HomeHome Metamath Proof Explorer
Theorem List (p. 244 of 315)
< 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
(1-21493)
  Hilbert Space Explorer  Hilbert Space Explorer
(21494-23016)
  Users' Mathboxes  Users' Mathboxes
(23017-31457)
 

Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfindreccl 24301* 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 24302* 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 24303 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 24304 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 24305 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 24306 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 24307 Extend class notation to include the gdc function.
 class  gcd OLD ( A ,  B )
 
Definitiondf-gcdOLD 24308*  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 24309 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 ) ) ) )
 
18.11  Mathbox for Wolf Lammen

Most of the theorems in the section "Logical implication" are about handling chains of implications:  ph  ->  ( ps  ->  ( ch  ->  .... With respect to chains, an rich set of rules clarify

- how to swap antecedents (com12, ...);

- how to drop antecedents (ax-mp, pm2.43, ...);

- how to add antecedents (a1i, ...)

- how to replace an antecedent (syl, ...);

- how to replace a consequent (ax-mp, syl, ...);

- what is, when an antecedent equals the consequent (ax-1, id, ...).

In all these cases, the operands of the chain have no inner structure, or it is of no importance. These chains are called "simple" here.

There is less support, when the operands are structured themselves. Some kinds of inner structure involving the  -. operator are best handled by the symmetric operators  /\ and  \/. But a nested, simple chain has no such convenient replacement. I can focus on antecedents here, since a consequent representing a chain is, in conjunction with its antecedents, just an extended simple chain again.

The following theorems show, how operations on nested chains appear somehow mirrored: The minor premises of the syllogisms look reverted, in comparison to their normal counterparts, and while adding an antecedent to a chain via a1i 10 is easy, in nested chains they can be easily dropped.

 
Theoremwl-jarri 24310 Dropping a nested antecedent. This theorem is one of two reversions of ja 153. Since ja 153 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 2101 is an instance of this idea.

 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theoremwl-jarli 24311 Dropping a nested consequent. This theorem is one of two reversions of ja 153. Since ja 153 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 2101 is an instance of this idea.

 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremwl-mps 24312 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ph  ->  ps )  ->  th )
 
Theoremwl-syls1 24313 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  ->  ch )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ph  ->  ps )  ->  th )
 
Theoremwl-syls2 24314 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ps  ->  ch )  ->  th )
 
Theoremwl-adnestant 24315 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantALT 24316) (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   =>    |-  ( ( ph  ->  ps )  ->  ch )
 
Theoremwl-adnestantALT 24316 Proof of wl-adnestant 24315 not based on ax-3 7. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   =>    |-  ( ( ph  ->  ps )  ->  ch )
 
Theoremwl-adnestantd 24317 Deduction version of wl-adnestant 24315. Generalization of a2i 12, imim12i 53, imim1i 54 and imim2i 13, which can be proved by specializing its hypotheses, and some trivial rearrangements. This theorem clarifies in a more general way, under what conditions a wff may be introduced as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantdALTOLD 24318). (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  th ) )
 
Theoremwl-adnestantdALTOLD 24318 Proof of wl-adnestantd 24317 not based on ax-3 7. (Contributed by Wolf Lammen, 4-Oct-2013.) (Moved to embantd 50 in main set.mm and may be deleted by mathbox owner, WL. --NM 14-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  th ) )
 
Theoremwl-bitr1 24319 Closed form of bitri 240. Place before bitri 240. [ +33] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ps  <->  ch )  ->  ( ph 
 <->  ch ) ) )
 
Theoremwl-bitri 24320 An inference from transitive law for logical equivalence. [ -5] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  <->  ch )
 
Theoremwl-bitrd 24321 Deduction form of bitri 240. [ -7] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  <->  th ) )
 
Theoremwl-bibi1 24322 Theorem *4.86 of [WhiteheadRussell] p. 122. Place this (and the following theorems) after bitr1. [ +22] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  <->  ( ps  <->  ch ) ) )
 
Theoremwl-bibi1i 24323 Inference adding a biconditional to the right in an equivalence. Move after bibi1. [ -8] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ps )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theoremwl-bibi1d 24324 Deduction adding a biconditional to the right in an equivalence. Move after bibi1i. [ -9] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  th ) ) )
 
