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Theorem List for Metamath Proof Explorer - 38901-39000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Axiomax-hgprmladder 38901 There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.)
 |-  E. d  e.  ( ZZ>= `  3 ) E. f  e.  (RePart `  d ) ( ( ( f `  0
 )  =  7  /\  ( f `  1
 )  = ; 1 3  /\  (
 f `  d )  =  (; 8 9  x.  ( 10 ^; 2 9 ) ) )  /\  A. i  e.  ( 0..^ d ) ( ( f `  i )  e.  ( Prime  \  { 2 } )  /\  ( ( f `  ( i  +  1 ) )  -  ( f `  i ) )  < 
 ( ( 4  x.  ( 10 ^; 1 8 ) )  -  4 )  /\  4  <  ( ( f `
  ( i  +  1 ) )  -  ( f `  i
 ) ) ) )
 
Theoremtgblthelfgott 38902 The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 38900, ax-hgprmladder 38901 and bgoldbtbnd 38898. (Contributed by AV, 4-Aug-2020.)
 |-  (
 ( N  e. Odd  /\  7  <  N  /\  N  <  (; 8 8  x.  ( 10 ^; 2 9 ) ) )  ->  N  e. GoldbachOddALTV  )
 
Theoremtgoldbachlt 38903* The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big  m greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 38902. (Contributed by AV, 4-Aug-2020.)
 |-  E. m  e.  NN  ( ( 8  x.  ( 10 ^; 3 0 ) )  <  m  /\  A. n  e. Odd  (
 ( 7  <  n  /\  n  <  m ) 
 ->  n  e. GoldbachOddALTV  ) )
 
Axiomax-tgoldbachgt 38904* The ternary Goldbach conjecture is valid for big odd numbers (i.e. for all odd numbers greater than a fixed  m). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for m = 10^27. Temporarily provided as "axiom". (Contributed by AV, 2-Aug-2020.)
 |-  E. m  e.  NN  ( m  <_  ( 10 ^; 2 7 )  /\  A. n  e. Odd  ( m  <  n  ->  n  e. GoldbachOddALTV  ) )
 
Theoremtgoldbach 38905 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 38903 and ax-tgoldbachgt 38904. (Contributed by AV, 2-Aug-2020.)
 |-  A. n  e. Odd  ( 7  <  n  ->  n  e. GoldbachOddALTV  )
 
21.33.5  Proth's theorem
 
Theoremmodexp2m1d 38906 The square of an integer which is -1 modulo a number greater than 1 is 1 modulo the same modulus. (Contributed by AV, 5-Jul-2020.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  1  <  E )   &    |-  ( ph  ->  ( A  mod  E )  =  ( -u 1  mod  E ) )   =>    |-  ( ph  ->  ( ( A ^ 2 )  mod  E )  =  1 )
 
Theoremproththdlem 38907 Lemma for proththd 38908. (Contributed by AV, 4-Jul-2020.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  P  =  ( ( K  x.  ( 2 ^ N ) )  +  1 ) )   =>    |-  ( ph  ->  ( P  e.  NN  /\  1  <  P  /\  (
 ( P  -  1
 )  /  2 )  e.  NN ) )
 
Theoremproththd 38908* Proth's theorem (1878). If P is a Proth number, i.e. a number of the form k2^n+1 with k less than 2^n, and if there exists an integer x for which x^((P-1)/2) is -1 modulo P, then P is prime. Such a prime is called a Proth prime. Like Pocklington's theorem (see pockthg 14843), Proth's theorem allows for a convenient method for verifying large primes. (Contributed by AV, 5-Jul-2020.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  P  =  ( ( K  x.  ( 2 ^ N ) )  +  1 ) )   &    |-  ( ph  ->  K  <  (
 2 ^ N ) )   &    |-  ( ph  ->  E. x  e.  ZZ  (
 ( x ^ (
 ( P  -  1
 )  /  2 )
 )  mod  P )  =  ( -u 1  mod  P ) )   =>    |-  ( ph  ->  P  e.  Prime )
 
