HomeHome Metamath Proof Explorer
Theorem List (p. 268 of 309)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30843)
 

Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-toplnd 26701* A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
 |- TopLnd  =  { x  e.  Top  |  A. y  e.  ~P  x ( U. x  =  U. y  ->  E. z  e.  ~P  x ( z  ~<_  om 
 /\  U. x  =  U. z ) ) }
 
16.16  Mathbox for Steve Rodriguez
 
16.16.1  Miscellanea
 
Theoremiso0 26702 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 26703  RR is a subset of both  RR and  CC. (Contributed by Steve Rodriguez, 22-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S )
 
Theoremseff 26704 Let set  S be the reals or complexes. 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 26705 The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 17786. (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 )
 
16.16.2  Function operations
 
Theoremcaofcan 26706* Transfer a cancellation law like mulcan 9285 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> T )   &    |-  ( ph  ->  G : A --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y )  =  ( x R z )  <->  y  =  z
 ) )   =>    |-  ( ph  ->  (
 ( F  o F R G )  =  ( F  o F R H )  <->  G  =  H ) )
 
Theoremofsubid 26707 Function analog of subid 8947. (Contributed by Steve Rodriguez, 5-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC )  ->  ( F  o F  -  F )  =  ( A  X.  { 0 } ) )
 
Theoremofmul12 26708 Function analog of mul12 8858. (Contributed by Steve Rodriguez, 13-Nov-2015.)
 |-  (
 ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A --> CC  /\  H : A --> CC )
 )  ->  ( F  o F  x.  ( G  o F  x.  H ) )  =  ( G  o F  x.  ( F  o F  x.  H ) ) )
 
Theoremofdivrec 26709 Function analog of divrec 9320, a division analog of ofnegsub 9624. (Contributed by Steve Rodriguez, 3-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC  \  { 0 } )
 )  ->  ( F  o F  x.  (
 ( A  X.  {
 1 } )  o F  /  G ) )  =  ( F  o F  /  G ) )
 
Theoremofdivcan4 26710 Function analog of divcan4 9329. (Contributed by Steve Rodriguez, 4-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC  \  { 0 } )
 )  ->  ( ( F  o F  x.  G )  o F  /  G )  =  F )
 
Theoremofdivdiv2 26711 Function analog of divdiv2 9352. (Contributed by Steve Rodriguez, 23-Nov-2015.)
 |-  (
 ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A --> ( CC  \  { 0 } )  /\  H : A --> ( CC  \  { 0 } )
 ) )  ->  ( F  o F  /  ( G  o F  /  H ) )  =  (
 ( F  o F  x.  H )  o F  /  G ) )
 
16.16.3  Calculus
 
Theoremlhe4.4ex1a 26712 Example of the Fundamental Theorem of Calculus, part two (ftc2 19223):  S. ( 1 (,) 2 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  /  3
). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 19223 as simply the "Fundamental Theorem of Calculus", then ftc1 19221 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)
 |-  S. ( 1 (,) 2
 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  / 
 3 )
 
Theoremdvsconst 26713 Derivative of a constant function on the reals or complexes. The function may return a complex  A even if  S is  RR. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  (
 ( S  e.  { RR ,  CC }  /\  A  e.  CC )  ->  ( S  _D  ( S  X.  { A }
 ) )  =  ( S  X.  { 0 } ) )
 
Theoremdvsid 26714 Derivative of the identity function on the reals or complexes. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  (  _I  |`  S ) )  =  ( S  X.  { 1 } ) )
 
Theoremdvsef 26715 Derivative of the exponential function on the reals or complexes. (Contributed by Steve Rodriguez, 12-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( exp  |`  S ) )  =  ( exp  |`  S ) )
 
Theoremexpgrowthi 26716* Exponential growth and decay model. See expgrowth 26718 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  Y  =  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t
 ) ) ) )   =>    |-  ( ph  ->  ( S  _D  Y )  =  ( ( S  X.  { K } )  o F  x.  Y ) )
 
Theoremdvconstbi 26717* The derivative of a function on  S is zero iff it is a constant function. Roughly a biconditional  S analog of dvconst 19098 and dveq0 19179. Corresponds to integration formula " S. 0  _d x  =  C " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  (  S  _D  Y )  =  S )   =>    |-  ( ph  ->  ( ( S  _D  Y )  =  ( S  X.  {
 0 } )  <->  E. c  e.  CC  Y  =  ( S  X.  { c } )
 ) )
 
Theoremexpgrowth 26718* Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 26716 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives.

Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant λ.

Here y' is given as  ( S  _D  Y
), C as  c, and ky as  ( ( S  X.  { K }
)  o F  x.  Y ).  ( S  X.  { K } ) is the constant function that maps any real or complex input to k and  o F  x. is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case.

Statements for this and expgrowthi 26716 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  (  S  _D  Y )  =  S )   =>    |-  ( ph  ->  (
 ( S  _D  Y )  =  ( ( S  X.  { K }
 )  o F  x.  Y )  <->  E. c  e.  CC  Y  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t ) ) ) ) ) )
 
16.17  Mathbox for Andrew Salmon
 
16.17.1  Principia Mathematica * 10
 
Theorempm10.12 26719* Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x ( ph  \/  ps )  ->  ( ph  \/  A. x ps )
 )
 
Theorempm10.14 26720 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( A. x ph  /\  A. x ps )  ->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps )
 )
 
Theorempm10.251 26721 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x  -.  ph  ->  -. 
 A. x ph )
 
Theorempm10.252 26722 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  E. x ph  <->  A. x  -.  ph )
 
Theorempm10.253 26723 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  A. x ph  <->  E. x  -.  ph )
 
Theoremalbitr 26724 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  <->  ps )  /\  A. x ( ps  <->  ch ) )  ->  A. x ( ph  <->  ch ) )
 
Theorempm10.42 26725 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( E. x ph  \/  E. x ps )  <->  E. x ( ph  \/  ps ) )
 
Theorempm10.52 26726* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  ->  ( A. x ( ph  ->  ps )  <->  ps ) )
 
Theorempm10.53 26727 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x ph  ->  A. x ( ph  ->  ps ) )
 
Theorempm10.541 26728* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  \/  ps )
 ) 
 <->  ( ch  \/  A. x ( ph  ->  ps ) ) )
 
Theorempm10.542 26729* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  ->  ps )
 ) 
 <->  ( ch  ->  A. x ( ph  ->  ps )
 ) )
 
Theorempm10.55 26730 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x (
 ph  /\  ps )  /\  A. x ( ph  ->  ps ) )  <->  ( E. x ph 
 /\  A. x ( ph  ->  ps ) ) )
 
Theorempm10.56 26731 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  ->  ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ps  /\  ch )
 )
 
Theorempm10.57 26732 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ps  \/  ch )
 )  ->  ( A. x ( ph  ->  ps )  \/  E. x ( ph  /\  ch )
 ) )
 
16.17.2  Principia Mathematica * 11
 
Theorem2alanimi 26733 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x A. y ph  /\  A. x A. y ps )  ->  A. x A. y ch )
 
Theorem2al2imi 26734 Removes two universal qunatifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x A. y ph  ->  ( A. x A. y ps  ->  A. x A. y ch ) )
 
Theoremstdpc4-2 26735 Theorem *11.1 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ph  ->  [ z  /  x ] [ w  /  y ] ph )
 
Theorempm11.11 26736 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ph   =>    |- 
 A. z A. w [ z  /  x ] [ w  /  y ] ph
 
Theorempm11.12 26737* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  ->  ( ph  \/  A. x A. y ps )
 )
 
Theorem2exnaln 26738 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ph  <->  -.  A. x A. y  -.  ph )
 
Theorem2nexaln 26739 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph 
 <-> 
 A. x A. y  -.  ph )
 
Theorem19.21vv 26740* Compare theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  ->  ph )  <->  ( ps  ->  A. x A. y ph ) )
 
