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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2pthfrgra 28401* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1 of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  ( V FriendGrph  E  ->  A. a  e.  V  A. b  e.  ( V  \  {
 a } ) E. f E. p ( f ( a ( V PathOn  E ) b ) p  /\  ( # `  f )  =  2 ) )
 
Theorem3cyclfrgrarn1 28402* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
 |-  (
 ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V )  /\  A  =/=  C ) 
 ->  E. b  e.  V  E. c  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  {
 c ,  A }  e.  ran  E ) )
 
Theorem3cyclfrgrarn 28403* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
 |-  (
 ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
 
Theorem3cyclfrgrarn2 28404* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( b  =/=  c  /\  ( {
 a ,  b }  e.  ran  E  /\  {
 b ,  c }  e.  ran  E  /\  {
 c ,  a }  e.  ran  E ) ) )
 
Theorem3cyclfrgra 28405* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  (
 ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. v  e.  V  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3  /\  ( p `  0 )  =  v ) )
 
Theorem4cycl2v2nb 28406 In a (maybe degenerated) 4-cycle, two vertices have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  (
 ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 /\  ( { C ,  D }  e.  ran  E 
 /\  { D ,  A }  e.  ran  E ) )  ->  ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran 
 E ) )
 
Theorem4cycl2vnunb 28407* In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  (
 ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 /\  ( { C ,  D }  e.  ran  E 
 /\  { D ,  A }  e.  ran  E ) 
 /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) )  ->  -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E )
 
Theoremn4cyclfrgra 28408 There is no 4-cycle in a friendship graph, see Proposition 1 of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  (
 ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `  F )  =/=  4 )
 
Theorem4cyclusnfrgra 28409 A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) ) 
 ->  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) 
 ->  -.  V FriendGrph  E ) )
 
Theoremfrgranbnb 28410 If two neighbors of a specific vertex have a common neighbor in a friendship graph, then this common neighbor must be the specific vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( ph  ->  X  e.  V )   &    |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  ( ph  ->  V FriendGrph  E )   =>    |-  ( ( ph  /\  ( U  e.  D  /\  W  e.  D )  /\  U  =/=  W ) 
 ->  ( ( { U ,  A }  e.  ran  E 
 /\  { W ,  A }  e.  ran  E ) 
 ->  A  =  X ) )
 
Theoremfrconngra 28411 A friendship graph is connected, see 1. remark after Proposition 1 of [MertziosUnger] p. 153 : "An arbitrary friendship graph has to be connected, ... ". (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  ( V FriendGrph  E  ->  V ConnGrph  E )
 
Theoremvdfrgra0 28412 A vertex in a friendship graph has degree 0 if the graph consists of only one vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  N  e.  V  /\  ( # `  V )  =  1 )  ->  ( ( V VDeg  E ) `  N )  =  0 )
 
Theoremvdgfrgra0 28413 A vertex in a friendship graph has degree 0 if the graph consists of only one vertex. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  N  e.  V  /\  ( # `  V )  =  1 )  ->  ( ( V VDeg  E ) `  N )  =  0 )
 
Theoremvdn0frgrav2 28414 A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  E  e.  Fin  /\  N  e.  V )  ->  ( 1  <  ( # `  V )  ->  ( ( V VDeg 
 E ) `  N )  =/=  0 ) )
 
Theoremvdgn0frgrav2 28415 A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  N  e.  V )  ->  (
 1  <  ( # `  V )  ->  ( ( V VDeg 
 E ) `  N )  =/=  0 ) )
 
Theoremvdn1frgrav2 28416 Any vertex in a friendship graph does not have degree 1, see 2. remark after Proposition 1 of [MertziosUnger] p. 153 : "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  E  e.  Fin  /\  N  e.  V )  ->  ( 1  <  ( # `  V )  ->  ( ( V VDeg 
 E ) `  N )  =/=  1 ) )
 
Theoremvdgn1frgrav2 28417 Any vertex in a friendship graph does not have degree 1, see 2. remark after Proposition 1 of [MertziosUnger] p. 153 : "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  N  e.  V )  ->  (
 1  <  ( # `  V )  ->  ( ( V VDeg 
 E ) `  N )  =/=  1 ) )
 