Theorem5tcu2e40 38909 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
 |-  (
 5  x.  ( 2 ^ 3 ) )  = ; 4 0
 
Theorem3exp4mod41 38910 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
 |-  (
 ( 3 ^ 4
 )  mod ; 4 1 )  =  ( -u 1  mod ; 4 1 )
 
Theorem41prothprmlem1 38911 Lemma 1 for 41prothprm 38913. (Contributed by AV, 4-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( ( P  -  1 )  /  2
 )  = ; 2 0
 
Theorem41prothprmlem2 38912 Lemma 2 for 41prothprm 38913. (Contributed by AV, 5-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( ( 3 ^
 ( ( P  -  1 )  /  2
 ) )  mod  P )  =  ( -u 1  mod  P )
 
Theorem41prothprm 38913 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
 |-  P  = ; 4 1   =>    |-  ( P  =  ( ( 5  x.  (
 2 ^ 3 ) )  +  1 ) 
 /\  P  e.  Prime )
 
21.33.6  Words over a set (extension)
 
21.33.6.1  Last symbol of a word (extension)
 
Theoremlswn0 38914 The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( (/) is the last symbol) and invalid cases ( (/) means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 ( W  e. Word  V  /\  (/)  e/  V  /\  ( # `  W )  =/=  0 )  ->  ( lastS  `  W )  =/=  (/) )
 
21.33.6.2  Concatenations with singleton words (extension)
 
Theoremccatw2s1cl 38915 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  (
 ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( W ++  <" X "> ) ++  <" Y "> )  e. Word  V )
 
21.33.6.3  Prefixes of a word

In https://www.allacronyms.com/prefix/abbreviated, "pfx" is proposed as abbreviation for "prefix". Regarding the meaning of "prefix", it is different in computer science (automata theory/formal languages) compared with linguistics: in linguistics, a prefix has a meaning (see Wikipedia "Prefix" https://en.wikipedia.org/wiki/Prefix), whereas in computer science, a prefix is an arbitrary substring/subword starting at the beginning of a string/word (see Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix), or https://math.stackexchange.com/questions/2190559/ is-there-standard-terminology-notation-for-the-prefix-of-a-word ).

 
Syntaxcpfx 38916 Syntax for the prefix operator.
 class prefix
 
Definitiondf-pfx 38917* Define an operation which extracts prefixes of words, i.e. subwords starting at the beginning of a word. Definition in section 9.1 of [AhoHopUll] p. 318. "pfx" is used as label fragment. (Contributed by AV, 2-May-2020.)
 |- prefix  =  ( s  e.  _V ,  l  e.  NN0  |->  ( s substr  <. 0 ,  l >. ) )
 
Theorempfxval 38918 Value of a prefix. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e.  V  /\  L  e.  NN0 )  ->  ( S prefix  L )  =  ( S substr  <. 0 ,  L >. ) )
 
Theorempfx00 38919 A zero length prefix. (Contributed by AV, 2-May-2020.)
 |-  ( S prefix  0 )  =  (/)
 
Theorempfx0 38920 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
 |-  ( (/) prefix  L )  =  (/)
 
Theorempfxcl 38921 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
 |-  ( S  e. Word  A  ->  ( S prefix  L )  e. Word  A )
 
Theorempfxmpt 38922* Value of the prefix extractor as mapping. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  =  ( x  e.  (
 0..^ L )  |->  ( S `  x ) ) )
 
Theorempfxres 38923 Value of the prefix extractor as restriction. Could replace swrd0val 12772. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  =  ( S  |`  ( 0..^ L ) ) )
 
Theorempfxf 38924 A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. Could replace swrd0f 12778. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) ) )  ->  ( W prefix  L ) : ( 0..^ L ) --> V )
 
Theorempfxfn 38925 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  V  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( S prefix  L )  Fn  ( 0..^ L ) )
 
Theorempfxlen 38926 Length of a prefix. Could replace swrd0len 12773. (Contributed by AV, 2-May-2020.)
 |-  (
 ( S  e. Word  A  /\  L  e.  ( 0
 ... ( # `  S ) ) )  ->  ( # `  ( S prefix  L ) )  =  L )
 