Theorem2alim 26741 Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( A. x A. y ph  ->  A. x A. y ps ) )
 
Theorem2albi 26742 Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( A. x A. y ph  <->  A. x A. y ps )
 )
 
Theorem2exim 26743 Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( E. x E. y ph  ->  E. x E. y ps ) )
 
Theorem2exbi 26744 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( E. x E. y ph  <->  E. x E. y ps )
 )
 
Theorema4sbce-2 26745 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  ->  E. x E. y ph )
 
Theorem19.33-2 26746 Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x A. y ph  \/  A. x A. y ps )  ->  A. x A. y ( ph  \/  ps ) )
 
Theorem19.36vv 26747* Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y (
 ph  ->  ps )  <->  ( A. x A. y ph  ->  ps )
 )
 
Theorem19.31vv 26748* Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  <->  (
 A. x A. y ph  \/  ps ) )
 
Theorem19.37vv 26749* Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ps  ->  ph )  <->  ( ps  ->  E. x E. y ph ) )
 
Theorem19.28vv 26750* Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  /\  ph )  <->  ( ps  /\  A. x A. y ph ) )
 
Theorempm11.52 26751 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  /\  ps )  <->  -. 
 A. x A. y
 ( ph  ->  -.  ps ) )
 
Theorem2exanali 26752 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y
 ( ph  /\  -.  ps ) 
 <-> 
 A. x A. y
 ( ph  ->  ps )
 )
 
Theoremaaanv 26753* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 1811. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ph  /\  A. y ps )  <->  A. x A. y
 ( ph  /\  ps )
 )
 
Theorempm11.57 26754* Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ph  <->  A. x A. y
 ( ph  /\  [ y  /  x ] ph )
 )
 
Theorempm11.58 26755* Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  <->  E. x E. y
 ( ph  /\  [ y  /  x ] ph )
 )
 
Theorempm11.59 26756* Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( A. x ( ph  ->  ps )  ->  A. y A. x ( ( ph  /\ 
 [ y  /  x ] ph )  ->  ( ps  /\  [ y  /  x ] ps ) ) )
 
Theorempm11.6 26757* Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x ( E. y
 ( ph  /\  ps )  /\  ch )  <->  E. y ( E. x ( ph  /\  ch )  /\  ps ) )
 
Theorempm11.61 26758* Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. y A. x (
 ph  ->  ps )  ->  A. x ( ph  ->  E. y ps ) )
 
Theorempm11.62 26759* Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ( ph  /\  ps )  ->  ch )  <->  A. x ( ph  ->  A. y ( ps 
 ->  ch ) ) )
 
Theorempm11.63 26760 Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph  ->  A. x A. y
 ( ph  ->  ps )
 )
 
Theorempm11.7 26761 Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  \/  ph )  <->  E. x E. y ph )
 
Theorempm11.71 26762* Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x ph  /\ 
 E. y ch )  ->  ( ( A. x ( ph  ->  ps )  /\  A. y ( ch 
 ->  th ) )  <->  A. x A. y
 ( ( ph  /\  ch )  ->  ( ps  /\  th ) ) ) )
 
16.17.3  Predicate Calculus
 
Theoremsbeqal1 26763* If  x  =  y always implies 
x  =  z, then  y  =  z is true. (Contributed by Andrew Salmon, 2-Jun-2011.)
 |-  ( A. x ( x  =  y  ->  x  =  z )  ->  y  =  z )
 
Theoremsbeqal1i 26764* Suppose you know  x  =  y implies  x  =  z, assuming  x and  z are distinct. Then,  y  =  z. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( x  =  y  ->  x  =  z )   =>    |-  y  =  z
 
Theoremsbeqal2i 26765* If  x  =  y implies  x  =  z, then we can infer  z  =  y. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( x  =  y  ->  x  =  z )   =>    |-  z  =  y
 
Theoremsbeqalbi 26766* When both  x and  z and  y and  z are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)
 |-  ( x  =  y  <->  A. z ( z  =  x  ->  z  =  y ) )
 
