Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-inf | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-inf | ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cB | . . 3 class 𝐵 | |
3 | cR | . . 3 class 𝑅 | |
4 | 1, 2, 3 | cinf 8905 | . 2 class inf(𝐴, 𝐵, 𝑅) |
5 | 3 | ccnv 5554 | . . 3 class ◡𝑅 |
6 | 1, 2, 5 | csup 8904 | . 2 class sup(𝐴, 𝐵, ◡𝑅) |
7 | 4, 6 | wceq 1537 | 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) |
Colors of variables: wff setvar class |
This definition is referenced by: infeq1 8940 infeq2 8943 infeq3 8944 infeq123d 8945 nfinf 8946 infexd 8947 eqinf 8948 infval 8950 infcl 8952 inflb 8953 infglb 8954 infglbb 8955 fiinfcl 8965 infltoreq 8966 inf00 8970 infempty 8971 infiso 8972 dfinfre 11622 infrenegsup 11624 tosglb 30657 rencldnfilem 39437 |
Copyright terms: Public domain | W3C validator |