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Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2ndpreima 24101 The preimage by  2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
 |-  ( A  C_  C  ->  ( `' ( 2nd  |`  ( B  X.  C ) )
 " A )  =  ( B  X.  A ) )
 
Theoremcurry2ima 24102* The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  B  /\  D  C_  A )  ->  ( G " D )  =  { y  |  E. x  e.  D  y  =  ( x F C ) } )
 
19.3.4.6  Supremum - misc additions
 
Theoremsupssd 24103* Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  B  C_  C )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   =>    |-  ( ph  ->  -.  sup ( C ,  A ,  R ) R sup ( B ,  A ,  R ) )
 
19.3.4.7  Countable Sets
 
Theoremnnct 24104  NN is countable (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  NN  ~<_  om
 
Theoremctex 24105 A countable set is a set (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  A  e.  _V )
 
Theoremssct 24106 The subset of a countable set is countable (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
 
Theoremxpct 24107 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( A  X.  B )  ~<_  om )
 
Theoremsnct 24108 A singleton is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  ( A  e.  V  ->  { A }  ~<_  om )
 
Theoremprct 24109 An unordered pair is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  { A ,  B } 
 ~<_  om )
 
Theoremfnct 24110 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  (
 ( F  Fn  A  /\  A  ~<_  om )  ->  F  ~<_  om )
 
Theoremdmct 24111 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  dom  A  ~<_  om )
 
Theoremcnvct 24112 If a set is countable, its converse is as well. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  `' A  ~<_  om )
 
Theoremrnct 24113 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ran  A  ~<_  om )
 
Theoremmptct 24114* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremmpt2cti 24115* An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  A. x  e.  A  A. y  e.  B  C  e.  V   =>    |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( x  e.  A ,  y  e.  B  |->  C )  ~<_ 
 om )
 
Theoremabrexct 24116* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  { y  |  E. x  e.  A  y  =  B }  ~<_  om )
 
Theoremmptctf 24117 A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremabrexctf 24118* An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  { y  |  E. x  e.  A  y  =  B } 
 ~<_  om )
 
19.3.5  Real and Complex Numbers
 
19.3.5.1  Complex addition - misc. additions
 
Theoremaddeq0 24119 Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  =  0  <->  A  =  -u B ) )
 
19.3.5.2  Ordering on reals - misc additions
 
Theoremlt2addrd 24120* If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  ( B  +  C )
 )   =>    |-  ( ph  ->  E. b  e.  RR  E. c  e. 
 RR  ( A  =  ( b  +  c
 )  /\  b  <  B 
 /\  c  <  C ) )
 
19.3.5.3  Extended reals - misc additions
 
Theoremxgepnf 24121 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR*  ->  (  +oo  <_  A  <->  A  =  +oo ) )
 
Theoremxlemnf 24122 An extended real which is less than minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  ( A  e.  RR*  ->  ( A  <_  -oo  <->  A  =  -oo ) )
 
Theoremxrlelttric 24123 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremxaddeq0 24124 Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A + e B )  =  0  <->  A  =  - e B ) )
 
Theoreminfxrmnf 24125 The infinimum of a set of extended reals containing minus infnity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  (
 ( A  C_  RR*  /\  -oo  e.  A )  ->  sup ( A ,  RR* ,  `'  <  )  =  -oo )
 
Theoremxrinfm 24126 The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  sup ( RR* ,  RR* ,  `'  <  )  =  -oo
 
Theoremle2halvesd 24127 A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  ( C  /  2 ) )   &    |-  ( ph  ->  B  <_  ( C  /  2 ) )   =>    |-  ( ph  ->  ( A  +  B )  <_  C )
 
Theoremxraddge02 24128 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  B  ->  A  <_  ( A + e B ) ) )
 
