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Theorem List for Metamath Proof Explorer - 37501-37600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxaddcomd 37501 The extended real addition operation is commutative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   =>    |-  ( ph  ->  ( A +e B )  =  ( B +e A ) )
 
Theoremsupxrre3 37502* The supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR*
 ,  <  )  e.  RR 
 <-> 
 E. x  e.  RR  A. y  e.  A  y 
 <_  x ) )
 
Theoremuzfissfz 37503* For any finite subset of the upper integers, there is a finite set of sequential integers that includes it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  A 
 C_  Z )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  E. k  e.  Z  A  C_  ( M ... k ) )
 
Theoremxleadd2d 37504 Addition of extended reals preserves the "less than or equal" relation, in the right slot (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( C +e A ) 
 <_  ( C +e B ) )
 
Theoremsuprltrp 37505* The supremum of a nonempty bounded set of reals can be approximated from below by elements of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  X  e.  RR+ )   =>    |-  ( ph  ->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  -  X )  < 
 z )
 
Theoremxleadd1d 37506 Addition of extended reals preserves the "less than or equal" relation, in the left slot (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( A +e C ) 
 <_  ( B +e C ) )
 
Theoremxreqled 37507 Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremxrgepnfd 37508 An extended real greater or equal to +oo is +oo (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  -> +oo  <_  A )   =>    |-  ( ph  ->  A  = +oo )
 
Theoremxrge0nemnfd 37509 A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  (
 0 [,] +oo ) )   =>    |-  ( ph  ->  A  =/= -oo )
 
Theoremsupxrgere 37510* If a real number can be approximated from below by members of a set, then it is smaller or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A 
 C_  RR* )   &    |-  ( ph  ->  B  e.  RR )   &    |-  (
 ( ph  /\  x  e.  RR+ )  ->  E. y  e.  A  ( B  -  x )  <  y )   =>    |-  ( ph  ->  B  <_  sup ( A ,  RR* ,  <  ) )
 
Theoremiuneqfzuzlem 37511* Lemma for iuneqfzuz 37512: here, inclusion is proven; aiuneqfzuz uses this lemma twice, to prove equality. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  Z  =  ( ZZ>= `  N )   =>    |-  ( A. m  e.  Z  U_ n  e.  ( N
 ... m ) A  =  U_ n  e.  ( N ... m ) B  ->  U_ n  e.  Z  A  C_  U_ n  e.  Z  B )
 
Theoremiuneqfzuz 37512* If two unions indexed by upper integers are equal if they agree on any partial indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  Z  =  ( ZZ>= `  N )   =>    |-  ( A. m  e.  Z  U_ n  e.  ( N
 ... m ) A  =  U_ n  e.  ( N ... m ) B  ->  U_ n  e.  Z  A  =  U_ n  e.  Z  B )
 
Theoremxle2addd 37513 Adding both side of two inequalities. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  D  e.  RR* )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  B 
 <_  D )   =>    |-  ( ph  ->  ( A +e B ) 
 <_  ( C +e D ) )
 
Theoremsupxrgelem 37514* If an extended real number can be approximated from below by members of a set, then it is smaller or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A 
 C_  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ( ph  /\  x  e.  RR+ )  ->  E. y  e.  A  B  <  ( y +e x ) )   =>    |-  ( ph  ->  B  <_  sup ( A ,  RR* ,  <  ) )
 
Theoremsupxrge 37515* If an extended real number can be approximated from below by members of a set, then it is smaller or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A 
 C_  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ( ph  /\  x  e.  RR+ )  ->  E. y  e.  A  B  <_  ( y +e x ) )   =>    |-  ( ph  ->  B  <_  sup ( A ,  RR* ,  <  ) )
 
Theoremsuplesup 37516* If any element of  A can be approximated from below by members of  B, then the supremum of  A is smaller or equal to the supremum of  B. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  C_  RR* )   &    |-  ( ph  ->  A. x  e.  A  A. y  e.  RR+  E. z  e.  B  ( x  -  y )  <  z )   =>    |-  ( ph  ->  sup ( A ,  RR* ,  <  )  <_  sup ( B ,  RR*
 ,  <  ) )
 
Theoreminfxrglb 37517* The infimum of a set of extended reals is less than an extended real if and only if the set contains a smaller number. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  (
 ( A  C_  RR*  /\  B  e.  RR* )  ->  (inf ( A ,  RR* ,  <  )  <  B  <->  E. x  e.  A  x  <  B ) )
 
Theoremxadd0ge2 37518 A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  A  <_  ( B +e A ) )
 