Theoremwl-bibi2d 24325 Deduction adding a biconditional to the left in an equivalence. Move after bibi1d. [ -25] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ( th  <->  ps )  <->  ( th  <->  ch ) ) )
 
Theoremwl-pm5.74lem 24326 Moving a common antecedent on one side of an equivalence. Place before pm5.74 235. [ +25] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( -.  ph  ->  ch )   =>    |-  (
 ( ph  ->  ps )  <->  ch )
 
Theoremwl-pm5.74 24327 Distribution of implication over biconditional. Theorem *5.74 of [ WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) Replace and move biimt 325.. albi 1551 before it. [ -22] (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
 
Theoremwl-pm5.32 24328 Distribution of implication over biconditional. Theorem *5.32 of [ WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Oct-2013.) Replace. [ -43] (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  /\ 
 ps )  <->  ( ph  /\  ch ) ) )
 
Theoremwl-bitr 24329 Theorem *4.22 of [WhiteheadRussell] p. 117. Replace. [ -4] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ps 
 <->  ch ) )  ->  ( ph  <->  ch ) )
 
Theoremwl-pm2.86i 24330 Inference based on pm2.86 94. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ph  ->  ch )
 )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremwl-dedlem0a 24331 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ( ( ch 
 ->  ph )  ->  ( ps  /\  ph ) ) ) )
 
18.12  Mathbox for Brendan Leahy
 
Theoremdvreasin 24332 Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.)
 |-  ( RR  _D  (arcsin  |`  ( -u 1 (,) 1 ) ) )  =  ( x  e.  ( -u 1 (,) 1 )  |->  ( 1 
 /  ( sqr `  (
 1  -  ( x ^ 2 ) ) ) ) )
 
Theoremdvreacos 24333 Real derivative of arccosine. (Contributed by Brendan Leahy, 3-Aug-2017.)
 |-  ( RR  _D  (arccos  |`  ( -u 1 (,) 1 ) ) )  =  ( x  e.  ( -u 1 (,) 1 )  |->  ( -u 1  /  ( sqr `  (
 1  -  ( x ^ 2 ) ) ) ) )
 
Theoremsspreima 24334 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
 |-  (
 ( Fun  F  /\  A  C_  B )  ->  ( `' F " A ) 
 C_  ( `' F " B ) )
 
Theoremareacirclem2 24335* Antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
 |-  ( R  e.  RR+  ->  ( RR  _D  ( t  e.  ( -u R (,) R )  |->  ( ( R ^ 2 )  x.  ( (arcsin `  (
 t  /  R )
 )  +  ( ( t  /  R )  x.  ( sqr `  (
 1  -  ( ( t  /  R ) ^ 2 ) ) ) ) ) ) ) )  =  ( t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  ( ( R ^
 2 )  -  (
 t ^ 2 ) ) ) ) ) )
 
Theoremareacirclem3 24336* Continuity of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
 |-  ( R  e.  RR+  ->  (
 t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  ( ( R ^
 2 )  -  (
 t ^ 2 ) ) ) ) )  e.  ( ( -u R (,) R ) -cn-> CC ) )
 
Theoremareacirclem4 24337* Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.)
 |-  (
 ( R  e.  RR  /\  0  <_  R )  ->  ( t  e.  ( -u R [,] R ) 
 |->  ( sqr `  (
 ( R ^ 2
 )  -  ( t ^ 2 ) ) ) )  e.  (
 ( -u R [,] R ) -cn-> CC ) )
 
Theoremareacirclem1 24338* Integrability of cross-section of circle. (Contributed by Brendan Leahy, 26-Aug-2017.)
 |-  (
 ( R  e.  RR  /\  0  <_  R )  ->  ( t  e.  ( -u R [,] R ) 
 |->  ( 2  x.  ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2
 ) ) ) ) )  e.  L ^1 )
 
Theoremareacirclem5 24339* Endpoint-inclusive continuity of antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 31-Aug-2017.)
 |-  ( R  e.  RR+  ->  (
 t  e.  ( -u R [,] R )  |->  ( ( R ^ 2
 )  x.  ( (arcsin `  ( t  /  R ) )  +  (
 ( t  /  R )  x.  ( sqr `  (
 1  -  ( ( t  /  R ) ^ 2 ) ) ) ) ) ) )  e.  ( (
 -u R [,] R ) -cn-> CC ) )
 