Theoremvdgfrgragt2 28418 Any vertex in a friendship graph (with more than one vertex - then, actually, the graph must have at least three vertices, because otherwise, it would not be a friendship graph) has at least degree 2, see 3. remark after Proposition 1 of [MertziosUnger] p. 153 : "It follows that deg(v) >= 2 for every node v of a friendship graph". (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  N  e.  V )  ->  (
 1  <  ( # `  V )  ->  2  <_  (
 ( V VDeg  E ) `  N ) ) )
 
Theoremfrgrancvvdeqlem1 28419* Lemma 1 for frgrancvvdeq 28431. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  Y  e.  ( V  \  { x } ) )
 
Theoremfrgrancvvdeqlem2 28420* Lemma 2 for frgrancvvdeq 28431. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  X  e/  N )
 
Theoremfrgrancvvdeqlem3 28421* Lemma 3 for frgrancvvdeq 28431. In a friendship graph, for each neighbor of a vertex there is exacly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  E! y  e.  N  { x ,  y }  e.  ran  E )
 
Theoremfrgrancvvdeqlem4 28422* Lemma 4 for frgrancvvdeq 28431. The restricted iota of a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  { ( iota_
 y  e.  N { x ,  y }  e.  ran  E ) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N ) )
 
Theoremfrgrancvvdeqlem5 28423* Lemma 5 for frgrancvvdeq 28431. The mapping of neighbors to neighbors is a function. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A : D
 --> N )
 
Theoremfrgrancvvdeqlem6 28424* Lemma 6 for frgrancvvdeq 28431. The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  { ( A `  x ) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N ) )
 
Theoremfrgrancvvdeqlem7 28425* Lemma 7 for frgrancvvdeq 28431. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  { x ,  ( A `  x ) }  e.  ran  E )
 
TheoremfrgrancvvdeqlemA 28426* Lemma A for frgrancvvdeq 28431. This corresponds to the following observation in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A. x  e.  D  ( A `  x )  =/=  X )
 
TheoremfrgrancvvdeqlemB 28427* Lemma B for frgrancvvdeq 28431. This corresponds to the following observation in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A : D -1-1-> ran  A )
 
TheoremfrgrancvvdeqlemC 28428* Lemma C for frgrancvvdeq 28431. This corresponds to the following observation in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A : D -onto-> N )
 
Theoremfrgrancvvdeqlem8 28429* Lemma 8 for frgrancvvdeq 28431. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  (
 <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A : D
 -1-1-onto-> N )
 
Theoremfrgrancvvdeqlem9 28430* Lemma 9 for frgrancvvdeq 28431. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
 |-  ( V FriendGrph  E  ->  A. x  e.  V  A. y  e.  ( V  \  { x } ) ( y 
 e/  ( <. V ,  E >. Neighbors  x )  ->  E. f  f : ( <. V ,  E >. Neighbors  x ) -1-1-onto-> ( <. V ,  E >. Neighbors  y ) ) )
 
Theoremfrgrancvvdeq 28431* In a finite friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to the first Lemma ("claim") of the proof of the (friendship) theorem in [Huneke] p. 1: "If x,y are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  (
 ( V FriendGrph  E  /\  E  e.  Fin )  ->  A. x  e.  V  A. y  e.  ( V  \  { x } ) ( y 
 e/  ( <. V ,  E >. Neighbors  x )  ->  (
 ( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
  y ) ) )
 
Theoremfrgrancvvdgeq 28432* In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to the first Lemma ("claim") of the proof of the (friendship) theorem in [Huneke] p. 1: "If x,y, are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( V FriendGrph  E  ->  A. x  e.  V  A. y  e.  ( V  \  { x } ) ( y 
 e/  ( <. V ,  E >. Neighbors  x )  ->  (
 ( V VDeg  E ) `  x )  =  ( ( V VDeg  E ) `
  y ) ) )
 
Theoremfrgrawopreglem1 28433* Lemma 1 for frgrawopreg 28438. In a friendship graph, the classes A and B are sets. (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  e.  _V  /\  B  e.  _V ) )
 
Theoremfrgrawopreglem2 28434* Lemma 2 for frgrawopreg 28438. In a friendship graph with at least 2 vertices, the degree of a vertex must be at least 2. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V  \  A )   =>    |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V )  /\  A  =/=  (/) )  -> 
 1  <  K )
 
Theoremfrgrawopreglem3 28435* Lemma 3 for frgrawopreg 28438. The vertices in the sets A and B have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V  \  A )   =>    |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( ( V VDeg 
 E ) `  X )  =/=  ( ( V VDeg 
 E ) `  Y ) )
 
