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Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrege105 36201 Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  Z  e.  V   =>    |-  ( ( -.  X ( t+ `  R ) Z  ->  Z  =  X )  ->  X ( ( t+ `  R )  u.  _I  ) Z )
 
Theoremfrege106 36202 Whatever follows  X in the  R-sequence belongs to the  R -sequence beginning with  X. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  Z  e.  V   =>    |-  ( X ( t+ `  R ) Z  ->  X (
 ( t+ `  R )  u.  _I  ) Z )
 
Theoremfrege107 36203 Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  V  e.  A   =>    |-  ( ( Z ( ( t+ `  R )  u.  _I  ) Y  ->  ( Y R V  ->  Z ( t+ `  R ) V ) )  ->  ( Z ( ( t+ `  R )  u.  _I  ) Y 
 ->  ( Y R V  ->  Z ( ( t+ `  R )  u.  _I  ) V ) ) )
 
Theoremfrege108 36204 If  Y belongs to the  R-sequence beginning with  Z, then every result of an application of the procedure  R to  Y belongs to the  R-sequence beginning with 
Z. Proposition 108 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  Z  e.  A   &    |-  Y  e.  B   &    |-  V  e.  C   &    |-  R  e.  D   =>    |-  ( Z ( ( t+ `  R )  u.  _I  ) Y 
 ->  ( Y R V  ->  Z ( ( t+ `  R )  u.  _I  ) V ) )
 
Theoremfrege109 36205 The property of belonging to the  R-sequence beginning with 
X is hereditary in the  R-sequence. Proposition 109 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  R  e.  V   =>    |-  R hereditary  ( ( ( t+ `
  R )  u. 
 _I  ) " { X } )
 
Theoremfrege110 36206* Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  A   &    |-  Y  e.  B   &    |-  M  e.  C   &    |-  R  e.  D   =>    |-  ( A. a ( Y R a  ->  X ( ( t+ `  R )  u.  _I  ) a )  ->  ( Y ( t+ `  R ) M  ->  X ( ( t+ `
  R )  u. 
 _I  ) M ) )
 
Theoremfrege111 36207 If  Y belongs to the  R-sequence beginning with  Z, then every result of an application of the procedure  R to  Y belongs to the  R-sequence beginning with 
Z or precedes  Z in the  R-sequence. Proposition 111 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Revised by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  Z  e.  A   &    |-  Y  e.  B   &    |-  V  e.  C   &    |-  R  e.  D   =>    |-  ( Z ( ( t+ `  R )  u.  _I  ) Y 
 ->  ( Y R V  ->  ( -.  V ( t+ `  R ) Z  ->  Z ( ( t+ `  R )  u.  _I  ) V ) ) )
 
Theoremfrege112 36208 Identity implies belonging to the 
R-sequence beginning with self. Proposition 112 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  Z  e.  V   =>    |-  ( Z  =  X  ->  X ( ( t+ `  R )  u.  _I  ) Z )
 
Theoremfrege113 36209 Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  Z  e.  V   =>    |-  ( ( Z ( ( t+ `  R )  u.  _I  ) X  ->  ( -.  Z ( t+ `  R ) X  ->  Z  =  X ) ) 
 ->  ( Z ( ( t+ `  R )  u.  _I  ) X 
 ->  ( -.  Z ( t+ `  R ) X  ->  X ( ( t+ `  R )  u.  _I  ) Z ) ) )
 
Theoremfrege114 36210 If  X belongs to the  R-sequence beginning with  Z, then  Z belongs to the  R-sequence beginning with  X or  X follows  Z in the  R-sequence. Proposition 114 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Z  e.  V   =>    |-  ( Z ( ( t+ `  R )  u.  _I  ) X 
 ->  ( -.  Z ( t+ `  R ) X  ->  X ( ( t+ `  R )  u.  _I  ) Z ) )
 
21.25.3.11  _Begriffsschrift_ Chapter III Single-valued procedures

 Fun  `' `' R means the relationship content of procedure  R is single-valued. The double converse allows us to simply apply this syntax in place of Frege's even though the original never explicitly limited discussion of propositional statments which vary on two variables to relations.

dffrege115 36211 through frege133 36229 develop this and how functions relate to transitive and transitive-reflexive closures.