Theoremax4567 26767 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1536 as the inference rule. This proof extends the idea of ax467 1752 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
 |-  (
 ( A. x A. y  -.  A. x A. y
 ( A. y ph  ->  ps )  ->  ( ph  ->  A. y ( A. y ph  ->  ps )
 ) )  ->  ( A. y ph  ->  A. y ps ) )
 
Theoremax4567to4 26768 Re-derivation of ax-4 1692 from ax4567 26767. Note that ax-9 1684 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  ph )
 
Theoremax4567to5 26769 Re-derivation of ax-5o 1694 from ax4567 26767. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ( A. x ph 
 ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremax4567to6 26770 Re-derivation of ax-6o 1697 from ax4567 26767. Note that neither ax-6o 1697 nor ax-7 1535 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax4567to7 26771 Re-derivation of ax-7 1535 from ax4567 26767. Note that ax-7 1535 is not required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax10ext 26772* This theorem shows that, given axext4 2237, we can derive a version of ax-10 1678. However, it is weaker than ax-10 1678 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
Theoremax10-16 26773* This theorem shows that, given ax-16 1926, we can derive a version of ax-10 1678. However, it is weaker than ax-10 1678 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
16.17.4  Principia Mathematica * 13 and * 14
 
Theorempm13.13a 26774 One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  A )  ->  [. A  /  x ]. ph )
 
Theorempm13.13b 26775 Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  x  =  A )  ->  ph )
 
Theorempm13.14 26776 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  -.  ph )  ->  x  =/=  A )
 
Theorempm13.192 26777* Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( A. x ( x  =  A  <->  x  =  y )  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.193 26778 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  x  =  y ) )
 
Theorempm13.194 26779 Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  ph  /\  x  =  y ) )
 
Theorempm13.195 26780* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 2945. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( y  =  A  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.196a 26781* Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( -.  ph  <->  A. y ( [
 y  /  x ] ph  ->  y  =/=  x ) )
 
Theorem2sbc6g 26782* Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theorem2sbc5g 26783* Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theoremiotain 26784 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( E! x ph  ->  |^| { x  |  ph }  =  (
 iota x ph ) )
 
Theoremiotaexeu 26785 The iota class exists. This theorem does not require ax-nul 4046 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e. 
 _V )
 
Theoremiotasbc 26786* Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define  iota in terms of a function of  ( iota x ph ). Their definition differs in that a function of  ( iota x ph ) evaluates to "false" when there isn't a single  x that satisfies  ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( A. x ( ph  <->  x  =  y
 )  /\  ps )
 ) )
 
Theoremiotasbc2 26787* Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  (
 ( E! x ph  /\ 
 E! x ps )  ->  ( [. ( iota
 x ph )  /  y ]. [. ( iota x ps )  /  z ]. ch  <->  E. y E. z
 ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z
 )  /\  ch )
 ) )
 
Theorempm14.12 26788* Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  A. x A. y ( ( ph  /\  [. y  /  x ].
 ph )  ->  x  =  y ) )
 
Theorempm14.122a 26789* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph )
 ) )
 
Theorempm14.122b 26790* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  ( ( A. x (
 ph  ->  x  =  A )  /\  [. A  /  x ].
 ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.122c 26791* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.123a 26792* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph ) ) )
 
Theorempm14.123b 26793* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.123c 26794* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.18 26795 Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ps  ->  [. ( iota x ph )  /  x ]. ps ) )
 
Theoremiotaequ 26796* Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x x  =  y )  =  y
 
Theoremiotavalb 26797* Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6154. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y
 ) 
 <->  ( iota x ph )  =  y )
 )
 
Theoremiotasbc5 26798* Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) ) )
 
Theorempm14.24 26799* Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  A. y
 ( [. y  /  x ].
 ph 
 <->  y  =  ( iota
 x ph ) ) )
 
Theoremiotavalsb 26800* Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  x  =  y
 )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps )
 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 30801-30843
  Copyright terms: Public domain < Previous  Next >