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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfnrnafv 27401* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 5530. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F' x ) } )
 
Theoremafvelrnb 27402* A member of a function's range is a value of the function, analogous to fvelrnb 5531 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  V ) 
 ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F' x )  =  B ) )
 
Theoremafvelrnb0 27403* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 5531. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
 |-  ( F  Fn  A  ->  ( B  e.  ran  F  ->  E. x  e.  A  ( F' x )  =  B ) )
 
Theoremdfaimafn 27404* Alternate definition of the image of a function, analogous to dfimafn 5532. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  C_  dom  F )  ->  ( F " A )  =  { y  |  E. x  e.  A  ( F' x )  =  y } )
 
Theoremdfaimafn2 27405* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5533. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  C_  dom  F )  ->  ( F " A )  =  U_ x  e.  A  { ( F' x ) } )
 
Theoremafvelima 27406* Function value in an image, analogous to fvelima 5535. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  ( F " B ) )  ->  E. x  e.  B  ( F' x )  =  A )
 
Theoremafvelrn 27407 A function's value belongs to its range, analogous to fvelrn 5622. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( F' A )  e.  ran  F )
 
Theoremfnafvelrn 27408 A function's value belongs to its range, analogous to fnfvelrn 5623. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( F' B )  e.  ran  F )
 
Theoremfafvelrn 27409 A function's value belongs to its codomain, analogous to ffvelrn 5624. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F : A --> B  /\  C  e.  A )  ->  ( F' C )  e.  B )
 
Theoremffnafv 27410* A function maps to a class to which all values belong, analogous to ffnfv 5646. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F' x )  e.  B )
 )
 
Theoremafvres 27411 The value of a restricted function, analogous to fvres 5502. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  e.  B  ->  ( ( F  |`  B )'
 A )  =  ( F' A ) )
 
Theoremtz6.12-afv 27412* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, , analogous to tz6.12-1 5504, but it is required for A to be a set. (Contributed by Alexander van der Vekens, 28-Jul-2017.)
 |-  ( A  e.  V  ->  ( ( A F y 
 /\  E! y  A F y )  ->  ( F'
 A )  =  y ) )
 
Theoremdmfcoafv 27413 Domains of a function composition, analogous to dmfco 5554. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( Fun  G  /\  A  e.  dom  G ) 
 ->  ( A  e.  dom  (  F  o.  G ) 
 <->  ( G' A )  e.  dom  F )
 )
 
Theoremafvco2 27414 Value of a function composition, analogous to fvco2 5555. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( G  Fn  A  /\  X  e.  A ) 
 ->  ( ( F  o.  G )' X )  =  ( F' ( G'
 X ) ) )
 
18.21.11  2.6 Alternative definition of the value of an operation
 
Syntaxcaov 27415 Extend class notation to include the value of an operation  F (such as  +) for two arguments  A and  B. Note that the syntax is simply three class symbols in a row surrounded by special parentheses (exclamation mark with underscore) in contrast to the current definition, see df-ov 5822.
 class (( A F B))
 
Definitiondf-aov 27416 Define the value of an operation. In contrast to df-ov 5822, the alternative definition for a function value ( see df-afv 27371) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation  F and its arguments  A and  B- will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |- (( A F B))  =  ( F' <. A ,  B >. )
 
Theoremaoveq123d 27417 Equality deduction for operation value, analogous to oveq123d 5840. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  -> (( A F C))  = (( B G D))  )
 
Theoremnfaov 27418 Bound-variable hypothesis builder for operation value, analogous to nfov 5842. To prove a deduction version of this analogous to nfovd 5841 is not quickly possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of alternative operation values is based on are not available (see nfafv 27376). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x (( A F B))
 
Theoremcsbaovg 27419 Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( A  e.  D  ->  [_ A  /  x ]_ (( B F C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )
 
Theoremaovfundmoveq 27420 If a class is a function restricted to an ordered pair of its domain, then the value of the operation on this pair is equal for both definitions. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( Fdef@ 
 <. A ,  B >.  -> (( A F B))  =  ( A F B ) )
 