 
Theoremdffrege115 36211* If from the the circumstance that  c is a result of an application of the procedure  R to  b, whatever  b may be, it can be inferred that every result of an application of the procedure  R to  b is the same as  c, then we say : "The procedure 
R is single-valued". Definition 115 of [Frege1879] p. 77. (Contributed by RP, 7-Jul-2020.)
 |-  ( A. c A. b ( b R c  ->  A. a ( b R a  ->  a  =  c ) )  <->  Fun  `' `' R )
 
Theoremfrege116 36212* One direction of dffrege115 36211. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   =>    |-  ( Fun  `' `' R  ->  A. b ( b R X  ->  A. a
 ( b R a 
 ->  a  =  X ) ) )
 
Theoremfrege117 36213* Lemma for frege118 36214. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   =>    |-  ( ( A. b
 ( b R X  ->  A. a ( b R a  ->  a  =  X ) )  ->  ( Y R X  ->  A. a ( Y R a  ->  a  =  X ) ) )  ->  ( Fun  `' `' R  ->  ( Y R X  ->  A. a ( Y R a  ->  a  =  X ) ) ) )
 
Theoremfrege118 36214* Simplified application of one direction of dffrege115 36211. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   =>    |-  ( Fun  `' `' R  ->  ( Y R X  ->  A. a
 ( Y R a 
 ->  a  =  X ) ) )
 
Theoremfrege119 36215* Lemma for frege120 36216. Proposition 119 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   =>    |-  (
 ( A. a ( Y R a  ->  a  =  X )  ->  ( Y R A  ->  A  =  X ) )  ->  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y R A  ->  A  =  X ) ) ) )
 
Theoremfrege120 36216 Simplified application of one direction of dffrege115 36211. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  A  e.  W   =>    |-  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y R A  ->  A  =  X ) ) )
 
Theoremfrege121 36217 Lemma for frege122 36218. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  A  e.  W   =>    |-  ( ( A  =  X  ->  X ( ( t+ `  R )  u.  _I  ) A )  ->  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y R A  ->  X ( ( t+ `
  R )  u. 
 _I  ) A ) ) ) )
 
Theoremfrege122 36218 If  X is a result of an application of the single-valued procedure  R to  Y, then every result of an application of the procedure  R to  Y belongs to the  R-sequence beginning with  X. Proposition 122 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  A  e.  W   =>    |-  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y R A  ->  X ( ( t+ `  R )  u.  _I  ) A ) ) )
 
Theoremfrege123 36219* Lemma for frege124 36220. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   =>    |-  (
 ( A. a ( Y R a  ->  X ( ( t+ `
  R )  u. 
 _I  ) a ) 
 ->  ( Y ( t+ `  R ) M  ->  X (
 ( t+ `  R )  u.  _I  ) M ) )  ->  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y ( t+ `  R ) M  ->  X (
 ( t+ `  R )  u.  _I  ) M ) ) ) )
 
Theoremfrege124 36220 If  X is a result of an application of the single-valued procedure  R to  Y and if  M follows  Y in the  R-sequence, then  M belongs to the  R-sequence beginning with  X. Proposition 124 of [Frege1879] p. 80. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  M  e.  W   &    |-  R  e.  S   =>    |-  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y ( t+ `
  R ) M 
 ->  X ( ( t+ `  R )  u.  _I  ) M ) ) )
 