Theoremareacirclem6 24340* Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  (
 ( x ^ 2
 )  +  ( y ^ 2 ) ) 
 <_  ( R ^ 2
 ) ) }   =>    |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e. 
 RR )  ->  ( S " { t }
 )  =  if (
 ( abs `  t )  <_  R ,  ( -u ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2
 ) ) ) [,] ( sqr `  (
 ( R ^ 2
 )  -  ( t ^ 2 ) ) ) ) ,  (/) ) )
 
Theoremareacirc 24341* The area of a circle of radius  R is  pi  x.  R ^ 2. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  (
 ( x ^ 2
 )  +  ( y ^ 2 ) ) 
 <_  ( R ^ 2
 ) ) }   =>    |-  ( ( R  e.  RR  /\  0  <_  R )  ->  (area `  S )  =  ( pi  x.  ( R ^ 2 ) ) )
 
18.13  Mathbox for Frédéric Liné

In the sequel "JFM" is the "Journal of Formalized Mathematics". http://mizar.uwb.edu.pl/JFM/mmlident.html

"CAT1"; means Bylinski Czeslaw, Introduction to Categories and Functors, Journal of Formalized Mathematics, 1990, volume 1, no 2, pages 409--420

"CAT2"; means Bylinski Czeslaw, Subcategories and Products of Categories, Journal of Formalized Mathematics, 1990, volume 1, no 4, pages 725--732

"CLASSES1" means Grzegorz Bancerek, Tarski's Classes and Ranks, Journal of Formalized Mathematics, 1990, volume 1, no 3, pages 563--567

"CLASSES2" means Bogdan Nowak and Grzegorz Bancerek, Universal Classes, Journal of Formalized Mathematics, may-august 1990, volume 1, nb 3, pages 595--600

"Bourbaki" means Bourbaki's treatise. The book General Topology is called TG (for Topologie Générale). The book Set Theory is called E (for théorie des Ensembles).

The treatise is translated in English.

More precisely, here are two examples of references:

"Bourbaki E II.32" means Set Theory, chapter II, 32nd page, "Bourbaki TG III.1" means General Topology, chapter III, 1st page.

The references are given according to the French edition.

"Viro" means O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov, Elementary topology. Available on the net.

"Goldblatt" means Robert Goldblatt, Topoi, the categorial analysis of logic, revised edition, Dover publications, Mineola, New-York, 2006

"Gallier" means Jean H. Gallier, "Logic For Computer Science -- Foundations of Automatic Theorem Proving". A new edition must be published in 2014 at Dover.

"Harju" means Tero Harju, "Lecture Notes on SemiGroups", unpublished, 1996. Available on the net.

In the following notices "experimental" means I have not yet sufficiently used a definition to be sure it is correct. Anyway I'm not the owner of the definition and you can use it as you wish if you think it is correct or replace it by a definition of your own if you think it is not.

 
18.13.1  Theorems from other workspaces
 
Theoremtpssg 24342 A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by FL, 17-May-2016.)
 |-  ( ph  ->  A  e.  E )   &    |-  ( ph  ->  B  e.  F )   &    |-  ( ph  ->  C  e.  G )   =>    |-  ( ph  ->  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D ) )
 
18.13.2  Propositional and predicate calculus
 
Theoremneleq12d 24343 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  e/  C  <->  B  e/  D ) )
 
Theoremr19.26-2a 24344 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 20-May-2016.)
 |-  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ( ph  /\  ps )  <->  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ph  /\  A. x  e.  A  A. y  e.  B  A. z  e.  C  ps ) )
 
Theoremreubidvag 24345* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by FL, 17-Nov-2014.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( E! x  e.  A  ps 
 <->  E! x  e.  B  ch ) )
 
Theoremintn3an1d 24346 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  ( ps  /\ 
 ch  /\  th )
 )
 
Theoremintn3an2d 24347 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  ( ch  /\ 
 ps  /\  th )
 )
 
Theoremintn3an3d 24348 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  ( ch  /\ 
 th  /\  ps )
 )
 
Theoremand4as 24349 A consequence of  /\ associativity in a triple conjunct. (Contributed by FL, 14-Jul-2007.)
 |-  (
 ( ph  /\  ps  /\  ( ch  /\  th )
 ) 
 <->  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )
 
