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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 9433 | . 2 class inf(𝐴, 𝐵, 𝑅) |
5 | 3 | ccnv 5675 | . . 3 class ◡𝑅 |
6 | 1, 2, 5 | csup 9432 | . 2 class sup(𝐴, 𝐵, ◡𝑅) |
7 | 4, 6 | wceq 1542 | 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) |
Colors of variables: wff setvar class |
This definition is referenced by: infeq1 9468 infeq2 9471 infeq3 9472 infeq123d 9473 nfinf 9474 infexd 9475 eqinf 9476 infval 9478 infcl 9480 inflb 9481 infglb 9482 infglbb 9483 fiinfcl 9493 infltoreq 9494 inf00 9498 infempty 9499 infiso 9500 dfinfre 12192 infrenegsup 12194 tosglb 32133 rencldnfilem 41544 |
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