Theoremfrege125 36221 Lemma for frege126 36222. Proposition 125 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  M  e.  W   &    |-  R  e.  S   =>    |-  (
 ( X ( ( t+ `  R )  u.  _I  ) M 
 ->  ( -.  X ( t+ `  R ) M  ->  M ( ( t+ `  R )  u.  _I  ) X ) )  ->  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y ( t+ `  R ) M  ->  ( -.  X ( t+ `
  R ) M 
 ->  M ( ( t+ `  R )  u.  _I  ) X ) ) ) ) )
 
Theoremfrege126 36222 If  M follows  Y in the  R-sequence and if the procedure  R is single-valued, then every result of an application of the procedure  R to  Y belongs to the  R-sequence beginning with  M or precedes  M in the  R-sequence. Proposition 126 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  M  e.  W   &    |-  R  e.  S   =>    |-  ( Fun  `' `' R  ->  ( Y R X  ->  ( Y ( t+ `
  R ) M 
 ->  ( -.  X ( t+ `  R ) M  ->  M ( ( t+ `  R )  u.  _I  ) X ) ) ) )
 
Theoremfrege127 36223 Communte antecedents of frege126 36222. Proposition 127 of [Frege1879] p. 82. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  M  e.  W   &    |-  R  e.  S   =>    |-  ( Fun  `' `' R  ->  ( Y ( t+ `  R ) M  ->  ( Y R X  ->  ( -.  X ( t+ `  R ) M  ->  M (
 ( t+ `  R )  u.  _I  ) X ) ) ) )
 
Theoremfrege128 36224 Lemma for frege129 36225. Proposition 128 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  M  e.  W   &    |-  R  e.  S   =>    |-  (
 ( M ( ( t+ `  R )  u.  _I  ) Y 
 ->  ( Y R X  ->  ( -.  X ( t+ `  R ) M  ->  M ( ( t+ `  R )  u.  _I  ) X ) ) ) 
 ->  ( Fun  `' `' R  ->  ( ( -.  Y ( t+ `
  R ) M 
 ->  M ( ( t+ `  R )  u.  _I  ) Y )  ->  ( Y R X  ->  ( -.  X ( t+ `
  R ) M 
 ->  M ( ( t+ `  R )  u.  _I  ) X ) ) ) ) )
 
Theoremfrege129 36225 If the procedure  R is single-valued and  Y belongs to the  R -sequence begining with  M or precedes  M in the  R-sequence, then every result of an application of the procedure  R to  Y belongs to the  R-sequence begining with 
M or precedes  M in the  R-sequence. Proposition 129 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  M  e.  W   &    |-  R  e.  S   =>    |-  ( Fun  `' `' R  ->  ( ( -.  Y ( t+ `  R ) M  ->  M (
 ( t+ `  R )  u.  _I  ) Y )  ->  ( Y R X  ->  ( -.  X ( t+ `
  R ) M 
 ->  M ( ( t+ `  R )  u.  _I  ) X ) ) ) )
 
Theoremfrege130 36226* Lemma for frege131 36227. Proposition 130 of [Frege1879] p. 84. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  M  e.  U   &    |-  R  e.  V   =>    |-  (
 ( A. b ( ( -.  b ( t+ `  R ) M  ->  M (
 ( t+ `  R )  u.  _I  )
 b )  ->  A. a
 ( b R a 
 ->  ( -.  a ( t+ `  R ) M  ->  M ( ( t+ `  R )  u.  _I  )
 a ) ) ) 
 ->  R hereditary  ( ( `' (
 t+ `  R ) " { M }
 )  u.  ( ( ( t+ `  R )  u.  _I  ) " { M } )
 ) )  ->  ( Fun  `' `' R  ->  R hereditary  ( ( `' ( t+ `  R ) " { M } )  u.  (
 ( ( t+ `
  R )  u. 
 _I  ) " { M } ) ) ) )
 
