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Theorem List for Metamath Proof Explorer - 25501-25600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiblsubnc 25501* Choice-free analogue of iblsub 19274. (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 25502* Choice-free analogue of itgsub 19278. (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 25503* Lemma for iblabsnc 25504; cf. iblabslem 19280. (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 25504* Choice-free analogue of iblabs 19281. As with ibladdnc 25497, 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 25505* Choice-free analogue of iblmulc2 19283. (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 25506* Lemma for itgmulc2nc 25508; cf. itgmulc2lem1 19284. (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 25507* Lemma for itgmulc2nc 25508; cf. itgmulc2lem2 19285. (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 25508* Choice-free analogue of itgmulc2 19286. (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 25509* Choice-free analogue of itgabs 19287. (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 25510* Choice-free proof of bddibl 19292. (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 25511 Choice-free proof of cniccibl 19293. (Contributed by Brendan Leahy, 2-Nov-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  F  e.  ( ( A [,] B ) -cn-> CC ) )  ->  F  e.  L ^1 )
 
Theoremitggt0cn 25512* itggt0 19294 holds for continuous functions in the absence of ax-cc 8148. (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 25513* Lemma for ftc1cnnc 25514; cf. ftc1lem4 19484. The stronger assumptions of ftc1cn 19488 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 25514* Choice-free proof of ftc1cn 19488. (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 25515 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 25516 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 25517* 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 25518* 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 25519* 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 25520* 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 25521* 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 25522* 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 25523* 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 Jeff Hankins
 
18.13.1  Miscellany
 
Theorema1i13 25524 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theorema1i4 25525 Add an antecedent to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i14 25526 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i24 25527 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i34 25528 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremimp5gOLD 25529 An importation inference. (Moved into main set.mm as imp5g 583 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ( ch  /\  th )  /\  ta )  ->  et )
 )
 
Theoremimp55OLD 25530 An importation inference. (Moved into main set.mm as imp55 584 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  /\  ta )  ->  et )
 
Theoremimp511OLD 25531 An importation inference. (Moved into main set.mm as imp511 585 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  (
 ( ps  /\  ( ch  /\  th ) ) 
 /\  ta ) )  ->  et )
 
Theoremexp5d 25532 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  ( ( th  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5g 25533 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ps )  ->  ( ( ( ch 
 /\  th )  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5j 25534 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ( ps  /\  ch )  /\  th )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5k 25535 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5l 25536 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp56 25537 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp58 25538 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ( ch 
 /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp510 25539 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ( ps  /\  ch )  /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp511 25540 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp512 25541 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ( ( ps  /\  ch )  /\  ( th  /\  ta ) ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
TheoremmtordOLD 25542 A modus tollens deduction involving disjunction. (Moved into main set.mm as mtord 641 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ps  ->  ( ch  \/  th )
 ) )   =>    |-  ( ph  ->  -.  ps )
 
Theorem3com12d 25543 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
 |-  ( ph  ->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ( ch  /\  ps  /\  th ) )
 
Theoremimp5p 25544 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch  /\  th 
 /\  ta )  ->  et )
 ) )
 
Theoremimp5q 25545 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ch 
 /\  th  /\  ta )  ->  et ) )
 
Theoremecase13d 25546 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ch  \/  ps 
 \/  th ) )   =>    |-  ( ph  ->  ps )
 
TheoremeqeuOLD 25547* A condition which implies existential uniqueness. (Moved into main set.mm as eqeu 3012 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps  /\ 
 A. x ( ph  ->  x  =  A ) )  ->  E! x ph )
 
Theoremsubtr 25548 Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x Y   &    |-  F/_ x Z   &    |-  ( x  =  A  ->  X  =  Y )   &    |-  ( x  =  B  ->  X  =  Z )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  Y  =  Z ) )
 
Theoremsubtr2 25549 Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/ x ps   &    |-  F/ x ch   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  =  B  ->  ( ps  <->  ch ) ) )
 
