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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrankeq1o 24801 The only set with rank  1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
 |-  (
 ( rank `  A )  =  1o  <->  A  =  { (/)
 } )
 
18.7.44  Hereditarily Finite Sets
 
Syntaxchf 24802 The constant Hf is a class.
 class Hf
 
Definitiondf-hf 24803 Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.)
 |- Hf  =  U. ( R1 " om )
 
Theoremelhf 24804* Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
 |-  ( A  e. Hf  <->  E. x  e.  om  A  e.  ( R1 `  x ) )
 
Theoremelhf2 24805 Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
 |-  A  e.  _V   =>    |-  ( A  e. Hf  <->  ( rank `  A )  e.  om )
 
Theoremelhf2g 24806 Hereditarily finiteness via rank. Closed form of elhf2 24805. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  ( A  e.  V  ->  ( A  e. Hf  <->  ( rank `  A )  e.  om )
 )
 
Theorem0hf 24807 The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
 |-  (/)  e. Hf
 
Theoremhfun 24808 The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  B )  e. Hf  )
 
Theoremhfsn 24809 The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  ( A  e. Hf  ->  { A }  e. Hf  )
 
Theoremhfadj 24810 Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  u.  { B } )  e. Hf  )
 
Theoremhfelhf 24811 Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  (
 ( A  e.  B  /\  B  e. Hf  )  ->  A  e. Hf  )
 
Theoremhftr 24812 The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  Tr Hf
 
Theoremhfext 24813* Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  (
 ( A  e. Hf  /\  B  e. Hf  )  ->  ( A  =  B  <->  A. x  e. Hf  ( x  e.  A  <->  x  e.  B ) ) )
 
Theoremhfuni 24814 The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  ( A  e. Hf  ->  U. A  e. Hf  )
 
Theoremhfpw 24815 The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  ( A  e. Hf  ->  ~P A  e. Hf  )
 
Theoremhfninf 24816  om is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
 |-  -.  om  e. Hf
 
18.8  Mathbox for Anthony Hart
 
18.8.1  Propositional Calculus
 
Theoremtb-ax1 24817 The first of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  ( ph  ->  ch ) ) )
 
Theoremtb-ax2 24818 The second of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ph ) )
 
Theoremtb-ax3 24819 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 8, tb-ax1 24817, and tb-ax2 24818, can be used to derive any theorem or rule that uses only  ->. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  (
 ( ( ph  ->  ps )  ->  ph )  ->  ph )
 
Theoremtbsyl 24820 The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremre1ax2lem 24821 Lemma for re1ax2 24822. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
 
Theoremre1ax2 24822 ax-2 6 rederived from the Tarski-Bernays axiom system. Often tb-ax1 24817 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theoremnaim1 24823 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ps  -/\  ch )  ->  ( ph  -/\  ch )
 ) )
 
Theoremnaim2 24824 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ch  -/\  ps )  ->  ( ch  -/\  ph )
 ) )
 
Theoremnaim1i 24825 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ps  -/\  ch )   =>    |-  ( ph  -/\  ch )
 
Theoremnaim2i 24826 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  -/\  ps )   =>    |-  ( ch  -/\  ph )
 
Theoremnaim12i 24827 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ps  -/\  th )   =>    |-  ( ph  -/\  ch )
 
Theoremnabi1 24828 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  -/\  ch )  <->  ( ps  -/\  ch )
 ) )
 
Theoremnabi2 24829 Constructor theorem for  -/\. (Contributed by Anthony Hart, 1-Sep-2011.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  -/\  ph )  <->  ( ch  -/\  ps )
 ) )
 
Theoremnabi1i 24830 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph 
 <->  ps )   &    |-  ( ps  -/\  ch )   =>    |-  ( ph  -/\  ch )
 
Theoremnabi2i 24831 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  -/\  ps )   =>    |-  ( ch  -/\  ph )
 
Theoremnabi12i 24832 Constructor rule for  -/\. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ps  -/\  th )   =>    |-  ( ph  -/\  ch )
 
Syntaxw3nand 24833 The double nand.
 wff  ( ph  -/\  ps  -/\  ch )
 