Theoremfrege131 36227 If the procedure  R is single-valued, then the property of belonging to the  R-sequence begining with 
M or preceeding  M in the  R-sequence is hereditary in the  R-sequence. Proposition 131 of [Frege1879] p. 85. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  M  e.  U   &    |-  R  e.  V   =>    |-  ( Fun  `' `' R  ->  R hereditary  ( ( `' ( t+ `  R ) " { M } )  u.  (
 ( ( t+ `
  R )  u. 
 _I  ) " { M } ) ) )
 
Theoremfrege132 36228 Lemma for frege133 36229. Proposition 132 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  M  e.  U   &    |-  R  e.  V   =>    |-  (
 ( R hereditary  ( ( `' ( t+ `  R ) " { M } )  u.  (
 ( ( t+ `
  R )  u. 
 _I  ) " { M } ) )  ->  ( X ( t+ `
  R ) M 
 ->  ( X ( t+ `  R ) Y  ->  ( -.  Y ( t+ `
  R ) M 
 ->  M ( ( t+ `  R )  u.  _I  ) Y ) ) ) ) 
 ->  ( Fun  `' `' R  ->  ( X ( t+ `  R ) M  ->  ( X ( t+ `  R ) Y  ->  ( -.  Y ( t+ `  R ) M  ->  M (
 ( t+ `  R )  u.  _I  ) Y ) ) ) ) )
 
Theoremfrege133 36229 If the procedure  R is single-valued and if  M and  Y follow  X in the  R-sequence, then  Y belongs to the  R-sequence beginning with  M or precedes  M in the  R-sequence. Proposition 133 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  M  e.  W   &    |-  R  e.  S   =>    |-  ( Fun  `' `' R  ->  ( X ( t+ `  R ) M  ->  ( X ( t+ `
  R ) Y 
 ->  ( -.  Y ( t+ `  R ) M  ->  M ( ( t+ `  R )  u.  _I  ) Y ) ) ) )
 
21.26  Mathbox for Stanislas Polu
 
Theoreminductionexd 36230 Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( N  e.  NN  ->  3 
 ||  ( ( 4 ^ N )  +  5 ) )
 
21.26.1  IMO Problems
 
21.26.1.1  IMO 1972 B2
 
Theoremwwlemuld 36231 Natural deduction form of lemul2d 11382. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  ( C  x.  A )  <_  ( C  x.  B ) )   &    |-  ( ph  ->  0  <  C )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremleeq1d 36232 Specialization of breq1d 4436 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  A  <_  C )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  B  <_  C )
 
Theoremleeq2d 36233 Specialization of breq2d 4438 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  A  <_  C )   &    |-  ( ph  ->  C  =  D )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  A  <_  D )
 
Theoremabsmulrposd 36234 Specialization of absmuld with absidd 13463. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( abs `  ( A  x.  B ) )  =  ( A  x.  ( abs `  B ) ) )
 
Theoremimadisjld 36235 Natural dduction form of one side of imadisj 5207. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  ( dom  A  i^i  B )  =  (/) )   =>    |-  ( ph  ->  ( A " B )  =  (/) )
 
Theoremimadisjlnd 36236 Natural deduction form of one negated side of imadisj 5207. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  ( dom  A  i^i  B )  =/=  (/) )   =>    |-  ( ph  ->  ( A " B )  =/=  (/) )
 
Theoremwnefimgd 36237 The image of a mapping from A is non empty if A is non empty. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( F " A )  =/=  (/) )
 
Theoremfco2d 36238 Natural deduction form of fco2 5757. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  ( F  |`  B ) : B --> C )   =>    |-  ( ph  ->  ( F  o.  G ) : A --> C )
 
Theoremsuprubd 36239* Natural deduction form of suprubd 36239. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  B 
 <_  sup ( A ,  RR ,  <  ) )
 
Theoremsuprcld 36240* Natural deduction form of suprcl 10569. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   =>    |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
 
Theoremfvco3d 36241 Natural deduction form of fvco3 5958. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  ( ( F  o.  G ) `  C )  =  ( F `  ( G `  C ) ) )
 