Theoremxlt2addrd 24129* If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  B  =/=  -oo )   &    |-  ( ph  ->  C  =/=  -oo )   &    |-  ( ph  ->  A  <  ( B + e C ) )   =>    |-  ( ph  ->  E. b  e.  RR*  E. c  e.  RR*  ( A  =  ( b + e
 c )  /\  b  <  B  /\  c  <  C ) )
 
Theoremxrsupssd 24130 Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  ( ph  ->  B  C_  C )   &    |-  ( ph  ->  C  C_  RR* )   =>    |-  ( ph  ->  sup ( B ,  RR* ,  <  ) 
 <_  sup ( C ,  RR*
 ,  <  ) )
 
Theoremxrofsup 24131 The supremum is preserved by extended addition set operation. (provided minus infinity is not involved as it does not behave well with addition) (Contributed by Thierry Arnoux, 20-Mar-2017.)
 |-  ( ph  ->  X  C_  RR* )   &    |-  ( ph  ->  Y  C_  RR* )   &    |-  ( ph  ->  sup ( X ,  RR*
 ,  <  )  =/=  -oo )   &    |-  ( ph  ->  sup ( Y ,  RR* ,  <  )  =/=  -oo )   &    |-  ( ph  ->  Z  =  ( + e "
 ( X  X.  Y ) ) )   =>    |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  =  ( sup ( X ,  RR*
 ,  <  ) + e sup ( Y ,  RR*
 ,  <  ) )
 )
 
Theoremsupxrnemnf 24132 The supremum of a non-empty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  (
 ( A  C_  RR*  /\  A  =/= 
 (/)  /\  -.  -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  =/=  -oo )
 
Theoremxrhaus 24133 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  (ordTop ` 
 <_  )  e.  Haus
 
19.3.5.4  Real number intervals - misc additions
 
Theoremicossicc 24134 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
 |-  ( A [,) B )  C_  ( A [,] B )
 
Theoremiocssicc 24135 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  ( A (,] B )  C_  ( A [,] B )
 
Theoremioossico 24136 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  ( A (,) B )  C_  ( A [,) B )
 
Theoremiocssioo 24137 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <  B ) )  ->  ( C (,] D )  C_  ( A (,) B ) )
 
Theoremicossioo 24138 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <_  B )
 )  ->  ( C [,) D )  C_  ( A (,) B ) )
 
Theoremioossioo 24139 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B )
 )  ->  ( C (,) D )  C_  ( A (,) B ) )
 
Theoremjoiniooico 24140 Disjoint joining an open interval with a closed below, open above interval to form a closed below, open above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( ( ( A (,) B )  i^i  ( B [,) C ) )  =  (/)  /\  (
 ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) ) )
 
Theoremiccgelb 24141 An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  A  <_  C )
 
Theoremsnunioc 24142 The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( { A }  u.  ( A (,] B ) )  =  ( A [,] B ) )
 
Theoremubico 24143 A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR* )  ->  -.  B  e.  ( A [,) B ) )
 
Theoremxeqlelt 24144 Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  -.  A  <  B ) ) )
 
Theoremeliccelico 24145 Relate elementhood to a closed interval with elementhood to the same closed-below, opened-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  ( A [,) B )  \/  C  =  B ) ) )
 
Theoremelicoelioo 24146 Relate elementhood to a closed-below, opened-above interval with elementhood to the same opened interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  ( C  e.  ( A [,) B )  <->  ( C  =  A  \/  C  e.  ( A (,) B ) ) ) )
 
Theoremiocinioc2 24147 Intersection between two opened below, closed above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  (
 ( A (,] C )  i^i  ( B (,] C ) )  =  ( B (,] C ) )
 
Theoremxrdifh 24148 Set difference of a half-opened interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
 |-  A  e.  RR*   =>    |-  ( RR*  \  ( A [,]  +oo ) )  =  (  -oo [,) A )
 
Theoremiocinif 24149 Relate intersection of two opened below, closed above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A (,] C )  i^i  ( B (,] C ) )  =  if ( A  <  B ,  ( B (,] C ) ,  ( A (,] C ) ) )
 