Theoremnepnfltpnf 37519 An extended real that is not +oo is less than +oo. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  =/= +oo )   &    |-  ( ph  ->  A  e.  RR* )   =>    |-  ( ph  ->  A  < +oo )
 
Theoremltadd12dd 37520 Addition to both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  B  <  D )   =>    |-  ( ph  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremnemnftgtmnft 37521 An extended real that is not minus infinity, is larger than minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  (
 ( A  e.  RR*  /\  A  =/= -oo )  -> -oo  <  A )
 
Theoremxrgtso 37522 'Greater than' is a strict ordering on the extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  `'  <  Or  RR*
 
Theoremrpex 37523 The positive reals form a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  RR+  e.  _V
 
Theoremxrge0ge0 37524 A nonnegative extended real is nonnegative. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( A  e.  ( 0 [,] +oo )  ->  0  <_  A )
 
Theoremxrssre 37525 A subset of extended reals that does not contain +oo and -oo is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  C_  RR* )   &    |-  ( ph  ->  -. +oo  e.  A )   &    |-  ( ph  ->  -. -oo  e.  A )   =>    |-  ( ph  ->  A  C_ 
 RR )
 
Theoremssuzfz 37526 A finite subset of the upper integers is a subset of a finite set of sequential integers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  A  C_  Z )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
 
Theoremabsfun 37527 The absolute value is a function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  Fun  abs
 
Theoreminfrpge 37528* The infimum of a non empty, bounded subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A 
 C_  RR* )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  x  <_  y )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  E. z  e.  A  z  <_  (inf ( A ,  RR* ,  <  ) +e B ) )
 
Theoremxrlexaddrp 37529* If an extended real number  A can be approximated from above, adding positive reals to  B, then  A is smaller or equal than  B. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ( ph  /\  x  e.  RR+ )  ->  A  <_  ( B +e x ) )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremsupsubc 37530* The supremum function distributes over subtraction in a sense similar to that in supaddc 10581. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B  e.  RR )   &    |-  C  =  { z  |  E. v  e.  A  z  =  ( v  -  B ) }   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  -  B )  =  sup ( C ,  RR ,  <  ) )
 
Theoremxralrple2 37531* Show that  A is less than  B by showing that there is no positive bound on the difference. A variant on xralrple 11505. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  ( 0 [,) +oo ) )   =>    |-  ( ph  ->  ( A  <_  B  <->  A. x  e.  RR+  A 
 <_  ( ( 1  +  x )  x.  B ) ) )
 
Theoremnnuzdisj 37532 The first  N elements of the set of nonnegative integers are distinct from any later members. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  (
 ( 1 ... N )  i^i  ( ZZ>= `  ( N  +  1 )
 ) )  =  (/)
 
Theoremltdivgt1 37533 Divsion by a number greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  (
 1  <  B  <->  ( A  /  B )  <  A ) )
 
Theoremxrltned 37534 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremnnsplit 37535 Express the set of positive integers as the disjoint (see nnuzdisj 37532) union of the first  N values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  ( N  e.  NN  ->  NN  =  ( ( 1
 ... N )  u.  ( ZZ>= `  ( N  +  1 ) ) ) )
 
21.30.4  Real intervals
 
Theoremgtnelioc 37536 A real number larger than the upper bound of a left open right closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  -.  C  e.  ( A (,] B ) )
 
Theoremioossioc 37537 An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A (,) B )  C_  ( A (,] B )
 
Theoremioondisj2 37538 A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B ) 
 /\  ( C  e.  RR*  /\  D  e.  RR*  /\  C  <  D ) )  /\  ( A  <  D  /\  D  <_  B ) ) 
 ->  ( ( A (,) B )  i^i  ( C (,) D ) )  =/=  (/) )
 
Theoremioondisj1 37539 A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B ) 
 /\  ( C  e.  RR*  /\  D  e.  RR*  /\  C  <  D ) )  /\  ( A  <_  C  /\  C  <  B ) ) 
 ->  ( ( A (,) B )  i^i  ( C (,) D ) )  =/=  (/) )
 
Theoremioosscn 37540 An open interval is a set of complex numbers (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A (,) B )  C_  CC
 
Theoremioogtlb 37541 An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A (,) B ) )  ->  A  <  C )
 
Theoremevthiccabs 37542* Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )   =>    |-  ( ph  ->  ( E. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( abs `  ( F `  y
 ) )  <_  ( abs `  ( F `  x ) )  /\  E. z  e.  ( A [,] B ) A. w  e.  ( A [,] B ) ( abs `  ( F `  z
 ) )  <_  ( abs `  ( F `  w ) ) ) )
 