Theoremfrgrawopreglem4 28436* Lemma 4 for frgrawopreg 28438. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to the observation in the proof of [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B." (Contributed by Alexander van der Vekens, 30-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V  \  A )   =>    |-  ( V FriendGrph  E  ->  A. a  e.  A  A. b  e.  B  { a ,  b }  e.  ran  E )
 
Theoremfrgrawopreglem5 28437* Lemma 5 for frgrawopreg 28438. If A as well as B contain at least two vertices in a friendship graph, there is a 4-cycle in the graph. This corresponds to the observation in the proof of [Huneke] p. 2: "... otherwise, there are two different vertices in A, and they have two common neighbors in B, ...". (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V  \  A )   =>    |-  ( ( V FriendGrph  E  /\  1  <  ( # `  A )  /\  1  <  ( # `
  B ) ) 
 ->  E. a  e.  A  E. x  e.  A  E. b  e.  B  E. y  e.  B  ( ( b  =/=  y  /\  a  =/= 
 x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )
 
Theoremfrgrawopreg 28438* 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 28439* 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 28440* 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 28441* 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 28442* 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 ) ) )
 
Theoremfrgraeu 28443* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  ->  E! b ( { A ,  b }  e.  ran  E 
 /\  { b ,  C }  e.  ran  E ) ) )
 
Theoremfrg2woteu 28444* For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices as ordered triple. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  (
 ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B ) 
 ->  E! c  e.  V  <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B ) )
 
Theoremfrg2wotn0 28445 In a friendship graph, there is always a path/walk of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  (
 ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B ) 
 ->  ( A ( V 2WalksOnOt  E ) B )  =/=  (/) )
 
Theoremfrg2wot1 28446 In a friendship graph, there is exactly one walk of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
 |-  (
 ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B ) 
 ->  ( # `  ( A ( V 2WalksOnOt  E ) B ) )  =  1 )
 
Theoremfrg2spot1 28447 In a friendship graph, there is exactly one simple path of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
 |-  (
 ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B ) 
 ->  ( # `  ( A ( V 2SPathOnOt  E ) B ) )  =  1 )
 
Theoremfrg2woteqm 28448 There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 20-Feb-2018.)
 |-  (
 ( V FriendGrph  E  /\  A  =/=  B )  ->  (
 ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  Q  =  P ) )
 
Theoremfrg2woteq 28449 There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 14-Feb-2018.)
 |-  (
 ( V FriendGrph  E  /\  A  =/=  B )  ->  (
 ( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) ) 
 /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P ) ) ) )
 
Theorem2spotdisj 28450* All simple paths of length 2 as ordered triple from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 4-Mar-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  -> Disj  b  e.  ( V 
 \  { A }
 ) ( A ( V 2SPathOnOt  E ) b ) )
 
Theorem2spotiundisj 28451* All simple paths of length 2 as ordered triple from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 -> Disj  a  e.  V U_ b  e.  ( V  \  { a } )
 ( a ( V 2SPathOnOt  E ) b ) )
 
Theoremfrghash2spot 28452 The number of simple paths of length 2 is n*(n-1) in a friendship graph with  n vertices. This corresponds to the proof of the third claim in the proof of the friendship theorem in [Huneke] p. 2: "... the paths of length two in G: by assumption there are ( n 2 ) such paths.". However, the order of vertices is not respected by Huneke, so he only counts half of the paths which are existing when respecting the order as it is the case for simple paths represented by orderes triples. (Contributed by Alexander van der Vekens, 6-Mar-2018.)
 |-  (
 ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/= 
 (/) ) )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V )  x.  ( ( # `  V )  -  1
 ) ) )
 
Theorem2spot0 28453 If there are no vertices, then there are no paths (of length 2), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  (
 ( V  =  (/)  /\  E  e.  X ) 
 ->  ( V 2SPathOnOt  E )  =  (/) )
 
Theoremusg2spot2nb 28454* The set of paths of length 2 with a given vertex in the middle for a finite graph is the union of all paths of length 2 from one neighbor to another neighbor of this vertex via this vertex. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
 |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
 ) )  =  a ) } )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V ) 
 ->  ( M `  N )  =  U_ x  e.  ( <. V ,  E >. Neighbors  N ) U_ y  e.  ( ( <. V ,  E >. Neighbors  N )  \  { x } ) { <. x ,  N ,  y >. } )
 
