Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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The reals are the quotient of rational Cauchy sequences by the maximal ideal of null sequences
Example
The reals are the quotient of rational Cauchy sequences by the maximal ideal of null sequences.
Facts & Assumptions
Given: The Cauchy-sequence ring and the null ideal .
The real numbers are defined as the quotient of rational Cauchy sequences by null sequences (The real numbers).
Rational Cauchy sequences form the ring (Cauchy sequences form a commutative ring).
is an ideal of (Null sequences form an ideal).
is maximal in (The null ideal is maximal).
Quotienting a commutative ring by a maximal ideal gives a field ( is a field if and only if is a maximal ideal).
The constructed real numbers form a field (The reals form a field).
Verification
By [L1], the underlying set and operations of are the quotient .
Since [L3] and [L4] make a maximal ideal, [L5] also identifies as a field.
This is the stated quotient realisation.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 41 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Tao, Analysis I (standard reference, not scraped)