Theoremwfximgfd 36242 The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( F `  C )  e.  ( F " A ) )
 
Theoremfvelimabd 36243* Natural deduction form of fvelimab 5937. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  B  C_  A )   =>    |-  ( ph  ->  ( C  e.  ( F " B )  <->  E. x  e.  B  ( F `  x )  =  C ) )
 
Theoremextoimad 36244* If |f(x)| <= C for all x then it applies to all x in the image of |f(x)| (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  A. y  e.  RR  ( abs `  ( F `  y ) )  <_  C )   =>    |-  ( ph  ->  A. x  e.  ( abs " ( F " RR ) ) x  <_  C )
 
Theoremimo72b2lem0 36245* Lemma for imo72b2 36256. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  G : RR --> RR )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( ( F `  ( A  +  B )
 )  +  ( F `
  ( A  -  B ) ) )  =  ( 2  x.  ( ( F `  A )  x.  ( G `  B ) ) ) )   &    |-  ( ph  ->  A. y  e.  RR  ( abs `  ( F `  y ) )  <_ 
 1 )   =>    |-  ( ph  ->  (
 ( abs `  ( F `  A ) )  x.  ( abs `  ( G `  B ) ) )  <_  sup ( ( abs " ( F
 " RR ) ) ,  RR ,  <  ) )
 
Theoremsuprleubrd 36246* Natural deduction form of specialized suprleub 10573. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A. z  e.  A  z  <_  B )   =>    |-  ( ph  ->  sup ( A ,  RR ,  <  )  <_  B )
 
Theoremimo72b2lem2 36247* Lemma for imo72b2 36256. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A. z  e.  RR  ( abs `  ( F `  z ) )  <_  C )   =>    |-  ( ph  ->  sup (
 ( abs " ( F
 " RR ) ) ,  RR ,  <  ) 
 <_  C )
 
Theoremsyldbl2 36248 Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  (
 ( ph  /\  ps )  ->  ( ps  ->  th )
 )   =>    |-  ( ( ph  /\  ps )  ->  th )
 
Theoremfunfvima2d 36249 A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ( ph  /\  x  e.  A )  ->  ( F `  x )  e.  ( F " A ) )
 
Theoremsuprlubrd 36250* Natural deduction form of specialized suprlub 10571. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  E. z  e.  A  B  <  z )   =>    |-  ( ph  ->  B  <  sup ( A ,  RR ,  <  ) )
 
Theoremimo72b2lem1 36251* Lemma for imo72b2 36256. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  E. x  e.  RR  ( F `  x )  =/=  0 )   &    |-  ( ph  ->  A. y  e.  RR  ( abs `  ( F `  y ) )  <_ 
 1 )   =>    |-  ( ph  ->  0  <  sup ( ( abs " ( F " RR ) ) ,  RR ,  <  ) )
 
Theoremlemuldiv3d 36252 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  ( B  x.  A )  <_  C )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  B  <_  ( C  /  A ) )
 
Theoremlemuldiv4d 36253 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  B  <_  ( C  /  A ) )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( B  x.  A )  <_  C )
 
Theoremhypstkd 36254 Natural deductionm, stacks an hypothesis. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  th )   &    |-  ( ph  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  th )
 
Theoremrspcdvinvd 36255* If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  A  e.  B )   &    |-  ( ph  ->  A. x  e.  B  ps )   =>    |-  ( ph  ->  ch )
 
Theoremimo72b2 36256* IMO 1972 B2. (14th International Mathemahics Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.)
 |-  ( ph  ->  F : RR --> RR )   &    |-  ( ph  ->  G : RR --> RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A. u  e.  RR  A. v  e. 
 RR  ( ( F `
  ( u  +  v ) )  +  ( F `  ( u  -  v ) ) )  =  ( 2  x.  ( ( F `
  u )  x.  ( G `  v
 ) ) ) )   &    |-  ( ph  ->  A. y  e. 
 RR  ( abs `  ( F `  y ) ) 
 <_  1 )   &    |-  ( ph  ->  E. x  e.  RR  ( F `  x )  =/=  0 )   =>    |-  ( ph  ->  ( abs `  ( G `  B ) )  <_ 
 1 )
 
21.26.2  INT Inequalities Proof Generator

This section formalizes theorems necessary to reproduce the equality and inequality generator described in "Neural Theorem Proving on Inequality Problems" http://aitp-conference.org/2020/abstract/paper_18.pdf.

