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Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremovoliunnfl 25001* ovoliun 18880 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 21-Nov-2017.)
 |-  (
 ( f  Fn  NN  /\ 
 A. n  e.  NN  ( ( f `  n )  C_  RR  /\  ( vol * `  (
 f `  n )
 )  e.  RR )
 )  ->  ( vol * `
  U_ m  e.  NN  ( f `  m ) )  <_  sup ( ran  seq  1 (  +  ,  ( m  e.  NN  |->  ( vol * `  (
 f `  m )
 ) ) ) , 
 RR* ,  <  ) )   =>    |-  ( ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremex-ovoliunnfl 25002* Demonstration of ovoliunnfl 25001. (Contributed by Brendan Leahy, 21-Nov-2017.)
 |-  (
 ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremitg2addnclem 25003* An alternate expression for the 
S.2 integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( E. y  e.  RR+  ( g  o F ( z  e.  RR ,  w  e.  RR  |->  if ( z  =  0 ,  0 ,  ( z  +  w ) ) ) ( RR  X.  { y } ) )  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }   =>    |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  = 
 sup ( L ,  RR*
 ,  <  ) )
 
Theoremitg2addnclem2 25004* Lemma for itg2addnc 25005. The function described is a simple function. (Contributed by Brendan Leahy, 29-Oct-2017.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   =>    |-  ( ( ( ph  /\  h  e.  dom  S.1 )  /\  v  e.  RR+ )  ->  ( x  e. 
 RR  |->  if ( ( ( ( ( |_ `  (
 ( F `  x )  /  ( v  / 
 3 ) ) )  -  1 )  x.  ( v  /  3
 ) )  <_  ( h `  x )  /\  ( h `  x )  =/=  0 ) ,  ( ( ( |_ `  ( ( F `  x )  /  (
 v  /  3 )
 ) )  -  1
 )  x.  ( v 
 /  3 ) ) ,  ( h `  x ) ) )  e.  dom  S.1 )
 
Theoremitg2addnc 25005 Alternate proof of itg2add 19130 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 19079, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 8077, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  G : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  ( S.2 `  G )  e.  RR )   =>    |-  ( ph  ->  ( S.2 `  ( F  o F  +  G ) )  =  (
 ( S.2 `  F )  +  ( S.2 `  G ) ) )
 
Theoremitg2gt0cn 25006* itg2gt0 19131 holds on functions continuous on an open interval in the absence of ax-cc 8077. The fourth hypothesis is made unnecessary by the continuity hypothesis. (Contributed by Brendan Leahy, 16-Nov-2017.)
 |-  ( ph  ->  X  <  Y )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  x  e.  ( X (,) Y ) ) 
 ->  0  <  ( F `
  x ) )   &    |-  ( ph  ->  ( F  |`  ( X (,) Y ) )  e.  (
 ( X (,) Y ) -cn-> CC ) )   =>    |-  ( ph  ->  0  <  ( S.2 `  F ) )
 
Theoremibladdnclem 25007* Lemma for ibladdnc 25008; cf ibladdlem 19190, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 25005. (Contributed by Brendan Leahy, 31-Oct-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  D  =  ( B  +  C ) )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ph  ->  (
 S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 ) ) )  e.  RR )   &    |-  ( ph  ->  ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) ) )  e. 
 RR )   =>    |-  ( ph  ->  ( S.2 `  ( x  e. 
 RR  |->  if ( ( x  e.  A  /\  0  <_  D ) ,  D ,  0 ) ) )  e.  RR )
 
Theoremibladdnc 25008* Choice-free analogue of itgadd 19195. A measurability hypothesis is necessitated by the loss of mbfadd 19032; for large classes of functions, such as continuous functions, it should be relatively easy to show. (Contributed by Brendan Leahy, 1-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e.  L ^1 )
 
Theoremitgaddnclem1 25009* Lemma for itgaddnc 25011; cf. itgaddlem1 19193. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  0  <_  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  C )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddnclem2 25010* Lemma for itgaddnc 25011; cf. itgaddlem2 19194. (Contributed by Brendan Leahy, 10-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddnc 25011* Choice-free analogue of itgadd 19195. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremiblsubnc 25012* Choice-free analogue of iblsub 19192. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  L ^1 )
 
Theoremitgsubnc 25013* Choice-free analogue of itgsub 19196. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  S. A ( B  -  C )  _d x  =  ( S. A B  _d x  -  S. A C  _d x ) )
 
Theoremiblabsnclem 25014* Lemma for iblabsnc 25015; cf. iblabslem 19198. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( abs `  ( F `  B ) ) ,  0 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  ( F `  B ) )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  B )  e. 
 RR )   =>    |-  ( ph  ->  ( G  e. MblFn  /\  ( S.2 `  G )  e.  RR ) )
 