TheoremcnvresimaOLD 25550 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) (Moved to cnvresima 5241 in main set.mm and may be deleted by mathbox owner, JGH. --NM 23-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( `' ( F  |`  A )
 " B )  =  ( ( `' F " B )  i^i  A )
 
Theoremtrer 25551* A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( A. a A. b A. c ( ( a 
 .<_  b  /\  b  .<_  c )  ->  a  .<_  c )  ->  (  .<_  i^i  `'  .<_  )  Er  dom  (  .<_  i^i  `'  .<_  ) )
 
Theoremelicc3 25552 An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  B  /\  ( C  =  A  \/  ( A  <  C  /\  C  <  B )  \/  C  =  B ) ) ) )
 
TheoremccidOLD 25553 A closed interval with identical lower and upper bounds is a singleton. (Moved into main set.mm as iccid 10790 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
 
TheoremioodisjOLD 25554 If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Moved into main set.mm as ioodisj 10854 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* ) )  /\  B  <_  C )  ->  (
 ( A (,) B )  i^i  ( C (,) D ) )  =  (/) )
 
Theoremfinminlem 25555* A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  Fin  ph  ->  E. x ( ph  /\  A. y
 ( ( y  C_  x  /\  ps )  ->  x  =  y )
 ) )
 
Theoremdivcan7OLD 25556 Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved to divcan7 9556 in main set.mm and may be deleted by mathbox owner, JGH. --NM 21-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) 
 /\  ( C  e.  CC  /\  C  =/=  0
 ) )  ->  (
 ( A  /  C )  /  ( B  /  C ) )  =  ( A  /  B ) )
 
TheoreminfleOLD 25557* If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.) (Moved to infmrlb 9822 in main set.mm and may be deleted by mathbox owner, JGH. --NM 21-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( S  C_  RR  /\ 
 E. x  e.  RR  A. y  e.  S  x  <_  y  /\  A  e.  S )  ->  sup ( S ,  RR ,  `'  <  )  <_  A )
 
Theoremgtinf 25558* Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  (
 ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. x  e.  RR  A. y  e.  S  x  <_  y
 )  /\  ( A  e.  RR  /\  sup ( S ,  RR ,  `'  <  )  <  A ) )  ->  E. z  e.  S  z  <  A )
 
Theoremopnrebl 25559* A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
 |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  E. y  e.  RR+  ( ( x  -  y ) (,) ( x  +  y )
 )  C_  A )
 )
 
Theoremopnrebl2 25560* A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
 |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_  y  /\  ( ( x  -  z ) (,) ( x  +  z )
 )  C_  A )
 ) )
 
Theoremdivides2OLD 25561 One nonzero integer divides another integer if and only if their quotient is an integer. (Moved to cnvresima 5241 in main set.mm and may be deleted by mathbox owner, JGH. --NM 28-Feb-2014.) (Contributed by Jeff Hankins, 29-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
 
Theoremnn0prpwlem 25562* Lemma for nn0prpw 25563. Use strong induction to show that every natural number has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.)
 |-  ( A  e.  NN  ->  A. k  e.  NN  (
 k  <  A  ->  E. p  e.  Prime  E. n  e.  NN  -.  ( ( p ^ n ) 
 ||  k  <->  ( p ^ n )  ||  A ) ) )
 
Theoremnn0prpw 25563* Two nonnegative integers are the same if and only if they are divisible by the same prime powers. (Contributed by Jeff Hankins, 29-Sep-2013.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  =  B  <->  A. p  e.  Prime  A. n  e.  NN  ( ( p ^ n )  ||  A 
 <->  ( p ^ n )  ||  B ) ) )
 
TheoremqredeqOLD 25564 Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved into main set.mm as qredeq 12876 and may be deleted by mathbox owner, JGH. --NM 13-Oct-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( M  e.  ZZ  /\  N  e.  NN  /\  ( M  gcd  N )  =  1 )  /\  ( P  e.  ZZ  /\  Q  e.  NN  /\  ( P  gcd  Q )  =  1 )  /\  ( M  /  N )  =  ( P  /  Q ) )  ->  ( M  =  P  /\  N  =  Q ) )
 