Other theorems required: 0red 9643 1red 9657 readdcld 9669 remulcld 9670 eqcomd 2437.

 
Theoremint-addcomd 36257 AdditionCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( B  +  C )  =  ( C  +  A ) )
 
Theoremint-addassocd 36258 AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( B  +  ( C  +  D ) )  =  ( ( A  +  C )  +  D ) )
 
Theoremint-addsimpd 36259 AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  0  =  ( A  -  B ) )
 
Theoremint-mulcomd 36260 MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( B  x.  C )  =  ( C  x.  A ) )
 
Theoremint-mulassocd 36261 MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( B  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  D ) )
 
Theoremint-mulsimpd 36262 MultiplicationSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  1  =  ( A  /  B ) )
 
Theoremint-leftdistd 36263 AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  (
 ( C  +  D )  x.  B )  =  ( ( C  x.  A )  +  ( D  x.  A ) ) )
 
Theoremint-rightdistd 36264 AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( B  x.  ( C  +  D ) )  =  ( ( A  x.  C )  +  ( A  x.  D ) ) )
 
Theoremint-sqdefd 36265 SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  x.  B )  =  ( A ^ 2
 ) )
 
Theoremint-mul11d 36266 First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  x.  1 )  =  B )
 
Theoremint-mul12d 36267 Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  (
 1  x.  A )  =  B )
 
Theoremint-add01d 36268 First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  +  0 )  =  B )
 
Theoremint-add02d 36269 Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  (
 0  +  A )  =  B )
 
Theoremint-sqgeq0d 36270 SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  0 
 <_  ( A  x.  B ) )
 
Theoremint-eqprincd 36271 PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  +  C )  =  ( B  +  D ) )
 
Theoremint-eqtransd 36272 EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A  =  C )
 
Theoremint-eqmvtd 36273 EquMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =  ( C  +  D ) )   =>    |-  ( ph  ->  C  =  ( B  -  D ) )
 
Theoremint-eqineqd 36274 EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  B  <_  A )
 
Theoremint-ineqmvtd 36275 IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  B  <_  A )   &    |-  ( ph  ->  A  =  ( C  +  D ) )   =>    |-  ( ph  ->  ( B  -  D )  <_  C )
 
Theoremint-ineq1stprincd 36276 FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  B  <_  A )   &    |-  ( ph  ->  D 
 <_  C )   =>    |-  ( ph  ->  ( B  +  D )  <_  ( A  +  C ) )
 
Theoremint-ineq2ndprincd 36277 SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <_  A )   &    |-  ( ph  ->  0  <_  C )   =>    |-  ( ph  ->  ( B  x.  C )  <_  ( A  x.  C ) )
 
Theoremint-ineqtransd 36278 InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <_  A )   &    |-  ( ph  ->  C  <_  B )   =>    |-  ( ph  ->  C  <_  A )
 
21.26.3  N-Digit Addition Proof Generator

This section formalizes theorems used in an n-digit addition proof generator.

Other theorems required: deccl 11065 addcomli 9824 00id 9807 addid1i 9819 addid2i 9820 eqid 2429 dec0h 11067 decadd 11092 decaddc 11093.