Theorempfxid 38927 A word is a prefix of itself. (Contributed by AV, 2-May-2020.)
 |-  ( S  e. Word  A  ->  ( S prefix  ( # `  S ) )  =  S )
 
Theorempfxrn 38928 The range of a prefix of a word is a subset of the set of symbols for the word. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) ) )  ->  ran  ( W prefix  L )  C_  V )
 
Theorempfxn0 38929 A prefix consisting of at least one symbol is not empty. Could replace swrdn0 12781. (Contributed by AV, 2-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN  /\  L  <_  ( # `  W ) )  ->  ( W prefix  L )  =/=  (/) )
 
Theorempfxnd 38930 The value of the prefix extractor is the empty set (undefined) if the argument is not within the range of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN0  /\  ( # `
  W )  <  L )  ->  ( W prefix  L )  =  (/) )
 
Theorempfxlen0 38931 Length of a prefix of a word reduced by a single symbol. Could replace swrd0len0 12787. TODO-AV: Really useful? swrd0len0 12787 is only used in wwlknred 25444. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  NN0  /\  ( # `
  W )  =  ( L  +  1 ) )  ->  ( # `
  ( W prefix  L ) )  =  L )
 
Theoremaddlenrevpfx 38932 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( # `  ( W substr 
 <. M ,  ( # `  W ) >. ) )  +  ( # `  ( W prefix  M ) ) )  =  ( # `  W ) )
 
Theoremaddlenpfx 38933 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( # `  ( W prefix  M ) )  +  ( # `  ( W substr  <. M ,  ( # `  W ) >. ) ) )  =  ( # `  W ) )
 
Theorempfxfv 38934 A symbol in a prefix of a word, indexed using the prefix' indices. Could replace swrd0fv 12790. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 0
 ... ( # `  W ) )  /\  I  e.  ( 0..^ L ) )  ->  ( ( W prefix  L ) `  I
 )  =  ( W `
  I ) )
 
Theorempfxfv0 38935 The first symbol in a prefix of a word. Could replace swrd0fv0 12791. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 1
 ... ( # `  W ) ) )  ->  ( ( W prefix  L ) `  0 )  =  ( W `  0
 ) )
 
Theorempfxtrcfv 38936 A symbol in a word truncated by one symbol. Could replace swrdtrcfv 12792. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/)  /\  I  e.  ( 0..^ ( ( # `  W )  -  1 ) ) ) 
 ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) `
  I )  =  ( W `  I
 ) )
 
Theorempfxtrcfv0 38937 The first symbol in a word truncated by one symbol. Could replace swrdtrcfv0 12793. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) `
  0 )  =  ( W `  0
 ) )
 
Theorempfxfvlsw 38938 The last symbol in a (not empty) prefix of a word. Could replace swrd0fvlsw 12794. (Contributed by AV, 3-May-2020.)
 |-  (
 ( W  e. Word  V  /\  L  e.  ( 1
 ... ( # `  W ) ) )  ->  ( lastS  `  ( W prefix  L ) )  =  ( W `  ( L  -  1 ) ) )
 
Theorempfxeq 38939* The prefixes of two words are equal iff they have the same length and the same symbols at each position. Could replace swrdeq 12795. (Contributed by AV, 4-May-2020.)
 |-  (
 ( ( W  e. Word  V 
 /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `
  W )  /\  N  <_  ( # `  U ) ) )  ->  ( ( W prefix  M )  =  ( U prefix  N )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( W `  i
 )  =  ( U `
  i ) ) ) )
 
Theorempfxtrcfvl 38940 The last symbol in a word truncated by one symbol. Could replace swrdtrcfvl 12801. (Contributed by AV, 5-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( lastS  `  ( W prefix  (
 ( # `  W )  -  1 ) ) )  =  ( W `
  ( ( # `  W )  -  2
 ) ) )
 