TheoremqredeuOLD 25565* Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.) (Moved into main set.mm as qredeq 12876 and may be deleted by mathbox owner, JGH. --NM 13-Oct-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  QQ  ->  E! x  e.  ( ZZ 
 X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) ) )
 
18.13.2  Basic topological facts
 
Theoremtopbnd 25566 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( cls `  J ) `  A )  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J ) `  A )  \  ( ( int `  J ) `  A ) ) )
 
Theoremopnbnd 25567 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( A  e.  J  <->  ( A  i^i  ( ( ( cls `  J ) `  A )  i^i  ( ( cls `  J ) `  ( X  \  A ) ) ) )  =  (/) ) )
 
Theoremcldbnd 25568 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  ( ( ( cls `  J ) `  A )  i^i  (
 ( cls `  J ) `  ( X  \  A ) ) )  C_  A ) )
 
Theoremntruni 25569* A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  O  C_  ~P X )  ->  U_ o  e.  O  ( ( int `  J ) `  o )  C_  ( ( int `  J ) `  U. O ) )
 
Theoremclsun 25570 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( ( cls `  J ) `  ( A  u.  B ) )  =  ( ( ( cls `  J ) `  A )  u.  ( ( cls `  J ) `  B ) ) )
 
Theoremclsint2 25571* The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  C  C_  ~P X )  ->  ( ( cls `  J ) `  |^| C )  C_  |^|_ c  e.  C  ( ( cls `  J ) `  c ) )
 
Theoremopnregcld 25572* A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( cls `  J ) `  (
 ( int `  J ) `  A ) )  =  A  <->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) ) )
 
Theoremcldregopn 25573* A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( ( ( int `  J ) `  (
 ( cls `  J ) `  A ) )  =  A  <->  E. c  e.  ( Clsd `  J ) A  =  ( ( int `  J ) `  c
 ) ) )
 
Theoremneiin 25574 Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)
 |-  (
 ( J  e.  Top  /\  M  e.  ( ( nei `  J ) `  A )  /\  N  e.  ( ( nei `  J ) `  B ) ) 
 ->  ( M  i^i  N )  e.  ( ( nei `  J ) `  ( A  i^i  B ) ) )
 
Theoremhmeoclda 25575 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Homeo  K ) )  /\  S  e.  ( Clsd `  J ) )  ->  ( F " S )  e.  ( Clsd `  K ) )
 
Theoremhmeocldb 25576 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J  Homeo  K ) )  /\  S  e.  ( Clsd `  K ) )  ->  ( `' F " S )  e.  ( Clsd `  J ) )
 
Theoremdfcon2OLD 25577* An alternate definition of connectedness. (Moved into main set.mm as dfcon2 17245 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 8-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  ( J  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( ( x  =/=  (/)  /\  y  =/=  (/)  /\  ( x  i^i  y
 )  =  (/) )  ->  X  =/=  ( x  u.  y ) ) ) )
 
TheoremconnsubOLD 25578* Two equivalent ways of saying that a subspace topology is connected. (Moved into main set.mm as connsub 17247 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X )  ->  ( ( Jt  S )  e.  Con  <->  A. x  e.  J  A. y  e.  J  ( ( ( x  i^i  S )  =/=  (/)  /\  (
 y  i^i  S )  =/= 
 (/)  /\  ( x  i^i  y )  C_  ( X 
 \  S ) ) 
 ->  -.  S  C_  ( x  u.  y ) ) ) )
 
18.13.3  Topology of the real numbers
 
TheoremreconnOLD 25579* A subset of the reals is connected iff it has the interval property. (Moved into main set.mm as reconn 18430 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  RR  ->  (
 ( ( topGen `  ran  (,) )t  A )  e.  Con  <->  A. x  e.  A  A. y  e.  A  ( x [,] y )  C_  A ) )
 
TheoremretopconOLD 25580 Corollary of reconn 18430. The set of real numbers is connected. (Moved into main set.mm as retopcon 18431 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( topGen `
  ran  (,) )  e. 
 Con
 