Theoremdifioo 24150 The difference between two opened intervals sharing the same lower bound (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  (
 ( A (,) C )  \  ( A (,) B ) )  =  ( B [,) C ) )
 
Theoremdifico 24151 The difference between two closed below, opened above intervals sharing the same upper bound (Contributed by Thierry Arnoux, 13-Oct-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  ->  ( ( A [,) C )  \  ( B [,) C ) )  =  ( A [,) B ) )
 
19.3.5.5  Finite intervals of integers - misc additions
 
Theoremfzssnn 24152 Finite sets of sequential integers starting from a natural are a subset of the natural numbers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
 |-  ( M  e.  NN  ->  ( M ... N ) 
 C_  NN )
 
Theoremssnnssfz 24153* For any finite subset of  NN, find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.)
 |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  E. n  e.  NN  A  C_  (
 1 ... n ) )
 
Theoremfzspl 24154 Split the last element of a finite set of sequential integers. (more generic than fzsuc 11101) (Contributed by Thierry Arnoux, 7-Nov-2016.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( M
 ... N )  =  ( ( M ... ( N  -  1
 ) )  u.  { N } ) )
 
Theoremfzsplit3 24155 Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
 |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M ... ( K  -  1
 ) )  u.  ( K ... N ) ) )
 
Theorembcm1n 24156 The proportion of one binomial coefficient to another with  N decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.)
 |-  (
 ( K  e.  (
 0 ... ( N  -  1 ) )  /\  N  e.  NN )  ->  ( ( ( N  -  1 )  _C  K )  /  ( N  _C  K ) )  =  ( ( N  -  K )  /  N ) )
 
19.3.5.6  Half-open integer ranges - misc additions
 
Theoremiundisjfi 24157* Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 19447 (Contributed by Thierry Arnoux, 15-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   =>    |-  U_ n  e.  ( 1..^ N ) A  =  U_ n  e.  ( 1..^ N ) ( A 
 \  U_ k  e.  (
 1..^ n ) B )
 
Theoremiundisj2fi 24158* A disjoint union is disjoint, finite version. Cf. iundisj2 19448 (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   =>    |- Disj  n  e.  ( 1..^ N ) ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisjcnt 24159* Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   =>    |-  ( ph  ->  U_ n  e.  N  A  =  U_ n  e.  N  ( A  \  U_ k  e.  ( 1..^ n ) B ) )
 
Theoremiundisj2cnt 24160* A countable disjoint union is disjoint. Cf. iundisj2 19448 (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   =>    |-  ( ph  -> Disj  n  e.  N ( A  \  U_ k  e.  ( 1..^ n ) B ) )
 
19.3.5.7  The ` # ` (finite set size) function - misc additions
 
Theoremhashresfn 24161 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( #  |`  A )  Fn  A
 
Theoremdmhashres 24162 Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 12-Jan-2017.)
 |-  dom  ( #  |`  A )  =  A
 
Theoremhashunif 24163* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 12609 (Contributed by Thierry Arnoux, 17-Feb-2017.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Fin )   &    |-  ( ph  -> Disj  x  e.  A x )   =>    |-  ( ph  ->  ( # `
  U. A )  = 
 sum_ x  e.  A  ( # `  x ) )
 
Theoremishashinf 24164* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 7325 (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  ( -.  A  e.  Fin  ->  A. n  e.  NN  E. x  e.  ~P  A ( # `  x )  =  n )
 
19.3.5.8  The greatest common divisor operator - misc. add
 
Theoremnumdenneg 24165 Numerator and denominator of the negative (Contributed by Thierry Arnoux, 27-Oct-2017.)
 |-  ( Q  e.  QQ  ->  ( (numer `  -u Q )  =  -u (numer `  Q )  /\  (denom `  -u Q )  =  (denom `  Q ) ) )
 