Theorempfxsuffeqwrdeq 38941 Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 12804. (Contributed by AV, 5-May-2020.)
 |-  (
 ( W  e. Word  V  /\  S  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  =  S  <->  ( ( # `  W )  =  ( # `  S )  /\  ( ( W prefix  I
 )  =  ( S prefix  I )  /\  ( W substr  <. I ,  ( # `  W ) >. )  =  ( S substr  <. I ,  ( # `  W )
 >. ) ) ) ) )
 
Theorempfxsuff1eqwrdeq 38942 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. Could replace 2swrd1eqwrdeq 12805. (Contributed by AV, 6-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  ( # `  U )  /\  ( ( W prefix  (
 ( # `  W )  -  1 ) )  =  ( U prefix  (
 ( # `  W )  -  1 ) ) 
 /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) ) )
 
Theoremdisjwrdpfx 38943* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. Could replace disjxwrd 12806. (Contributed by AV, 6-May-2020.)
 |- Disj  y  e.  W  { x  e. Word  V  |  ( x prefix  N )  =  y }
 
Theoremccatpfx 38944 Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
 |-  (
 ( S  e. Word  A  /\  Y  e.  ( 0
 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) )  ->  ( ( S prefix  Y ) ++  ( S substr  <. Y ,  Z >. ) )  =  ( S prefix  Z )
 )
 
Theorempfxccat1 38945 Recover the left half of a concatenated word. Could replace swrdccat1 12808. (Contributed by AV, 6-May-2020.)
 |-  (
 ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S ++  T ) prefix  ( # `  S ) )  =  S )
 
Theorempfx1 38946 A prefix of length 1. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( W prefix  1 )  =  <" ( W `
  0 ) "> )
 
Theorempfx2 38947 A prefix of length 2. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  V  /\  2  <_  ( # `  W ) )  ->  ( W prefix  2 )  =  <" ( W `
  0 ) ( W `  1 ) "> )
 
Theorempfxswrd 38948 A prefix of a subword. Could replace swrd0swrd 12812. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  M  e.  ( 0 ... N ) )  ->  ( L  e.  ( 0 ... ( N  -  M ) )  ->  ( ( W substr  <. M ,  N >. ) prefix  L )  =  ( W substr  <. M ,  ( M  +  L ) >. ) ) )
 
Theoremswrdpfx 38949 A subword of a prefix. Could replace swrdswrd0 12813. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) ) 
 ->  ( ( W prefix  N ) substr 
 <. K ,  L >. )  =  ( W substr  <. K ,  L >. ) ) )
 
Theorempfxpfx 38950 A prefix of a prefix. Could replace swrd0swrd0 12814. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  L  e.  ( 0 ... N ) )  ->  ( ( W prefix  N ) prefix  L )  =  ( W prefix  L ) )
 
Theorempfxpfxid 38951 A prefix of a prefix with the same length is the prefix. Could replace swrd0swrdid 12815. (Contributed by AV, 8-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( W prefix  N ) prefix  N )  =  ( W prefix  N ) )
 
Theorempfxcctswrd 38952 The concatenation of the prefix of a word and the rest of the word yields the word itself. Could replace wrdcctswrd 12816. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( W prefix  M ) ++  ( W substr  <. M ,  ( # `  W )
 >. ) )  =  W )
 
Theoremlenpfxcctswrd 38953 The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. Could replace lencctswrd 12817. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( # `  ( ( W prefix  M ) ++  ( W substr  <. M ,  ( # `  W ) >. ) ) )  =  ( # `  W ) )
 
Theoremlenrevpfxcctswrd 38954 The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. Could replace lenrevcctswrd 12818. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( # `  ( ( W substr  <. M ,  ( # `
  W ) >. ) ++  ( W prefix  M )
 ) )  =  ( # `  W ) )
 
Theorempfxlswccat 38955 Reconstruct a nonempty word from its prefix and last symbol. Could replace wrdeqcats1OLD 12825 resp. swrdccatwrd 12819. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( ( W prefix  (
 ( # `  W )  -  1 ) ) ++ 
 <" ( lastS  `  W ) "> )  =  W )
 