 
Theoremunitadd 36279 Theorem used in conjunction with decaddc 11093 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  ( A  +  B )  =  F   &    |-  ( C  +  1 )  =  B   &    |-  A  e.  NN0   &    |-  C  e.  NN0   =>    |-  ( ( A  +  C )  +  1
 )  =  F
 
Theorem5p5e10b 36280 5 + 5 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 5  +  5 )  = ; 1 0
 
Theorem6p4e10b 36281 6 + 4 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 6  +  4 )  = ; 1 0
 
Theorem7p3e10b 36282 7 + 3 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 7  +  3 )  = ; 1 0
 
Theorem8p2e10b 36283 8 + 2 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 8  +  2 )  = ; 1 0
 
Theorem9p1e10b 36284 9 + 1 = 10. Decimal form. (Contributed by Stanislas Polu, 7-Apr-2020.)
 |-  (
 9  +  1 )  = ; 1 0
 
21.26.4  AM-GM (for k = 2,3,4)
 
Theoremgsumws3 36285 Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Mnd  /\  ( S  e.  B  /\  ( T  e.  B  /\  U  e.  B ) ) )  ->  ( G  gsumg 
 <" S T U "> )  =  ( S  .+  ( T 
 .+  U ) ) )
 
Theoremgsumws4 36286 Valuation of a length 4 word in a monoid (Contributed by Stanislas Polu, 10-Sep-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Mnd  /\  ( S  e.  B  /\  ( T  e.  B  /\  ( U  e.  B  /\  V  e.  B ) ) ) )  ->  ( G  gsumg 
 <" S T U V "> )  =  ( S  .+  ( T  .+  ( U  .+  V ) ) ) )
 
Theoremamgm2d 36287 Arithmetic-geometric mean inequality for  n  =  2, derived from amgmlem 23780. (Contributed by Stanislas Polu, 8-Sep-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  B )  ^c  ( 1 
 /  2 ) ) 
 <_  ( ( A  +  B )  /  2
 ) )
 
Theoremamgm3d 36288 Arithmetic-geometric mean inequality for  n  =  3. (Contributed by Stanislas Polu, 11-Sep-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  ( B  x.  C ) ) 
 ^c  ( 1 
 /  3 ) ) 
 <_  ( ( A  +  ( B  +  C ) )  /  3
 ) )
 
Theoremamgm4d 36289 Arithmetic-geometric mean inequality for  n  =  4. (Contributed by Stanislas Polu, 11-Sep-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR+ )   =>    |-  ( ph  ->  (
 ( A  x.  ( B  x.  ( C  x.  D ) ) ) 
 ^c  ( 1 
 /  4 ) ) 
 <_  ( ( A  +  ( B  +  ( C  +  D )
 ) )  /  4
 ) )
 
21.27  Mathbox for Steve Rodriguez
 
21.27.1  Miscellanea
 
Theoremnanorxor 36290 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
 |-  (
 ( ph  -/\  ps )  <->  ( ( ph  \/  ps ) 
 <->  ( ph  \/_  ps ) ) )
 
Theoremundisjrab 36291 Union of two disjoint restricted class abstractions; compare unrab 3750. (Contributed by Steve Rodriguez, 28-Feb-2020.)
 |-  (
 ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  (/)  <->  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  { x  e.  A  |  ( ph  \/_  ps ) } )
 
Theoremiso0 36292 The empty set is an  R ,  S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
 |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
 
Theoremssrecnpr 36293  RR is a subset of both  RR and  CC. (Contributed by Steve Rodriguez, 22-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S )
 
Theoremseff 36294 Let set  S be the real or complex numbers. Then the exponential function restricted to  S is a mapping from  S to  S. (Contributed by Steve Rodriguez, 6-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   =>    |-  ( ph  ->  ( exp  |`  S ) : S --> S )
 
Theoremsblpnf 36295 The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 21343. (Contributed by Steve Rodriguez, 8-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )   =>    |-  ( ( ph  /\  P  e.  S ) 
 ->  ( P ( ball `  D ) +oo )  =  S )
 