Theoremltnelicc 37543 A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  C  <  A )   =>    |-  ( ph  ->  -.  C  e.  ( A [,] B ) )
 
Theoremeliood 37544 Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  C  <  B )   =>    |-  ( ph  ->  C  e.  ( A (,) B ) )
 
Theoremiooabslt 37545 An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  ( ( A  -  B ) (,) ( A  +  B ) ) )   =>    |-  ( ph  ->  ( abs `  ( A  -  C ) )  <  B )
 
Theoremgtnelicc 37546 A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  -.  C  e.  ( A [,] B ) )
 
Theoremiooinlbub 37547 An open interval has empty intersection with its bounds (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A (,) B )  i^i  { A ,  B } )  =  (/)
 
Theoremiocgtlb 37548 An element of a left open right closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A (,] B ) )  ->  A  <  C )
 
Theoremiocleub 37549 An element of a left open right closed interval is smaller or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A (,] B ) )  ->  C  <_  B )
 
Theoremeliccd 37550 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  C )   &    |-  ( ph  ->  C  <_  B )   =>    |-  ( ph  ->  C  e.  ( A [,] B ) )
 
Theoremiccssred 37551 A closed real interval is a set of reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A [,] B )  C_  RR )
 
Theoremeliccre 37552 A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR )
 
Theoremeliooshift 37553 Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   =>    |-  ( ph  ->  ( A  e.  ( B (,) C )  <->  ( A  +  D )  e.  (
 ( B  +  D ) (,) ( C  +  D ) ) ) )
 
Theoremeliocd 37554 Membership in a left open, right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  C  <_  B )   =>    |-  ( ph  ->  C  e.  ( A (,] B ) )
 
Theoremsnunioo2 37555 The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  (
 ( A (,) B )  u.  { B }
 )  =  ( A (,] B ) )
 
Theoremicoltub 37556 An element of a left closed right open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,) B ) )  ->  C  <  B )
 
Theoremtgiooss 37557 The restriction of the complex topology to a subset of reals, is a restriction of the standard topology on reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): Use rerest 21820 instead in fourierdlem48 37958, fourierdlem49 37959, fourierdlem62 37972 then delete this.
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  (
 topGen `  ran  (,) )   =>    |-  ( A  C_  RR  ->  ( Jt  A )  =  ( Kt  A ) )
 
Theoremeliocre 37558 A member of a left open, right closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( B  e.  RR  /\  C  e.  ( A (,] B ) ) 
 ->  C  e.  RR )
 
Theoremiooltub 37559 An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A (,) B ) )  ->  C  <  B )
 
Theoremioontr 37560 The interior of an interval in the standard topology on  RR is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( int `  ( topGen `  ran  (,) ) ) `  ( A (,) B ) )  =  ( A (,) B )
 
Theoremeliccxr 37561 A member of a closed interval is a an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  ( B [,] C )  ->  A  e.  RR* )
 
Theoremsnunioo1 37562 The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  (
 ( A (,) B )  u.  { A }
 )  =  ( A [,) B ) )
 
Theoremlbioc 37563 An left open right closed interval doesn't contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  -.  A  e.  ( A (,] B )
 
Theoremioomidp 37564 The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( ( A  +  B )  /  2
 )  e.  ( A (,) B ) )
 
Theoremiccdifioo 37565 If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
 ( A [,] B )  \  ( A (,) B ) )  =  { A ,  B }
 )
 
Theoremiccdifprioo 37566 An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A [,] B )  \  { A ,  B } )  =  ( A (,) B ) )
 
Theoremioossioobi 37567 Biconditional form of ioossioo 11733 (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  D  e.  RR* )   &    |-  ( ph  ->  C  <  D )   =>    |-  ( ph  ->  ( ( C (,) D )  C_  ( A (,) B )  <->  ( A  <_  C 
 /\  D  <_  B ) ) )
 
Theoremiccshift 37568* A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  T  e.  RR )   =>    |-  ( ph  ->  ( ( A  +  T ) [,] ( B  +  T ) )  =  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) } )
 
Theoremiccsuble 37569 An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  ( A [,] B ) )   &    |-  ( ph  ->  D  e.  ( A [,] B ) )   =>    |-  ( ph  ->  ( C  -  D )  <_  ( B  -  A ) )
 