Theoremccats1pfxeq 38956 The last symbol of a word concatenated with the word with the last symbol removed having results in the word itself. Could replace ccats1swrdeq 12820. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) ) 
 ->  U  =  ( W ++ 
 <" ( lastS  `  U ) "> ) ) )
 
Theoremccats1pfxeqrex 38957* There exists a symbol such that its concatenation with the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. Could replace ccats1swrdeqrex 12830. (Contributed by AV, 9-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) ) 
 ->  E. s  e.  V  U  =  ( W ++  <" s "> ) ) )
 
Theorempfxccatin12lem1 38958 Lemma 1 for pfxccatin12 38960. Could replace swrdccatin12lem2b 12837. (Contributed by AV, 9-May-2020.)
 |-  (
 ( M  e.  (
 0 ... L )  /\  N  e.  ( L ... X ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( K  -  ( L  -  M ) )  e.  ( 0..^ ( N  -  L ) ) ) )
 
Theorempfxccatin12lem2 38959 Lemma 2 for pfxccatin12 38960. Could replace swrdccatin12lem2 12840. (Contributed by AV, 9-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ...
 L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( (
 ( A ++  B ) substr  <. M ,  N >. ) `
  K )  =  ( ( B prefix  ( N  -  L ) ) `
  ( K  -  ( # `  ( A substr  <. M ,  L >. ) ) ) ) ) )
 
Theorempfxccatin12 38960 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 12842. (Contributed by AV, 9-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ...
 L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A ++  B ) substr 
 <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) ++  ( B prefix  ( N  -  L ) ) ) ) )
 
Theorempfxccat3 38961 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 12843. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ...
 N )  /\  N  e.  ( 0 ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A ++  B ) substr 
 <. M ,  N >. )  =  if ( N 
 <_  L ,  ( A substr  <. M ,  N >. ) ,  if ( L 
 <_  M ,  ( B substr  <. ( M  -  L ) ,  ( N  -  L ) >. ) ,  ( ( A substr  <. M ,  L >. ) ++  ( B prefix 
 ( N  -  L ) ) ) ) ) ) )
 
Theorempfxccatpfx1 38962 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V  /\  N  e.  ( 0
 ... L ) ) 
 ->  ( ( A ++  B ) prefix  N )  =  ( A prefix  N ) )
 
Theorempfxccatpfx2 38963 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   &    |-  M  =  ( # `  B )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V  /\  N  e.  ( ( L  +  1 )
 ... ( L  +  M ) ) ) 
 ->  ( ( A ++  B ) prefix  N )  =  ( A ++  ( B prefix  ( N  -  L ) ) ) )
 
Theorempfxccat3a 38964 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. Could replace swrdccat3a 12845. (Contributed by AV, 10-May-2020.)
 |-  L  =  ( # `  A )   &    |-  M  =  ( # `  B )   =>    |-  ( ( A  e. Word  V 
 /\  B  e. Word  V )  ->  ( N  e.  ( 0 ... ( L  +  M )
 )  ->  ( ( A ++  B ) prefix  N )  =  if ( N 
 <_  L ,  ( A prefix  N ) ,  ( A ++  ( B prefix  ( N  -  L ) ) ) ) ) )
 
Theorempfxccatid 38965 A prefix of a concatenation of length of the first concatenated word is the first word itself. Could replace swrdccatid 12848. (Contributed by AV, 10-May-2020.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V  /\  N  =  ( # `  A ) )  ->  ( ( A ++  B ) prefix  N )  =  A )
 
Theoremccats1pfxeqbi 38966 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. Could replace ccats1swrdeqbi 12849. (Contributed by AV, 10-May-2020.)
 |-  (
 ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  ->  ( W  =  ( U prefix  ( # `  W ) )  <->  U  =  ( W ++  <" ( lastS  `  U ) "> ) ) )
 