Definitiondf-3nand 24834 The double nand. This definition allows us to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  (
 ( ph  -/\  ps  -/\  ch )  <->  (
 ph  ->  ( ps  ->  -. 
 ch ) ) )
 
Theoremdf3nandALT1 24835 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  (
 ( ph  -/\  ps  -/\  ch )  <->  (
 ph  -/\  ( ( ps  -/\  ch )  -/\  ( ps  -/\  ch ) ) ) )
 
Theoremdf3nandALT2 24836 The double nand expressed in terms of negation and and. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  (
 ( ph  -/\  ps  -/\  ch )  <->  -.  ( ph  /\  ps  /\ 
 ch ) )
 
Theoremandnand1 24837 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  (
 ( ph  /\  ps  /\  ch )  <->  ( ( ph  -/\ 
 ps  -/\  ch )  -/\  ( ph  -/\  ps  -/\  ch )
 ) )
 
Theoremimnand2 24838 An  -> nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
 |-  (
 ( -.  ph  ->  ps )  <->  ( ( ph  -/\  ph )  -/\  ( ps  -/\  ps ) ) )
 
18.8.2  Predicate Calculus
 
Theoremquantriv 24839* Any wff can be trivially quantified, so long as the quantifier's set is distinct from said wff.

See also 19.9v 1663. (Contributed by Anthony Hart, 13-Sep-2011.)

 |-  ( A. x ph  <->  ph )
 
Theoremallt 24840 For all sets,  T. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  A. x  T.
 
Theoremalnof 24841 For all sets,  F. is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  A. x  -.  F.
 
Theoremnalf 24842 Not all sets hold  F. as true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  A. x  F.
 
Theoremextt 24843 There exists a set that holds  T. as true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  E. x  T.
 
Theoremnextnt 24844 There does not exist a set, such that  T. is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E. x  -.  T.
 
Theoremnextf 24845 There does not exist a set, such that  F. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E. x  F.
 
Theoremunnf 24846 There does not exist exactly one set, such that  F. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E! x  F.
 
Theoremunnt 24847 There does not exist exactly one set, such that  T. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E! x  T.
 
Theoremmont 24848 There does not exist at most one set, such that  T. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E* x  T.
 
Theoremmof 24849 There exist at most one set, such that  F. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  E* x  F.
 
18.8.3  Misc. Single Axiom Systems
 
Theoremmeran1 24850 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
 |-  ( -.  ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  th  \/  ph )  \/  ( ch 
 \/  ( ta  \/  ph ) ) ) )
 
Theoremmeran2 24851 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
 |-  ( -.  ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  ta  \/  th )  \/  ( ch 
 \/  ( ph  \/  th ) ) ) )
 
Theoremmeran3 24852 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
 |-  ( -.  ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  ch  \/  ph )  \/  ( ta 
 \/  ( th  \/  ph ) ) ) )
 
Theoremwaj-ax 24853 A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)
 |-  (
 ( ph  -/\  ( ps  -/\  ch ) )  -/\  ( ( ( th  -/\ 
 ch )  -/\  (
 ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) )  -/\  ( ph  -/\  ( ph  -/\  ps )
 ) ) )
 
Theoremlukshef-ax2 24854 A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  (
 ( ph  -/\  ( ps  -/\  ch ) )  -/\  ( ( ph  -/\  ( ch  -/\  ph ) )  -/\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) ) ) )
 
Theoremarg-ax 24855 ? (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  (
 ( ph  -/\  ( ps  -/\  ch ) )  -/\  ( ( ph  -/\  ( ps  -/\  ch ) ) 
 -/\  ( ( th  -/\ 
 ch )  -/\  (
 ( ch  -/\  th )  -/\  ( ph  -/\  th )
 ) ) ) )
 
18.8.4  Connective Symmetry
 
Theoremnegsym1 24856 In the paper "On Variable Functors of Propositional Arguments", Lukasiewicz introduced a system that can handle variable connectives. This was done by introducing a variable, marked with a lowercase delta, which takes a wff as input. In the system, "delta  ph " means that "something is true of 
ph." "delta  ph " can be substituted with  -.  ph,  ps  /\ 
ph,  A. x ph, etc.

Later on, Meredith discovered a single axiom, in the form of  ( delta delta  F.  -> delta  ph  ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus.