Theoremaovnfundmuv 27421 If an ordered pair is not in the domain of a class or the class is not a function restricted to the ordered pair, then the operation value for this pair is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  Fdef@  <. A ,  B >.  -> (( A F B))  =  _V )
 
Theoremndmaov 27422 The value of an operation outside its domain, analogous to ndmafv 27380. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  <. A ,  B >.  e.  dom  F  -> (( A F B))  =  _V )
 
Theoremndmaovg 27423 The value of an operation outside its domain, analogous to ndmovg 5964. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( dom  F  =  ( R  X.  S ) 
 /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> (( A F B))  =  _V )
 
Theoremaovvdm 27424 If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  e.  C  -> 
 <. A ,  B >.  e. 
 dom  F )
 
Theoremnfunsnaov 27425 If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  Fun  ( F  |`  { <. A ,  B >. } )  -> (( A F B))  =  _V )
 
Theoremaovvfunressn 27426 If the operation value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  e.  C  ->  Fun  ( F  |`  { <. A ,  B >. } )
 )
 
Theoremaovprc 27427 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 5846. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> (( A F B))  =  _V )
 
Theoremaovrcl 27428 Reverse closure for an operation value, analogous to afvvv 27385. In contrast to ovrcl 5849, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( (( A F B))  e.  C  ->  ( A  e.  _V  /\  B  e.  _V ) )
 
Theoremaovpcov0 27429 If the alternative value of the operation on an ordered pair is the universal class, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  =  _V  ->  ( A F B )  =  (/) )
 
Theoremaovnuoveq 27430 The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  =/=  _V  -> (( A F B))  =  ( A F B ) )
 
Theoremaovvoveq 27431 The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  e.  C  -> (( A F B))  =  ( A F B ) )
 
Theoremaov0ov0 27432 If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (( A F B))  =  (/)  ->  ( A F B )  =  (/) )
 
Theoremaovovn0oveq 27433 If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( A F B )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
 
Theoremaov0nbovbi 27434 The operation's value on an ordered pair is an element of a set if and only if the alternative value of the operation on this ordered pair is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( (/)  e/  C  ->  ( (( A F B))  e.  C  <->  ( A F B )  e.  C ) )
 
Theoremaovov0bi 27435 The operation's value on an ordered pair is the empty set if and only if the alternative value of the operation on this ordered pair is either the empty set or the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( A F B )  =  (/)  <->  ( (( A F B))  =  (/)  \/ (( A F B))  =  _V ) )
 
Theoremrspceaov 27436* A frequently used special case of rspc2ev 2893 for operation values, analogous to rspceov 5854. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( C  e.  A  /\  D  e.  B  /\  S  = (( C F D))  )  ->  E. x  e.  A  E. y  e.  B  S  = (( x F y))  )
 
Theoremfnotaovb 27437 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5525. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( (( C F D))  =  R  <->  <. C ,  D ,  R >.  e.  F ) )
 
Theoremffnaov 27438* An operation maps to a class to which all values belong, analogous to ffnov 5909. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B ) 
 /\  A. x  e.  A  A. y  e.  B (( x F y))  e.  C ) )
 
Theoremfaovcl 27439 Closure law for an operation, analogous to fovcl 5910. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F : ( R  X.  S ) --> C   =>    |-  ( ( A  e.  R  /\  B  e.  S )  -> (( A F B))  e.  C )
 
Theoremaovmpt4g 27440* Value of a function given by the "maps to" notation, analogous to ovmpt4g 5931. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  -> (( x F y))  =  C )
 
Theoremaoprssdm 27441* Domain of closure of an operation. In contrast to oprssdm 5963, no additional property for S (
-.  (/)  e.  S) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( x  e.  S  /\  y  e.  S )  -> (( x F y))  e.  S )   =>    |-  ( S  X.  S )  C_  dom  F
 
Theoremndmaovcl 27442 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 5966 where it is required that the domain contains the empty set ( (/) 
e.  S). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   &    |-  ( ( A  e.  S  /\  B  e.  S )  -> (( A F B))  e.  S )   &    |- (( A F B))  e.  _V   =>    |- (( A F B))  e.  S
 