Theoremand4com 24350 A consequence of  /\ associativity in a triple conjunct. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  th )
 ) 
 <->  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )
 
Theoremanddi2 24351 Conjunction of triple disjunctions. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  (
 ( ( ph  \/  ps 
 \/  ch )  /\  ( th  \/  ta  \/  et ) )  <->  ( ( (
 ph  /\  th )  \/  ( ph  /\  ta )  \/  ( ph  /\  et ) )  \/  (
 ( ps  /\  th )  \/  ( ps  /\  ta )  \/  ( ps 
 /\  et ) )  \/  ( ( ch  /\  th )  \/  ( ch 
 /\  ta )  \/  ( ch  /\  et ) ) ) )
 
Theoremcondis 24352 Proof by contradiction combined with a disjunction. (Contributed by FL, 20-Apr-2011.)
 |-  ( ph  ->  ps )   &    |-  ( -.  ph  ->  ch )   =>    |-  ( ps  \/  ch )
 
Theoremcondisd 24353 Proof by contradiction combined with a disjunction. (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 -.  ps )  ->  th )   =>    |-  ( ph  ->  ( ch  \/  th ) )
 
Theoremeeeeanv 24354* Rearrange existential quantifiers. (Contributed by FL, 14-Jul-2007.)
 |-  ( E. w E. x E. y E. z ( (
 ph  /\  ps  /\  ch )  /\  th )  <->  ( ( E. w ph  /\  E. x ps  /\  E. y ch )  /\  E. z th ) )
 
Theoremnegcmpprcal1 24355 Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
 |-  ( -.  E. x  e.  A  A. y  e.  B  (
 ph  ->  ps )  <->  A. x  e.  A  E. y  e.  B  ( ph  /\  -.  ps ) )
 
Theoremnegcmpprcal2 24356 Negation of a complex predicated inequality. (Contributed by FL, 31-Jul-2009.)
 |-  ( -.  E. x  e.  A  A. y  e.  B  C  =/=  D  <->  A. x  e.  A  E. y  e.  B  C  =  D )
 
Theoremeqriv2 24357 Infer equality of classes from equivalence of membership. (Contributed by FL, 22-Nov-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  ( x  e.  A  <->  x  e.  B )   =>    |-  A  =  B
 
Theoremaltdftru 24358 Alternate definition of true. In fact any tautology is a definition of true. (Contributed by FL, 23-Mar-2011.)
 |-  (  T. 
 <->  ( ph  \/  -.  ph ) )
 
Theoremtrant 24359 A true antecedent can be removed. (Contributed by FL, 16-Apr-2012.)
 |-  (
 (  T.  ->  ph )  <->  ph )
 
Theoremvutr 24360 Vacuous universal quantification is true. (Contributed by FL, 16-Apr-2012.)
 |-  (  T. 
 <-> 
 A. x  e.  (/)  ph )
 
Theoremtrcrm 24361 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.)
 |-  (
 (  T.  /\  ph )  <->  ph )
 
Theoremtnf 24362 True is not false. (Contributed by FL, 20-Mar-2011.)
 |-  (  T. 
 <->  -.  F.  )
 
Theoremfacrm 24363 False can be removed from a disjunction. (Contributed by FL, 20-Mar-2011.)
 |-  (
 (  F.  \/  ph )  <->  ph )
 
Theoremfordisxex 24364 If  ( ph  \/  ps ) is true for all  x and  ps is not true for all  x then  ph is true for some  x. (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( A. x  e.  A  ( ph  \/  ps )  /\  -.  A. x  e.  A  ps )  ->  E. x  e.  A  ph )
 
Theoremfates 24365* Equivalence of  A. and  E. in the case of quantifiers restricted to a singleton. (Contributed by FL, 1-Jun-2011.)
 |-  A  e.  B   =>    |-  ( A. x  e. 
 { A } ph  <->  E. x  e.  { A } ph )
 
Theoremfatesg 24366* Equivalence of  A. and  E. in the case of quantifiers restricted to a singleton. (Contributed by FL, 1-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x  e.  { A } ph  <->  E. x  e.  { A } ph ) )
 
Theoremr19.2zr 24367* Quantifying a hypothesis with a universal restricted quantifier. (Contributed by FL, 19-Sep-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  ps )
 