Theoremusgreghash2spotv 28455* According to the proof of the third claim in the proof of the friendship theorem in [Huneke] p. 2: "For each vertex v, there are exactly ( k 2 ) paths with length two having v in the middle, ..." in a finite k-regular graph. For simple paths of length 2 represented by ordered triples, we have again k*(k-1) such paths. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
 ) )  =  a ) } )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin )  ->  A. v  e.  V  ( ( ( V VDeg 
 E ) `  v
 )  =  K  ->  ( # `  ( M `  v ) )  =  ( K  x.  ( K  -  1 ) ) ) )
 
Theoremusgreg2spot 28456* In a finite k-regular graph the set of all paths of length two is the union of all the paths of length 2 over the vertices which are in the the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
 ) )  =  a ) } )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin )  ->  ( A. v  e.  V  ( ( V VDeg 
 E ) `  v
 )  =  K  ->  ( V 2SPathOnOt  E )  =  U_ x  e.  V  ( M `  x ) ) )
 
Theorem2spotmdisj 28457* The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
 ) )  =  a ) } )   =>    |-  ( V  e.  _V 
 -> Disj 
 x  e.  V ( M `  x ) )
 
Theoremusgreghash2spot 28458* In a finite k-regular graph with N vertices there are N times " k choose 2 " paths with length 2, according to the proof of the third claim in the proof of the friendship theorem in [Huneke] p. 2: "... giving n * ( k 2 ) total paths of length two.", if the direction of traversing the path is not respected. For simple paths of length 2 represented by ordered triples, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  V  e.  Fin  /\  V  =/= 
 (/) )  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v )  =  K  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V )  x.  ( K  x.  ( K  -  1
 ) ) ) ) )
 
Theoremfrgregordn0 28459* If a nonempty friendship graph is k-regular, its order is k(k-1)+1. This corresponds to the third claim in the proof of the friendship theorem in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  (
 ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
 ( V VDeg  E ) `  v )  =  K  ->  ( # `  V )  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
 
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 28460 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1762. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbidd 28461 An identity theorem for substitution. See sbid 1947. 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 28462 An identity theorem for substitution. See sbid 1947. 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 28463 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 28465.
 class  >_
 
Syntaxcgt 28464 Extend wff notation to include the 'greater than' relation, see df-gt 28466.
 class  >
 
Definitiondf-gte 28465 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 9126.

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 28466), 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 28467 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 28466 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 9003. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 28465 for a discussion on why this approach is used for the definition. See gt-lt 28468 and gt-lth 28470 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 28467 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 28468 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 28469 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 28470 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 28471 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 28472 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 28473 Extend class notation to include the hyperbolic sine function, see df-sinh 28476.
 class sinh
 
Syntaxccosh 28474 Extend class notation to include the hyperbolic cosine function. see df-cosh 28477.
 class cosh
 
Syntaxctanh 28475 Extend class notation to include the hyperbolic tangent function, see df-tanh 28478.
 class tanh
 
Definitiondf-sinh 28476 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4266, which requires the form  (
x  e.  A  |->  B ). See sinhval-named 28479 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 28482 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 28477 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4266, 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 28478 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4266, 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 28479 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 28476. See sinhval 12755 for a theorem to convert this further. See sinh-conventional 28482 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 28480 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 28477. See coshval 12756 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 28481 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 28478. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  ( `'cosh " ( CC  \  {
 0 } ) ) 
 ->  (tanh `  A )  =  ( ( tan `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremsinh-conventional 28482 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 28483 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 28484 Extend class notation to include the secant function, see df-sec 28487.
 class  sec
 
Syntaxccsc 28485 Extend class notation to include the cosecant function, see df-csc 28488.
 class  csc
 
Syntaxccot 28486 Extend class notation to include the cotangent function, see df-cot 28489.
 class  cot
 
Definitiondf-sec 28487* Define the secant function. We define it this way for cmpt 4266, 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 28488* Define the cosecant function. We define it this way for cmpt 4266, 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 28489* Define the cotangent function. We define it this way for cmpt 4266, 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 28490 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 28491 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 28492 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 28493 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 28494 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 28495 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 28496 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 28497 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 28498 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 28499 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 28500 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 ) ) )
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