Theorempfxccatin12d 38967 The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12d 12852. (Contributed by AV, 10-May-2020.)
 |-  ( ph  ->  ( # `  A )  =  L )   &    |-  ( ph  ->  ( A  e. Word  V 
 /\  B  e. Word  V ) )   &    |-  ( ph  ->  M  e.  ( 0 ...
 L ) )   &    |-  ( ph  ->  N  e.  ( L ... ( L  +  ( # `  B ) ) ) )   =>    |-  ( ph  ->  ( ( A ++  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) ++  ( B prefix  ( N  -  L ) ) ) )
 
Theoremreuccatpfxs1lem 38968* Lemma for reuccatpfxs1 38969. Could replace reuccats1lem 12831. (Contributed by AV, 9-May-2020.)
 |-  (
 ( ( W  e. Word  V 
 /\  U  e.  X )  /\  A. s  e.  V  ( ( W ++ 
 <" s "> )  e.  X  ->  S  =  s )  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  ->  ( W  =  ( U prefix  ( # `  W ) )  ->  U  =  ( W ++  <" S "> ) ) )
 
Theoremreuccatpfxs1 38969* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 12832. (Contributed by AV, 10-May-2020.)
 |-  (
 ( W  e. Word  V  /\  A. x  e.  X  ( x  e. Word  V  /\  ( # `  x )  =  ( ( # `  W )  +  1 ) ) )  ->  ( E! v  e.  V  ( W ++  <" v "> )  e.  X  ->  E! w  e.  X  W  =  ( w prefix  ( # `  W ) ) ) )
 
Theoremsplvalpfx 38970 Value of the substring replacement operator. (Contributed by AV, 11-May-2020.)
 |-  (
 ( S  e.  V  /\  ( F  e.  NN0  /\  T  e.  X  /\  R  e.  Y )
 )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S prefix  F ) ++  R ) ++  ( S substr  <. T ,  ( # `  S ) >. ) ) )
 
Theoremrepswpfx 38971 A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
 |-  (
 ( S  e.  V  /\  N  e.  NN0  /\  L  e.  ( 0 ... N ) )  ->  ( ( S repeatS  N ) prefix  L )  =  ( S repeatS  L ) )
 
Theoremcshword2 38972 Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ZZ )  ->  ( W cyclShift  N )  =  ( ( W substr  <. ( N 
 mod  ( # `  W ) ) ,  ( # `
  W ) >. ) ++  ( W prefix  ( N  mod  ( # `  W ) ) ) ) )
 
Theorempfxco 38973 Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
 |-  (
 ( W  e. Word  A  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  F : A
 --> B )  ->  ( F  o.  ( W prefix  N ) )  =  (
 ( F  o.  W ) prefix  N ) )
 
21.33.7  Auxiliary theorems for graph theory

Additional theorems for classical first-order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory.

 
21.33.7.1  Negated equality and membership - extension
 
Theoremelnelall 38974 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  e.  B  ->  ( A  e/  B  ->  ph ) )
 
21.33.7.2  Subclasses and subsets - extension
 
Theoremclel5 38975* Alternate definition of class membership: a class  X is an element of another class  A iff there is an element of  A equal to  X. (Contributed by AV, 13-Nov-2020.)
 |-  ( X  e.  A  <->  E. x  e.  A  X  =  x )
 
Theoremdfss7 38976* Alternate definition of subclass relationship: a class  A is a subclass of another class  B iff each element of  A is equal to an element of  B. (Contributed by AV, 13-Nov-2020.)
 |-  ( A  C_  B  <->  A. x  e.  A  E. y  e.  B  x  =  y )
 
Theoremsssseq 38977 If a class is a subclass of another class, the classes are equal iff the other class is a subclass of the first class. (Contributed by AV, 23-Dec-2020.)
 |-  ( B  C_  A  ->  ( A  C_  B  <->  A  =  B ) )
 
Theoremprcssprc 38978 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
 |-  (
 ( A  C_  B  /\  A  e/  _V )  ->  B  e/  _V )
 
21.33.7.3  The empty set - extension
 
Theoremralnralall 38979* A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
 |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
 /\  A. x  e.  A  -.  ph )  ->  ps )
 )
 