Theoremprmunb2 36296* The primes are unbounded. This generalizes prmunb 14821 to real  A with arch 10866 and lttrd 9795: every real is less than some positive integer, itself less than some prime. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( A  e.  RR  ->  E. p  e.  Prime  A  <  p )
 
Theoremisprm7 36297* One need only check prime divisors of  P up to  sqr P in order to ensure primality. This version of isprm5 14622 combines the primality and bound on  z into a finite interval of prime numbers. (Contributed by Steve Rodriguez, 20-Jan-2020.)
 |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  (
 ( 2 ... ( |_ `  ( sqr `  P ) ) )  i^i 
 Prime )  -.  z  ||  P ) )
 
21.27.2  Ratio test for infinite series convergence and divergence
 
Theoremdvgrat 36298* Ratio test for divergence of a complex infinite series. See e.g. remark "if  ( abs `  (
( a `  (
n  +  1 ) )  /  ( a `
 n ) ) )  >_  1 for all large n..." in https://en.wikipedia.org/wiki/Ratio_test#The_test. (Contributed by Steve Rodriguez, 28-Feb-2020.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  ( ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  W )  ->  ( F `  k )  =/=  0 )   &    |-  ( ( ph  /\  k  e.  W ) 
 ->  ( abs `  ( F `  k ) ) 
 <_  ( abs `  ( F `  ( k  +  1 ) ) ) )   =>    |-  ( ph  ->  seq M (  +  ,  F )  e/  dom  ~~>  )
 
Theoremcvgdvgrat 36299* Ratio test for convergence and divergence of a complex infinite series. If the ratio  R of the absolute values of successive terms in an infinite sequence  F converges to less than one, then the infinite sum of the terms of  F converges to a complex number; and if  R converges greater then the sum diverges. This combined form of cvgrat 13917 and dvgrat 36298 directly uses the limit of the ratio.

(It also demonstrates how to use climi2 13553 and absltd 13470 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191, and how to use r19.29a 2977 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 2924 at https://groups.google.com/forum/#!topic/metamath/2RPikOiXLMo.) (Contributed by Steve Rodriguez, 28-Feb-2020.)

 |-  Z  =  ( ZZ>= `  M )   &    |-  W  =  ( ZZ>= `  N )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  W )  ->  ( F `  k )  =/=  0 )   &    |-  R  =  ( k  e.  W  |->  ( abs `  ( ( F `  ( k  +  1 ) )  /  ( F `  k ) ) ) )   &    |-  ( ph  ->  R  ~~>  L )   &    |-  ( ph  ->  L  =/=  1
 )   =>    |-  ( ph  ->  ( L  <  1  <->  seq M (  +  ,  F )  e.  dom  ~~>  ) )
 
Theoremradcnvrat 36300* Let  L be the limit, if one exists, of the ratio  ( abs `  (
( A `  (
k  +  1 ) )  /  ( A `
 k ) ) ) (as in the ratio test cvgdvgrat 36299) as  k increases. Then the radius of convergence of power series  sum_ n  e.  NN0 ( ( A `  n )  x.  (
x ^ n ) ) is  ( 1  /  L ) if  L is nonzero. Proof "The limit involved in the ratio test..." in https://en.wikipedia.org/wiki/Radius_of_convergence —a few lines that evidently hide quite an involved process to confirm. (Contributed by Steve Rodriguez, 8-Mar-2020.)
 |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^ n ) ) ) )   &    |-  ( ph  ->  A : NN0 --> CC )   &    |-  R  =  sup ( { r  e.  RR  |  seq 0
 (  +  ,  ( G `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )   &    |-  D  =  ( k  e.  NN0  |->  ( abs `  (
 ( A `  (
 k  +  1 ) )  /  ( A `
  k ) ) ) )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( A `  k )  =/=  0 )   &    |-  ( ph  ->  D  ~~>  L )   &    |-  ( ph  ->  L  =/=  0 )   =>    |-  ( ph  ->  R  =  ( 1  /  L ) )
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