Theoremr19.2zrr 24368* Removing a universal restricted quantifier when the variable doesn't occur in the proposition. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( A  =/=  (/)  /\  A. x  e.  A  ph )  -> 
 ph )
 
Theoremrexlimib 24369* Removal of an universal restricted quantifier in an antecedent. See also reximdva0 3466. (Contributed by FL, 19-Apr-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ps   &    |-  ( x  e.  A  ->  ( ph  ->  ps ) )   =>    |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  ps )
 
Theoremeqint 24370* To prove that a set  A is the finest one that has the property  ph, prove that  A is a part of all sets that has this property and that  A has also that property. (Contributed by FL, 21-Apr-2012.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ps   &    |-  ( ph  ->  A  C_  x )   =>    |-  ( A  e.  V  ->  A  =  |^| { x  |  ph } )
 
Theoremeqintg 24371* To prove that a set  A is the finest one that has the property  ph prove that  A is a part of all sets that has this property and that  A has also that property. (Contributed by FL, 15-Oct-2012.)
 |-  ( x  =  A  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\ 
 ps )  ->  A  C_  x )   =>    |-  ( ( ph  /\  A  e.  V )  ->  A  =  |^| { x  |  ps } )
 
Theoremalexeqd 24372* Two ways to express substitution of 
A for  x in  ph. (Contributed by FL, 4-Jun-2012.)
 |-  ( A  e.  V  ->  (
 A. x ( x  =  A  ->  ph )  <->  E. x ( x  =  A  /\  ph )
 ) )
 
Theoremrspc2edv 24373* 2-variable restricted existential specialization, using implicit substitution. (rspc2ev 2892 with an antecedent.) (Contributed by FL, 2-Jul-2012.)
 |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  (
 y  =  B  ->  ( th  <->  ch ) )   =>    |-  ( ( ph  /\  A  e.  C  /\  B  e.  D )  ->  ( ch  ->  E. x  e.  C  E. y  e.  D  ps ) )
 
Theorempm11.53g 24374 Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by FL, 27-Oct-2013.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x A. y (
 ph  ->  ps )  <->  ( E. x ph 
 ->  A. y ps )
 )
 
Theoremeqvinopb 24375* A variable introduction law for ordered triples. See eqvinop 4249. (Contributed by FL, 6-Nov-2013.)
 |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =  <. <. B ,  C >. ,  D >.  <->  E. x E. y E. z ( A  =  <.
 <. x ,  y >. ,  z >.  /\  <. <. x ,  y >. ,  z >.  = 
 <. <. B ,  C >. ,  D >. ) )
 
Theoremcopsexgb 24376* Substitution of class  A for ordered triple  <. <. x ,  y >. ,  z
>.. See copsexg 4250. (Contributed by FL, 10-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( A  =  <. <. x ,  y >. ,  z >.  ->  ( ph  <->  E. x E. y E. z ( A  =  <.
 <. x ,  y >. ,  z >.  /\  ph )
 ) )
 
Theoremdifeq12dOLD 24377 Deduction adding difference to the right in a class equality. (Moved into main set.mm as difeq12d 3295 and may be deleted by mathbox owner, FL. --NM 2-Jul-2014.) (Contributed by FL, 29-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theorem3netr3 24378 Inequality. (Contributed by FL, 30-May-2014.)
 |-  A  =/=  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  =/=  D
 
Theoremsbcbidv2 24379* Formula-building deduction rule for class substitution with different classes. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. B  /  x ].
 ch ) )
 
18.13.3  Linear temporal logic

Propositional Linear temporal logic (LTL) is a kind of modal logic. It is composed of the axioms of classical logic plus the axioms ax-ltl1 24384, ax-ltl2 24385, ax-ltl3 24386, ax-ltl4 , ax-lmp 24388, and ax-nmp 24389. In classical logic, propositions don't depend on the time. In LTL the "world" evolves. We will imagine the world as a sequence of states with a first state and future states. Instead of state I will also use the term "step" to emphasize that LTL is used to formalize the evolution of process in a computer. A proposition that is true in one state of the "world" may be false in the next one. The proposition  [.] ph means  ph is true in every state of the world, in the first state as well as in the future states. It is read "
ph is always true " or " ph always holds ". The proposition  () ph means  ph is true in the next state of the world. The proposition 
<> ph means that  ph is true in one state of the world at least but we don't know exactly which one. It can be the first state, it can be a future state. It is read " ph is eventually true " or " ph eventually holds". When no operator is used in front of a proposition, it means that  ph is unconditionnaly true or that it is true in the current state ( depending on the context).  ph  until  ps means  ph is true in every state of the world until  ps is true.