Theoremfalseral0 38980* A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)
 |-  (
 ( A. x  -.  ph  /\ 
 A. x  e.  A  ph )  ->  A  =  (/) )
 
Theoremralralimp 38981* Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)
 |-  (
 ( ph  /\  A  =/=  (/) )  ->  ( A. x  e.  A  (
 ( ph  ->  ( th  \/  ta ) )  /\  -. 
 th )  ->  ta )
 )
 
Theoremn0rex 38982* There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)
 |-  ( A  =/=  (/)  ->  E. x  e.  A  x  e.  A )
 
Theoremssn0rex 38983* There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)
 |-  (
 ( A  C_  B  /\  A  =/=  (/) )  ->  E. x  e.  B  x  e.  A )
 
21.33.7.4  Unordered and ordered pairs - extension
 
Theoremelpwdifsn 38984 A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.)
 |-  (
 ( S  e.  W  /\  S  C_  V  /\  A  e/  S )  ->  S  e.  ~P ( V  \  { A }
 ) )
 
Theorempr1eqbg 38985 A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =  C 
 <->  { A ,  B }  =  { B ,  C } ) )
 
Theorempr1nebg 38986 A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =/=  C  <->  { A ,  B }  =/=  { B ,  C } ) )
 
Theoremprelpw 38987 A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ( A  e.  C  /\  B  e.  C ) 
 <->  { A ,  B }  e.  ~P C ) )
 
Theoremrexdifpr 38988 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
 |-  ( E. x  e.  ( A  \  { B ,  C } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  x  =/=  C  /\  ph )
 )
 
Theoremissn 38989* A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)
 |-  ( E. x  e.  A  A. y  e.  A  x  =  y  ->  E. z  A  =  { z } )
 
Theoremn0snor2el 38990* A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.)
 |-  ( A  =/=  (/)  ->  ( E. x  e.  A  E. y  e.  A  x  =/=  y  \/  E. z  A  =  { z } )
 )
 
Theoremopidg 38991 The ordered pair  <. A ,  A >. in Kuratowski's representation. Closed form of opid 4184. (Contributed by AV, 18-Sep-2020.)
 |-  ( A  e.  _V  ->  <. A ,  A >.  =  { { A } } )
 
Theoremsnopeqop 38992 Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( { <. A ,  B >. }  =  <. C ,  D >. 
 <->  ( A  =  B  /\  C  =  D  /\  C  =  { A } ) )
 
Theorempropeqop 38993 Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   =>    |-  ( { <. A ,  B >. ,  <. C ,  D >. }  =  <. E ,  F >. 
 <->  ( ( A  =  C  /\  E  =  { A } )  /\  (
 ( A  =  B  /\  F  =  { A ,  D } )  \/  ( A  =  D  /\  F  =  { A ,  B } ) ) ) )
 
Theorempropssopi 38994 If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   =>    |-  ( { <. A ,  B >. ,  <. C ,  D >. }  C_  <. E ,  F >.  ->  A  =  C )
 
Theoremssprss 38995 A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( { A ,  B }  C_  { C ,  D }  <->  ( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D )
 ) ) )
 
Theoremssprsseq 38996 A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  ( { A ,  B }  C_  { C ,  D }  <->  { A ,  B }  =  { C ,  D } ) )
 
Theoremelpr2elpr 38997* For an element of an unordered pair which is a subset of a given set, there is another (maybe the same) element of the given set being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)
 |-  (
 ( X  e.  V  /\  Y  e.  V  /\  A  e.  { X ,  Y } )  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b } )
 
21.33.7.5  Indexed union and intersection - extension
 
Theoremiunxprg 38998* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A ,  B } C  =  ( D  u.  E ) )
 
Theoremiunopeqop 38999* Equivalence for an ordered pair equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.)
 |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  (/)  ->  ( U_ x  e.  A  { <. x ,  B >. }  =  <. C ,  D >.  ->  E. z  A  =  { z } )
 )
 
TheoremotiunsndisjX 39000* The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  ( B  e.  X  -> Disj  a  e.  V  U_ c  e.  W  { <. a ,  B ,  c >. } )
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40527
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