Theoremdivnumden2 24166 Calculate the reduced form of a quotient using  gcd. This version extends divnumden 13145 for the negative integers. (Contributed by Thierry Arnoux, 25-Oct-2017.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  -u B  e.  NN )  ->  ( (numer `  ( A  /  B ) )  =  -u ( A  /  ( A  gcd  B ) )  /\  (denom `  ( A  /  B ) )  =  -u ( B  /  ( A  gcd  B ) ) ) )
 
19.3.5.9  Integers
 
Theoremltesubnnd 24167 Subtracting an integer number from another number decreases it. See ltsubrpd 10681 (Contributed by Thierry Arnoux, 18-Apr-2017.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  (
 ( M  +  1 )  -  N ) 
 <_  M )
 
19.3.5.10  Division in the extended real number system
 
Syntaxcxdiv 24168 Extend class notation to include division of extended reals.
 class /𝑒
 
Definitiondf-xdiv 24169* Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |- /𝑒  =  ( x  e.  RR* ,  y  e.  ( RR  \  {
 0 } )  |->  (
 iota_ z  e.  RR* (
 y x e z )  =  x ) )
 
Theoremxdivval 24170* Value of division: the (unique) element  x such that  ( B  x.  x )  =  A. This is meaningful only when  B is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
 iota_ x  e.  RR* ( B x e x )  =  A ) )
 
Theoremxrecex 24171* Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  A  =/=  0 ) 
 ->  E. x  e.  RR  ( A x e x )  =  1 )
 
Theoremxmulcand 24172 Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  C  =/=  0
 )   =>    |-  ( ph  ->  (
 ( C x e A )  =  ( C x e B )  <->  A  =  B )
 )
 
Theoremxreceu 24173* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B x e x )  =  A )
 
Theoremxdivcld 24174 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  ( A /𝑒 
 B )  e.  RR* )
 
Theoremxdivcl 24175 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  e.  RR* )
 
Theoremxdivmul 24176 Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  ( ( A /𝑒  C )  =  B  <->  ( C x e B )  =  A ) )
 
Theoremrexdiv 24177 The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A  /  B ) )
 
Theoremxdivrec 24178 Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A x e ( 1 /𝑒 
 B ) ) )
 
Theoremxdivid 24179 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  A  =/=  0 ) 
 ->  ( A /𝑒  A )  =  1 )
 
Theoremxdiv0 24180 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  A  =/=  0 ) 
 ->  ( 0 /𝑒  A )  =  0 )
 
Theoremxdiv0rp 24181 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR+  ->  (
 0 /𝑒  A )  =  0
 )
 
Theoremeliccioo 24182 Membership in a closed interval of extended reals vs. the same open interval (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  <->  ( C  =  A  \/  C  e.  ( A (,) B )  \/  C  =  B ) ) )
 
Theoremelxrge02 24183 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  ( 0 [,]  +oo )  <->  ( A  =  0  \/  A  e.  RR+  \/  A  =  +oo )
 )
 
Theoremxdivpnfrp 24184 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR+  ->  (  +oo /𝑒  A )  =  +oo )
 
Theoremrpxdivcld 24185 Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A /𝑒  B )  e.  RR+ )
 
Theoremxrpxdivcld 24186 Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( ph  ->  A  e.  (
 0 [,]  +oo ) )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A /𝑒  B )  e.  ( 0 [,]  +oo ) )
 
19.3.6  Structure builders
 
19.3.6.1  Structure builder restriction operator
 
Theoremress0g 24187  0g is unaffected by restriction. This is a bit more generic than submnd0 14730 (Contributed by Thierry Arnoux, 23-Oct-2017.)
 |-  S  =  ( Rs  A )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  =  ( 0g `  S ) )
 