A symmetry with  -.. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  ( -.  -.  F.  ->  -.  ph )
 
Theoremimsym1 24857 A symmetry with  ->.

See negsym1 24856 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  ->  ( ps  ->  F.  ) )  ->  ( ps  ->  ph )
 )
 
Theorembisym1 24858 A symmetry with 
<->.

See negsym1 24856 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  <->  ( ps  <->  F.  ) )  ->  ( ps  <->  ph ) )
 
Theoremconsym1 24859 A symmetry with  /\.

See negsym1 24856 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  /\  ( ps  /\  F.  ) ) 
 ->  ( ps  /\  ph )
 )
 
Theoremdissym1 24860 A symmetry with  \/.

See negsym1 24856 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  \/  ( ps  \/  F.  ) ) 
 ->  ( ps  \/  ph ) )
 
Theoremnandsym1 24861 A symmetry with  -/\.

See negsym1 24856 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  -/\  ( ps  -/\  F.  ) ) 
 ->  ( ps  -/\  ph )
 )
 
Theoremunisym1 24862 A symmetry with  A..

See negsym1 24856 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

 |-  ( A. x A. x  F.  ->  A. x ph )
 
Theoremexisym1 24863 A symmetry with  E..

See negsym1 24856 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  ( E. x E. x  F.  ->  E. x ph )
 
Theoremunqsym1 24864 A symmetry with  E!.

See negsym1 24856 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)

 |-  ( E! x E! x  F.  ->  E! x ph )
 
Theoremamosym1 24865 A symmetry with  E*.

See negsym1 24856 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

 |-  ( E* x E* x  F.  ->  E* x ph )
 
Theoremsubsym1 24866 A symmetry with  [ x  / 
y ].

See negsym1 24856 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)

 |-  ( [ x  /  y ] [ x  /  y ]  F.  ->  [ x  /  y ] ph )
 
18.9  Mathbox for Chen-Pang He
 
18.9.1  Ordinal topology
 
Theoremontopbas 24867 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
 |-  ( B  e.  On  ->  B  e.  TopBases )
 
Theoremonsstopbas 24868 The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)
 |-  On  C_  TopBases
 
Theoremonpsstopbas 24869 The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.)
 |-  On  C.  TopBases
 
Theoremontgval 24870 The topology generated from an ordinal number  B is 
suc  U. B. (Contributed by Chen-Pang He, 10-Oct-2015.)
 |-  ( B  e.  On  ->  (
 topGen `  B )  = 
 suc  U. B )
 
Theoremontgsucval 24871 The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)
 |-  ( A  e.  On  ->  (
 topGen `  suc  A )  =  suc  A )
 
Theoremonsuctop 24872 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
 |-  ( A  e.  On  ->  suc 
 A  e.  Top )
 
Theoremonsuctopon 24873 One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.)
 |-  ( A  e.  On  ->  suc 
 A  e.  (TopOn `  A ) )
 
Theoremordtoplem 24874 Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
 |-  ( U. A  e.  On  ->  suc  U. A  e.  S )   =>    |-  ( Ord  A  ->  ( A  =/=  U. A  ->  A  e.  S ) )
 
Theoremordtop 24875 An ordinal is a topology iff it is not its supermum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.)
 |-  ( Ord  J  ->  ( J  e.  Top  <->  J  =/=  U. J ) )
 
Theoremonsucconi 24876 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
 |-  A  e.  On   =>    |- 
 suc  A  e.  Con
 
Theoremonsuccon 24877 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
 |-  ( A  e.  On  ->  suc 
 A  e.  Con )
 
Theoremordtopcon 24878 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
 |-  ( Ord  J  ->  ( J  e.  Top  <->  J  e.  Con ) )
 
Theoremonintopsscon 24879 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
 |-  ( On  i^i  Top )  C_  Con
 
Theoremonsuct0 24880 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
 |-  ( A  e.  On  ->  suc 
 A  e.  Kol2 )
 
Theoremordtopt0 24881 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
 |-  ( Ord  J  ->  ( J  e.  Top  <->  J  e.  Kol2 )
 )
 
Theoremonsucsuccmpi 24882 The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.)
 |-  A  e.  On   =>    |- 
 suc  suc  A  e.  Comp
 