TheoremiccconnOLD 25581 A closed interval is connected. (Moved into main set.mm as iccconn 18432 and may be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B ) )  e.  Con )
 
TheoremivthALT 25582* An alternate proof of the Intermediate Value Theorem ivth 18912 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR  /\  U  e.  RR )  /\  A  <  B  /\  ( ( A [,] B )  C_  D  /\  D  C_  CC  /\  ( F  e.  ( D -cn->
 CC )  /\  ( F " ( A [,] B ) )  C_  RR  /\  U  e.  ( ( F `  A ) (,) ( F `  B ) ) ) ) )  ->  E. x  e.  ( A (,) B ) ( F `  x )  =  U )
 
18.13.4  Refinements
 
Syntaxcfne 25583 Extend class definition to include the "finer than" relation.
 class  Fne
 
Syntaxcref 25584 Extend class definition to include the refinement relation.
 class  Ref
 
Syntaxcptfin 25585 Extend class definition to include the class of point-finite covers.
 class  PtFin
 
Syntaxclocfin 25586 Extend class definition to include the class of locally finite covers.
 class  LocFin
 
Definitiondf-fne 25587* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  Fne  =  { <. x ,  y >.  |  ( U. x  =  U. y  /\  A. z  e.  x  z  C_ 
 U. ( y  i^i 
 ~P z ) ) }
 
Definitiondf-ref 25588* Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010.)
 |-  Ref  =  { <. x ,  y >.  |  ( U. x  =  U. y  /\  A. z  e.  y  E. w  e.  x  z  C_  w ) }
 
Definitiondf-ptfin 25589* Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  PtFin  =  { x  |  A. y  e. 
 U. x { z  e.  x  |  y  e.  z }  e.  Fin }
 
Definitiondf-locfin 25590* Define "locally finite." (Contributed by Jeff Hankins, 21-Jan-2010.)
 |-  LocFin  =  ( x  e.  Top  |->  { y  |  ( U. x  = 
 U. y  /\  A. p  e.  U. x E. n  e.  x  ( p  e.  n  /\  { s  e.  y  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) ) } )
 
Theoremfnerel 25591 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  Rel  Fne
 
Theoremisfne 25592* The predicate " B is finer than  A." This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  x  C_ 
 U. ( B  i^i  ~P x ) ) ) )
 
Theoremisfne4 25593 The predicate " B is finer than  A " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Fne B  <->  ( X  =  Y  /\  A  C_  ( topGen `  B ) ) )
 
Theoremisfne4b 25594 A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  V  ->  ( A Fne B  <->  ( X  =  Y  /\  ( topGen `  A )  C_  ( topGen `  B )
 ) ) )
 
Theoremisfne2 25595* The predicate " B is finer than  A." (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  A. y  e.  x  E. z  e.  B  (
 y  e.  z  /\  z  C_  x ) ) ) )
 
Theoremisfne3 25596* The predicate " B is finer than  A." (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( B  e.  C  ->  ( A Fne B  <->  ( X  =  Y  /\  A. x  e.  A  E. y ( y  C_  B  /\  x  =  U. y ) ) ) )
 
Theoremfnebas 25597 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  X  =  U. A   &    |-  Y  =  U. B   =>    |-  ( A Fne B  ->  X  =  Y )
 
Theoremfnetg 25598 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( A Fne B  ->  A  C_  ( topGen `  B )
 )
 
Theoremfnessex 25599* If  B is finer than  A and  S is an element of  A, every point in  S is an element of a subset of  S which is in  B. (Contributed by Jeff Hankins, 28-Sep-2009.)
 |-  (
 ( A Fne B  /\  S  e.  A  /\  P  e.  S )  ->  E. x  e.  B  ( P  e.  x  /\  x  C_  S ) )
 
Theoremfneuni 25600* If  B is finer than  A, every element of  A is a union of elements of  B. (Contributed by Jeff Hankins, 11-Oct-2009.)
 |-  (
 ( A Fne B  /\  S  e.  A ) 
 ->  E. x ( x 
 C_  B  /\  S  =  U. x ) )
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