Theoremiocopn 37570 A left open right closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  B  e.  RR )   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  J  =  ( Kt  ( A (,] B ) )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( C (,] B )  e.  J )
 
Theoremeliccelioc 37571 Membership in a closed interval and in a left open right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR* )   =>    |-  ( ph  ->  ( C  e.  ( A (,] B )  <->  ( C  e.  ( A [,] B ) 
 /\  C  =/=  A ) ) )
 
Theoremiooshift 37572* An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  T  e.  RR )   =>    |-  ( ph  ->  ( ( A  +  T ) (,) ( B  +  T ) )  =  { w  e.  CC  |  E. z  e.  ( A (,) B ) w  =  ( z  +  T ) } )
 
Theoremiccintsng 37573 Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  ->  ( ( A [,] B )  i^i  ( B [,] C ) )  =  { B }
 )
 
Theoremicoiccdif 37574 Left closed, right open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A [,) B )  =  ( ( A [,] B )  \  { B } ) )
 
Theoremicoopn 37575 A left closed right open interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  J  =  ( Kt  ( A [,) B ) )   &    |-  ( ph  ->  C 
 <_  B )   =>    |-  ( ph  ->  ( A [,) C )  e.  J )
 
Theoremicoub 37576 A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  e.  RR*  ->  -.  B  e.  ( A [,) B ) )
 
Theoremeliccxrd 37577 Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  C 
 <_  B )   =>    |-  ( ph  ->  C  e.  ( A [,] B ) )
 
Theorempnfel0pnf 37578 +oo is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |- +oo  e.  ( 0 [,] +oo )
 
Theoremge0nemnf2 37579 A nonnegative extended real is not -oo (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  e.  ( 0 [,] +oo )  ->  A  =/= -oo )
 
Theoremeliccnelico 37580 An element of a closed interval that is not a member of the left closed right open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  ( A [,] B ) )   &    |-  ( ph  ->  -.  C  e.  ( A [,) B ) )   =>    |-  ( ph  ->  C  =  B )
 
Theoremeliccelicod 37581 A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  ( A [,] B ) )   &    |-  ( ph  ->  C  =/=  B )   =>    |-  ( ph  ->  C  e.  ( A [,) B ) )
 
Theoremge0xrre 37582 A nonnegative extended real that is not +oo is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( A  e.  (
 0 [,] +oo )  /\  A  =/= +oo )  ->  A  e.  RR )
 
Theoremelicores 37583* Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( A  e.  ran  ( [,)  |`  ( RR  X.  RR ) )  <->  E. x  e.  RR  E. y  e.  RR  A  =  ( x [,) y
 ) )
 
Theoreminficc 37584 The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  S 
 C_  ( A [,] B ) )   &    |-  ( ph  ->  S  =/=  (/) )   =>    |-  ( ph  -> inf ( S ,  RR* ,  <  )  e.  ( A [,] B ) )
 
21.30.5  Finite sums
 
Theoremsumeq2ad 37585* Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  C )
 
Theoremfsumclf 37586* Closure of a finite sum of complex numbers  A ( k ). A version of fsumcl 13798 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  CC )
 
Theoremfsumsplitf 37587* Split a sum into two parts. A version of fsumsplit 13805 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ( ph  /\  k  e.  U )  ->  C  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  U  C  =  (
 sum_ k  e.  A  C  +  sum_ k  e.  B  C ) )
 
Theoremfsummulc1f 37588* Closure of a finite sum of complex numbers  A ( k ). A version of fsummulc1 13845 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  ( sum_ k  e.  A  B  x.  C )  =  sum_ k  e.  A  ( B  x.  C ) )
 
Theoremsumsnf 37589* A sum of a singleton is the term. A version of sumsn 13806 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/_ k B   &    |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  V  /\  B  e.  CC )  ->  sum_ k  e.  { M } A  =  B )
 
Theoremfsumsplitsn 37590* Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ k ph   &    |-  F/_ k D   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ( ph  /\  k  e.  A )  ->  C  e.  CC )   &    |-  ( k  =  B  ->  C  =  D )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  sum_
 k  e.  ( A  u.  { B }
 ) C  =  (
 sum_ k  e.  A  C  +  D )
 )
 
Theoremfsumnncl 37591* Closure of a non empty, finite sum of positive integers (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  NN )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  NN )
 
Theoremfsumsplit1 37592* Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ k ph   &    |-  F/_ k D   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  A )   &    |-  (
 k  =  C  ->  B  =  D )   =>    |-  ( ph  ->  sum_
 k  e.  A  B  =  ( D  +  sum_ k  e.  ( A  \  { C } ) B ) )
 