Theoremndmaovrcl 27443 Reverse closure law, in contrast to ndmovrcl 5967 where it is required that the operation's domain doesn't contain the empty set ( -.  (/)  e.  S), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( (( A F B))  e.  S  ->  ( A  e.  S  /\  B  e.  S ) )
 
Theoremndmaovcom 27444 Any operation is commutative outside its domain, analogous to ndmovcom 5968. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S )  -> (( A F B))  = (( B F A))  )
 
Theoremndmaovass 27445 Any operation is associative outside its domain. In contrast to ndmaovass 27445 where it is required that the operation's domain doesn't contain the empty set ( -.  (/)  e.  S), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  -> (( (( A F B))  F C))  = (( A F (( B F C)) ))  )
 
Theoremndmaovdistr 27446 Any operation is distributive outside its domain. In contrast to ndmaovdistr 27446 where it is required that the operation's domain doesn't contain the empty set (
-.  (/)  e.  S), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   &    |-  dom  G  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) 
 -> (( A G (( B F C)) ))  = (( (( A G B))  F (( A G C)) ))  )
 
18.22  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/.

 
18.22.1  Natural deduction
 
Theorem19.8ad 27447 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1719. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbidd 27448 An identity theorem for substitution. See sbid 1864. 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 27449 An identity theorem for substitution. See sbid 1864. 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 ) )
 
18.22.2  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 27450 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 27452.
 class  >_
 
Syntaxcgt 27451 Extend wff notation to include the 'greater than' relation, see df-gt 27453.
 class  >
 
Definitiondf-gte 27452 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 8868.

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 27453), 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 27454 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 27453 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 8745. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 27452 for a discussion on why this approach is used for the definition. See gt-lt 27455 and gt-lth 27457 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 27454 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 27455 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 27456 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 27457 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 27458 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 27459 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
 
18.22.3  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 27460 Extend class notation to include the hyperbolic sine function, see df-sinh 27463.
 class sinh
 
Syntaxccosh 27461 Extend class notation to include the hyperbolic cosine function. see df-cosh 27464.
 class cosh
 
Syntaxctanh 27462 Extend class notation to include the hyperbolic tangent function, see df-tanh 27465.
 class tanh
 
Definitiondf-sinh 27463 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4078, which requires the form  (
x  e.  A  |->  B ). See sinhval-named 27466 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 27469 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 27464 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4078, 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 27465 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4078, 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 27466 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 27463. See sinhval 12428 for a theorem to convert this further. See sinh-conventional 27469 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 27467 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 27464. See coshval 12429 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 27468 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 27465. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  ( `'cosh " ( CC  \  {
 0 } ) ) 
 ->  (tanh `  A )  =  ( ( tan `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremsinh-conventional 27469 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 27470 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 ) )
 
18.22.4  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 27471 Extend class notation to include the secant function, see df-sec 27474.
 class  sec
 
Syntaxccsc 27472 Extend class notation to include the cosecant function, see df-csc 27475.
 class  csc
 
Syntaxccot 27473 Extend class notation to include the cotangent function, see df-cot 27476.
 class  cot
 
Definitiondf-sec 27474* Define the secant function. We define it this way for cmpt 4078, 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 27475* Define the cosecant function. We define it this way for cmpt 4078, 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 27476* Define the cotangent function. We define it this way for cmpt 4078, 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 27477 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 27478 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 27479 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 27480 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 27481 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 27482 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 27483 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 27484 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 27485 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 27486 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 27487 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 27488 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 27489 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 27490 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 27491 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 27492 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
 ) )
 
18.22.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 27493 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 3573 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
18.22.6  Not-member-of
 
TheoremAnelBC 27494 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 }
 
18.22.7  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 27498 and df-dp2 27497 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10120.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 27495 Constant used for decimal fraction constructor. See df-dp2 27497.
 class _ A B
 
Syntaxcdp 27496 Decimal point operator. See df-dp 27498.
 class  period
 
Definitiondf-dp2 27497 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10120. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 27498* 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 27499 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 27500 Define the value of the decimal point operator. See df-dp 27498. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
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