Theoremiblabsnc 25015* Choice-free analogue of iblabs 19199. As with ibladdnc 25008, a measurability hypothesis is needed. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B )
 )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B ) )  e.  L ^1 )
 
Theoremiblmulc2nc 25016* Choice-free analogue of iblmulc2 19201. (Contributed by Brendan Leahy, 17-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e.  L ^1 )
 
Theoremitgmulc2nclem1 25017* Lemma for itgmulc2nc 25019; cf. itgmulc2lem1 19202. (Contributed by Brendan Leahy, 17-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  0 
 <_  C )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgmulc2nclem2 25018* Lemma for itgmulc2nc 25019; cf. itgmulc2lem2 19203. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgmulc2nc 25019* Choice-free analogue of itgmulc2 19204. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgabsnc 25020* Choice-free analogue of itgabs 19205. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B )
 )  e. MblFn )   &    |-  ( ph  ->  ( x  e.  A  |->  ( ( * `  S. A B  _d x )  x.  B ) )  e. MblFn )   =>    |-  ( ph  ->  ( abs `  S. A B  _d x )  <_  S. A ( abs `  B )  _d x )
 
Theorembddiblnc 25021* Choice-free proof of bddibl 19210. (Contributed by Brendan Leahy, 2-Nov-2017.) (Revised by Brendan Leahy, 6-Nov-2017.)
 |-  (
 ( F  e. MblFn  /\  ( vol `  dom  F )  e.  RR  /\  E. x  e.  RR  A. y  e.  dom  F ( abs `  ( F `  y
 ) )  <_  x )  ->  F  e.  L ^1 )
 
Theoremcnicciblnc 25022 Choice-free proof of cniccibl 19211. (Contributed by Brendan Leahy, 2-Nov-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  F  e.  ( ( A [,] B ) -cn-> CC ) )  ->  F  e.  L ^1 )
 
Theoremitggt0cn 25023* itggt0 19212 holds for continuous functions in the absence of ax-cc 8077. (Contributed by Brendan Leahy, 16-Nov-2017.)
 |-  ( ph  ->  X  <  Y )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  ( X (,) Y ) )  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( X (,) Y )
 -cn-> CC ) )   =>    |-  ( ph  ->  0  <  S. ( X (,) Y ) B  _d x )
 
Theoremftc1cnnclem 25024* Lemma for ftc1cnnc 25025; cf. ftc1lem4 19402. The stronger assumptions of ftc1cn 19406 are exploited to make use of weaker theorems. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  c  e.  ( A (,) B ) )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { c }
 )  |->  ( ( ( G `  z )  -  ( G `  c ) )  /  ( z  -  c
 ) ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  y  e.  ( A (,) B ) ) 
 ->  ( ( abs `  (
 y  -  c ) )  <  R  ->  ( abs `  ( ( F `  y )  -  ( F `  c ) ) )  <  E ) )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( X  -  c ) )  <  R )   &    |-  ( ph  ->  Y  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( Y  -  c ) )  <  R )   =>    |-  ( ( ph  /\  X  <  Y )  ->  ( abs `  ( ( ( ( G `  Y )  -  ( G `  X ) )  /  ( Y  -  X ) )  -  ( F `  c ) ) )  <  E )
 
Theoremftc1cnnc 25025* Choice-free proof of ftc1cn 19406. (Contributed by Brendan Leahy, 20-Nov-2017.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  F  e.  L ^1 )   =>    |-  ( ph  ->  ( RR  _D  G )  =  F )
 
Theoremdvreasin 25026 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 25027 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 ) ) ) ) )
 
Theoremareacirclem2 25028* 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 25029* 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 25030* 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 25031* 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 25032* 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 25033* 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 25034* 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 25035 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 25036 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 25037 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 25038* 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 25039 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 25040 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 25041 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 25042 A consequence of  /\ associativity in a triple conjunct. (Contributed by FL, 14-Jul-2007.)
 |-  (
 ( ph  /\  ps  /\  ( ch  /\  th )
 ) 
 <->  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )
 
Theoremand4com 25043 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 25044 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 25045 Proof by contradiction combined with a disjunction. (Contributed by FL, 20-Apr-2011.)
 |-  ( ph  ->  ps )   &    |-  ( -.  ph  ->  ch )   =>    |-  ( ps  \/  ch )
 
Theoremcondisd 25046 Proof by contradiction combined with a disjunction. (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 -.  ps )  ->  th )   =>    |-  ( ph  ->  ( ch  \/  th ) )
 
