27877  _e is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  _e  =/=  1
 
Theoremelogb 27878 Using  _e as the base is the same as  log. (Contributed by David A. Wheeler, 17-Oct-2017.)
 |-  ( A  e.  ( CC  \  { 0 } )  ->  ( _elogb A )  =  ( log `  A ) )
 
19.23.11  Logarithm laws generalized to an arbitrary base - log_

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear.

This supports the notational form  ( (log_ `  B ) `  X
); that looks a little more like traditional notation, but is different from other 2-parameter functions. E.G.,  ( (log_ `  10 ) ` ;; 1 0 0 )  =  2

This form is less convenient to work with inside metamath as compared to the  ( Blogb X
) form defined separately.

 
Syntaxclog_ 27879 Extend class notation to include the logarithm generalized to an arbitrary base.
 class log_
 
Definitiondf-log_ 27880* Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as  ( (log_ `  B ) `  X ) for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.)
 |- log_  =  ( b  e.  ( CC  \  { 0 ,  1 } )  |->  ( x  e.  ( CC  \  { 0 } )  |->  ( ( log `  x )  /  ( log `  b
 ) ) ) )
 
19.23.12  Miscellaneous

Miscellaneous proofs.

 
Theorem5m4e1 27881 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 5  -  4 )  =  1
 
Theorem2p2ne5 27882 Prove that  2  +  2  =/=  5. In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase  2  +  2  =  5 has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.)
 |-  (
 2  +  2 )  =/=  5
 
Theoremresolution 27883 Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.)
 |-  (
 ( ( ph  /\  ps )  \/  ( -.  ph  /\ 
 ch ) )  ->  ( ps  \/  ch )
 )
 
19.24  Mathbox for Alan Sare
 
19.24.1  Supplementary "adant" deductions
 
Theoremad4ant13 27884 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant14 27885 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant123 27886 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ta )  ->  th )
 
Theoremad4ant124 27887 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ta )  /\  ch )  ->  th )
 
Theoremad4ant134 27888 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ta )  /\  ps )  /\  ch )  ->  th )
 
Theoremad4ant23 27889 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant24 27890 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant234 27891 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ta 
 /\  ph )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant12 27892 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  ps )  /\  th )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant13 27893 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant14 27894 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant15 27895 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant23 27896 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant24 27897 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant25 27898 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant245 27899 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  et )  /\  ps )  /\  ch )  ->  th )
 
Theoremad5ant234 27900 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  th )   =>    |-  (
 ( ( ( ( ta  /\  ph )  /\  ps )  /\  ch )  /\  et )  ->  th )
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