Theoremfsumge0cl 37593* The finite sum of nonnegative reals is a nonnegative real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  (
 0 [,) +oo ) )
 
Theoremfsumf1of 37594* Re-index a finite sum using a bijection. Same as fsumf1o 13788, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ k ph   &    |-  F/ n ph   &    |-  (
 k  =  G  ->  B  =  D )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ph  ->  F : C -1-1-onto-> A )   &    |-  ( ( ph  /\  n  e.  C ) 
 ->  ( F `  n )  =  G )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ k  e.  A  B  =  sum_ n  e.  C  D )
 
Theoremfsumiunss 37595* Sum over a disjoint indexed union, intersected with a finite set  D. Similar to fsumiun 13880, but here  A and 
B need not be finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  (
 ( ph  /\  x  e.  A  /\  k  e.  B )  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  Fin )   =>    |-  ( ph  ->  sum_
 k  e.  U_  x  e.  A  ( B  i^i  D ) C  =  sum_ x  e.  { x  e.  A  |  ( B  i^i  D )  =/=  (/) } sum_ k  e.  ( B  i^i  D ) C )
 
Theoremfsumreclf 37596* Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  RR )
 
Theoremfsumlessf 37597* A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  F/ k ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  k  e.  A ) 
 ->  0  <_  B )   &    |-  ( ph  ->  C  C_  A )   =>    |-  ( ph  ->  sum_ k  e.  C  B  <_  sum_ k  e.  A  B )
 
21.30.6  Finite multiplication of numbers and finite multiplication of functions
 
Theoremfmul01 37598* Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ i B   &    |- 
 F/ i ph   &    |-  A  =  seq L (  x.  ,  B )   &    |-  ( ph  ->  L  e.  ZZ )   &    |-  ( ph  ->  M  e.  ( ZZ>= `  L ) )   &    |-  ( ph  ->  K  e.  ( L ... M ) )   &    |-  ( ( ph  /\  i  e.  ( L
 ... M ) ) 
 ->  ( B `  i
 )  e.  RR )   &    |-  (
 ( ph  /\  i  e.  ( L ... M ) )  ->  0  <_  ( B `  i ) )   &    |-  ( ( ph  /\  i  e.  ( L
 ... M ) ) 
 ->  ( B `  i
 )  <_  1 )   =>    |-  ( ph  ->  ( 0  <_  ( A `  K ) 
 /\  ( A `  K )  <_  1 ) )
 
Theoremfmulcl 37599* If ' Y ' is closed under the multiplication of two functions, then Y is closed under the multiplication ( ' X ' ) of a finite number of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) ) )   &    |-  X  =  (  seq 1
 ( P ,  U ) `  N )   &    |-  ( ph  ->  N  e.  (
 1 ... M ) )   &    |-  ( ph  ->  U :
 ( 1 ... M )
 --> Y )   &    |-  ( ( ph  /\  f  e.  Y  /\  g  e.  Y )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) )  e.  Y )   &    |-  ( ph  ->  T  e.  _V )   =>    |-  ( ph  ->  X  e.  Y )
 
Theoremfmuldfeqlem1 37600* induction step for the proof of fmuldfeq 37601. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ f ph   &    |-  F/ g ph   &    |-  F/_ t Y   &    |-  P  =  ( f  e.  Y ,  g  e.  Y  |->  ( t  e.  T  |->  ( ( f `
  t )  x.  ( g `  t
 ) ) ) )   &    |-  F  =  ( t  e.  T  |->  ( i  e.  ( 1 ... M )  |->  ( ( U `
  i ) `  t ) ) )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  U : ( 1 ...
 M ) --> Y )   &    |-  ( ( ph  /\  f  e.  Y  /\  g  e.  Y )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  Y )   &    |-  ( ph  ->  N  e.  ( 1 ... M ) )   &    |-  ( ph  ->  ( N  +  1 )  e.  ( 1 ...
 M ) )   &    |-  ( ph  ->  ( (  seq 1 ( P ,  U ) `  N ) `  t )  =  (  seq 1 (  x.  ,  ( F `
  t ) ) `
  N ) )   &    |-  ( ( ph  /\  f  e.  Y )  ->  f : T --> RR )   =>    |-  ( ( ph  /\  t  e.  T ) 
 ->  ( (  seq 1
 ( P ,  U ) `  ( N  +  1 ) ) `  t )  =  (  seq 1 (  x.  ,  ( F `  t ) ) `  ( N  +  1 ) ) )
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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 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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