Theoremeeeeanv 25047* 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 25048 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 25049 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 25050 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 25051 Alternate definition of true. In fact any tautology is a definition of true. (Contributed by FL, 23-Mar-2011.)
 |-  (  T. 
 <->  ( ph  \/  -.  ph ) )
 
Theoremtrant 25052 A true antecedent can be removed. (Contributed by FL, 16-Apr-2012.)
 |-  (
 (  T.  ->  ph )  <->  ph )
 
Theoremvutr 25053 Vacuous universal quantification is true. (Contributed by FL, 16-Apr-2012.)
 |-  (  T. 
 <-> 
 A. x  e.  (/)  ph )
 
Theoremtrcrm 25054 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.)
 |-  (
 (  T.  /\  ph )  <->  ph )
 
Theoremtnf 25055 True is not false. (Contributed by FL, 20-Mar-2011.)
 |-  (  T. 
 <->  -.  F.  )
 
Theoremfacrm 25056 False can be removed from a disjunction. (Contributed by FL, 20-Mar-2011.)
 |-  (
 (  F.  \/  ph )  <->  ph )
 
Theoremfordisxex 25057 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 25058* 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 25059* 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 25060* 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 25061* 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 25062* Removal of a universal restricted quantifier in an antecedent. See also reximdva0 3479. (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 25063* 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 25064* 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 25065* 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 25066* 2-variable restricted existential specialization, using implicit substitution. (rspc2ev 2905 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 25067 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 25068* A variable introduction law for ordered triples. See eqvinop 4267. (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 25069* Substitution of class  A for ordered triple  <. <. x ,  y >. ,  z
>.. See copsexg 4268. (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 25070 Deduction adding difference to the right in a class equality. (Moved into main set.mm as difeq12d 3308 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 25071 Inequality. (Contributed by FL, 30-May-2014.)
 |-  A  =/=  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  =/=  D
 
Theoremsbcbidv2 25072* 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 25077, ax-ltl2 25078, ax-ltl3 25079, ax-ltl4 , ax-lmp 25081, and ax-nmp 25082. 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 25073 An always true proposition is well-formed.
 wff  [.] ph
 
Syntaxwdia 25074 An eventually true proposition is well-formed.
 wff  <> ph
 
Syntaxwcirc 25075 A proposition true in the next step is well-formed.
 wff  ()
 ph
 
Syntaxwunt 25076 The proposition " ph is true until  ps is true " is well-formed.
 wff  ( ph  until  ps )
 
Axiomax-ltl1 25077 If  ( ph  ->  ps ) and  ph always hold then  ps always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( [.] ( ph  ->  ps )  ->  ( [.] ph  ->  [.]
 ps ) )
 
Axiomax-ltl2 25078  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 25079 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 25080 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 25081 If  ph is a theorem, then it always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ph   =>    |- 
 [.] ph
 
Axiomax-nmp 25082 If  ph is a theoremm then it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ph   =>    |- 
 () ph
 
Definitiondf-dia 25083  ph eventually holds iff it is not true that  -.  ph always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( <> ph  <->  -. 
 [.]  -.  ph )
 
Theoremimpbox 25084 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 25085 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 25086 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 25087 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 25088  ( 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 25089  ( 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 25090  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 25091 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 25092 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 25093 It's false that  ph eventually holds iff  -.  ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -. 
 <> ph  <->  [.]  -.  ph )
 
Theoremnotal 25094 It's false that  ph always holds iff  -.  ph eventually holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  [.] ph  <->  <>  -.  ph )
 
Theoremltl4ev 25095 The contrapositive of ax-ltl4 25080. If the truth of  ph in each step implies it is true in the previous step, and  ph is eventually true, then  ph is true in the first step. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  (
 ( [.] ( () ph  -> 
 ph )  /\  <> ph )  -> 
 ph )
 
Axiomax-ltl5 25096  ph holds until  ps iff  ps holds in the current step or  ph holds in the current step and in the next step  ph holds until  ps. (Contributed by FL, 27-Feb-2011.)
 |-  (
 ( ph  until  ps )  <->  ( ps  \/  ( ph  /\ 
 () ( ph  until  ps )
 ) ) )
 
Axiomax-ltl6 25097 If  ph holds until  ps then eventually  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( ph  until  ps )  -> 
 <> ps )
 
Theoremnopsthph 25098 If  ps doesn't hold in the first step and  ph holds until  ps then  ph holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  (
 ( -.  ps  /\  ( ph  until  ps ) )  ->  ph )
 
Theoremphthps 25099 If  ph doesn't hold in the current step and  ph holds until  ps then  ps holds in the current step. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( -.  ph  /\  ( ph  until  ps ) )  ->  ps )
 
Theoremimunt 25100 If  ps is true, then  ph is true until  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  ( ph  until  ps )
 )
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