Theoremonsucsuccmp 24883 The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.)
 |-  ( A  e.  On  ->  suc 
 suc  A  e.  Comp )
 
Theoremlimsucncmpi 24884 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
 |-  Lim  A   =>    |-  -. 
 suc  A  e.  Comp
 
Theoremlimsucncmp 24885 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
 |-  ( Lim  A  ->  -.  suc  A  e.  Comp )
 
Theoremordcmp 24886 Iff an ordinal topology is compact, the underlying set is its supermum (union) only when the ordinal is  1o. (Contributed by Chen-Pang He, 1-Nov-2015.)
 |-  ( Ord  A  ->  ( A  e.  Comp 
 <->  ( U. A  =  U.
 U. A  ->  A  =  1o ) ) )
 
Theoremssoninhaus 24887 The ordinal topologies  1o and  2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
 |-  { 1o ,  2o }  C_  ( On  i^i  Haus )
 
Theoremonint1 24888 The ordinal T1 spaces are  1o and  2o, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.)
 |-  ( On  i^i  Fre )  =  { 1o ,  2o }
 
Theoremoninhaus 24889 The ordinal Hausdorff spaces are 
1o and  2o. (Contributed by Chen-Pang He, 10-Nov-2015.)
 |-  ( On  i^i  Haus )  =  { 1o ,  2o }
 
18.10  Mathbox for Jeff Hoffman
 
18.10.1  Inferences for finite induction on generic function values
 
Theoremfveleq 24890 Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
 |-  ( A  =  B  ->  ( ( ph  ->  ( F `  A )  e.  P )  <->  ( ph  ->  ( F `  B )  e.  P ) ) )
 
Theoremfindfvcl 24891* Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
 |-  ( ph  ->  ( F `  (/) )  e.  P )   &    |-  ( y  e.  om  ->  (
 ph  ->  ( ( F `
  y )  e.  P  ->  ( F ` 
 suc  y )  e.  P ) ) )   =>    |-  ( A  e.  om  ->  (
 ph  ->  ( F `  A )  e.  P ) )
 
Theoremfindreccl 24892* Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
 |-  (
 z  e.  P  ->  ( G `  z )  e.  P )   =>    |-  ( C  e.  om 
 ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C )  e.  P ) )
 
Theoremfindabrcl 24893* Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  (
 z  e.  P  ->  ( G `  z )  e.  P )   =>    |-  ( ( C  e.  om  /\  A  e.  P )  ->  (
 ( x  e.  _V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  e.  P )
 
18.10.2  gdc.mm
 
Theoremnnssi2 24894 Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  NN  C_  D   &    |-  ( B  e.  NN  ->  ph )   &    |-  ( ( A  e.  D  /\  B  e.  D  /\  ph )  ->  ps )   =>    |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ps )
 
Theoremnnssi3 24895 Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  NN  C_  D   &    |-  ( C  e.  NN  ->  ph )   &    |-  ( ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  /\  ph )  ->  ps )   =>    |-  (
 ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ps )
 
Theoremnndivsub 24896 Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  (
 ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A  /  C )  e.  NN  /\  A  <  B ) )  ->  ( ( B  /  C )  e. 
 NN 
 <->  ( ( B  -  A )  /  C )  e.  NN ) )
 
Theoremnndivlub 24897 A factor of a natural number cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  (
 ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  /  B )  e.  NN  ->  B  <_  A )
 )
 
SyntaxcgcdOLD 24898 Extend class notation to include the gdc function.
 class  gcd OLD ( A ,  B )
 
Definitiondf-gcdOLD 24899*  gcd OLD ( A ,  B ) is the largest natural number that evenly divides both  A and  B. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  gcd OLD ( A ,  B )  =  sup ( { x  e.  NN  |  ( ( A  /  x )  e.  NN  /\  ( B  /  x )  e. 
 NN ) } ,  NN ,  <  )
 
Theoremee7.2aOLD 24900 Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as  A mod  B. Here, just one subtraction step is proved to preserve the  gcd OLD. The  rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  ->  gcd OLD ( A ,  B )  = 
 gcd OLD ( A ,  ( B  -  A ) ) ) )
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