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

Definition df-inf 8232
 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 8230 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 5037 . . 3 class 𝑅
61, 2, 5csup 8229 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1475 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
 Colors of variables: wff setvar class This definition is referenced by:  infeq1  8265  infeq2  8268  infeq3  8269  infeq123d  8270  nfinf  8271  infexd  8272  eqinf  8273  infval  8275  infcl  8277  inflb  8278  infglb  8279  infglbb  8280  fiinfcl  8290  infltoreq  8291  inf00  8294  infempty  8295  infiso  8296  lbinf  10855  dfinfre  10881  infrenegsup  10883  tosglb  29001  rencldnfilem  36402
 Copyright terms: Public domain W3C validator