MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-inf Structured version   Unicode version

Definition df-inf 7960
Description: Define the infimum of class  A. It is meaningful when  R is a relation that strictly orders 
B and when the infimum exists. For example,  R could be 'less than',  B could be the set of real numbers, and  A could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
Assertion
Ref Expression
df-inf  |- inf ( A ,  B ,  R
)  =  sup ( A ,  B ,  `' R )

Detailed syntax breakdown of Definition df-inf
StepHypRef Expression
1 cA . . 3  class  A
2 cB . . 3  class  B
3 cR . . 3  class  R
41, 2, 3cinf 7958 . 2  class inf ( A ,  B ,  R
)
53ccnv 4849 . . 3  class  `' R
61, 2, 5csup 7957 . 2  class  sup ( A ,  B ,  `' R )
74, 6wceq 1437 1  wff inf ( A ,  B ,  R
)  =  sup ( A ,  B ,  `' R )
Colors of variables: wff setvar class
This definition is referenced by:  infeq1  7995  infeq2  7998  infeq3  7999  infeq123d  8000  nfinf  8001  infexd  8002  eqinf  8003  infval  8005  infcl  8007  inflb  8008  infglb  8009  infglbb  8010  fiinfcl  8020  infltoreq  8021  inf00  8024  infempty  8025  infiso  8026  lbinf  10560  dfinfre  10589  infrenegsup  10592  tosglb  28426  rencldnfilem  35582
  Copyright terms: Public domain W3C validator