 
Syntaxwbox 24380 An always true proposition is well formed.
 wff  [.] ph
 
Syntaxwdia 24381 An eventually true proposition is well formed.
 wff  <> ph
 
Syntaxwcirc 24382 A proposition true in the next step is well formed.
 wff  ()
 ph
 
Syntaxwunt 24383 The proposition " ph is true until  ps is true " is well formed.
 wff  ( ph  until  ps )
 
Axiomax-ltl1 24384 If  ( ph  ->  ps ) and  ph always hold then  ps always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( [.] ( ph  ->  ps )  ->  ( [.] ph  ->  [.]
 ps ) )
 
Axiomax-ltl2 24385  ph doesn't hold in the next step iff in the next step 
-.  ph holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  () ph  <->  ()  -.  ph )
 
Axiomax-ltl3 24386 If, in the next step,  ph  ->  ps and  ph hold then, in the next step,  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( () ( ph  ->  ps )  ->  ( () ph  ->  ()
 ps ) )
 
Axiomax-ltl4 24387 Suppose that it is always true that if  ph is true in the current step then  ph is true in the next step. Suppose that  ph is true in the first step. Then  ph is always true. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( [.] ( ph  ->  ()
 ph )  /\  ph )  ->  [.] ph )
 
Axiomax-lmp 24388 If  ph is a theorem then it always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ph   =>    |- 
 [.] ph
 
Axiomax-nmp 24389 If  ph is a theorem then it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ph   =>    |- 
 () ph
 
Definitiondf-dia 24390  ph eventually holds iff it is not true that  -.  ph always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( <> ph  <->  -. 
 [.]  -.  ph )
 
Theoremimpbox 24391 If  ph  ->  ps is unconditionally true and if  ph is always true then  ps is always true. (Contributed by FL, 22-Feb-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( [.] ph  ->  [.]
 ps )
 
Theorembibox 24392 If  ph  <->  ps is unconditionally true then  ph is always true is equivalent to  ps is always true. (Contributed by FL, 22-Feb-2011.)
 |-  ( ph 
 <->  ps )   =>    |-  ( [.] ph  <->  [.] ps )
 
Theoremimpxt 24393 If  ph  ->  ps holds unconditionally and if  ph holds in the next state then  ps holds in the next state. (Contributed by FL, 20-Mar-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( () ph  ->  ()
 ps )
 
Theorembinxt 24394 If  ph  <->  ps holds unconditionally then  ph holds in the next state of the world iff  ps holds in the next state. (Contributed by FL, 20-Mar-2011.)
 |-  ( ph 
 <->  ps )   =>    |-  ( () ph  <->  () ps )
 
Theoremnxtor 24395  ( ph  \/  ps ) holds in the next step iff  ph holds in the next step or  ps holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( () ( ph  \/  ps ) 
 <->  ( () ph  \/  () ps ) )
 
Theoremnxtand 24396  ( ph  /\ 
ps ) holds in the next step iff  ph holds in the next step and  ps holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( () ( ph  /\  ps ) 
 <->  ( () ph  /\  () ps ) )
 
Theoremboxeq 24397  ph holds now and will always hold in the future iff it is not true that  -.  ph holds now or sometimes in the future. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( [.] ph  <->  -.  <>  -.  ph )
 
Theoremdiaimi 24398 If  ph implies  ps unconditionally, then if  ph eventually holds so does  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph  ->  ps )   =>    |-  ( <> ph  ->  <> ps )
 
Theorembidia 24399 If  ph  <->  ps holds then  ph eventually holds iff  ps eventually holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   =>    |-  ( <> ph  <->  <> ps )
 
Theoremnotev 24400 It's false that  ph eventually holds iff  -.  ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -. 
 <> ph  <->  [.]  -.  ph )
    < 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 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31457
  Copyright terms: Public domain < Previous  Next >