Theoremressplusf 24188 The group operation function  + f of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
 |-  B  =  ( Base `  G )   &    |-  H  =  ( Gs  A )   &    |-  .+^  =  ( +g  `  G )   &    |-  .+^  Fn  ( B  X.  B )   &    |-  A  C_  B   =>    |-  ( + f `  H )  =  (  .+^  |`  ( A  X.  A ) )
 
Theoremressnm 24189 The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.)
 |-  H  =  ( Gs  A )   &    |-  B  =  (
 Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( norm `  G )   =>    |-  ( ( G  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  ( N  |`  A )  =  ( norm `  H ) )
 
Theoremabvpropd2 24190 Weaker version of abvpropd 15935. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  ( ph  ->  ( Base `  K )  =  ( Base `  L ) )   &    |-  ( ph  ->  ( +g  `  K )  =  ( +g  `  L ) )   &    |-  ( ph  ->  ( .r `  K )  =  ( .r `  L ) )   =>    |-  ( ph  ->  (AbsVal `  K )  =  (AbsVal `  L ) )
 
19.3.6.2  Posets
 
Theoremtospos 24191 A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  ( F  e. Toset  ->  F  e.  Poset
 )
 
Theoremresspos 24192 The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  (
 ( F  e.  Poset  /\  A  e.  V ) 
 ->  ( Fs  A )  e.  Poset )
 
Theoremresstos 24193 The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  (
 ( F  e. Toset  /\  A  e.  V )  ->  ( Fs  A )  e. Toset )
 
Theoremtleile 24194 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  \/  Y  .<_  X ) )
 
Theoremtltnle 24195 In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 14428. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   =>    |-  ( ( K  e. Toset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  -.  Y  .<_  X ) )
 
Theoremtoslub 24196* In a toset, the lowest upper bound 
lub, defined for partial orders is the supremum,  sup ( A ,  B ,  .<  ), defined for total orders, if this supremum exists (these are the set.mm definition: lowest upper bound and supremum are normally synonymous) Note that the two definitions do not have the same value when there is no such supremum. In that case,  sup ( A ,  B ,  .<  ) will take the value  (/), but  ( ( lub `  K ) `  A ) takes the value  ( Undef `  B
), therefore, we restrict this theorem only to the case where this supremum exists, which is expressed by the last assumption. (Contributed by Thierry Arnoux, 15-Feb-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  ( ph  ->  K  e. Toset )   &    |-  ( ph  ->  A 
 C_  B )   &    |-  ( ph  ->  E. a  e.  B  ( A. b  e.  A  -.  a  .<  b  /\  A. b  e.  B  ( b  .<  a  ->  E. d  e.  A  b 
 .<  d ) ) )   =>    |-  ( ph  ->  ( ( lub `  K ) `  A )  =  sup ( A ,  B ,  .<  ) )
 
Theoremtosglb 24197* Same theorem as toslub 24196, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  ( ph  ->  K  e. Toset )   &    |-  ( ph  ->  A 
 C_  B )   &    |-  ( ph  ->  E. a  e.  B  ( A. b  e.  A  -.  b  .<  a  /\  A. b  e.  B  ( a  .<  b  ->  E. d  e.  A  d 
 .<  b ) ) )   =>    |-  ( ph  ->  ( ( glb `  K ) `  A )  =  sup ( A ,  B ,  `'  .<  ) )
 
19.3.6.3  Complete lattices
 
Theoremclatp0ex 24198 The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  .0.  =  ( 0. `  W )   =>    |-  ( W  e.  CLat  ->  .0.  e.  B )
 
Theoremclatp1ex 24199 The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
 |-  B  =  ( Base `  W )   &    |-  .1.  =  ( 1. `  W )   =>    |-  ( W  e.  CLat  ->  .1.  e.  B )
 
19.3.6.4  Extended reals Structure - misc additions
 
Axiomax-xrssca 24200 Assume the scalar component of the extended real structure is the field of the real numbers (this has to be defined in the main body of set.mm) (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  (flds  RR )  =  (Scalar `  RR* s )
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