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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-fldgen | Structured version Visualization version GIF version |
Description: Define a function generating the smallest sub-division-ring of a given ring containing a given set. If the base structure is a division ring, then this is also a division ring (see fldgensdrg 32907). If the base structure is a field, this is a subfield (see fldgenfld 32913 and fldsdrgfld 20649). In general this will be used in the context of fields, hence the name fldGen. (Contributed by Saveliy Skresanov and Thierry Arnoux, 9-Jan-2025.) |
Ref | Expression |
---|---|
df-fldgen | β’ fldGen = (π β V, π β V β¦ β© {π β (SubDRingβπ) β£ π β π}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfldgen 32903 | . 2 class fldGen | |
2 | vf | . . 3 setvar π | |
3 | vs | . . 3 setvar π | |
4 | cvv 3468 | . . 3 class V | |
5 | 3 | cv 1532 | . . . . . 6 class π |
6 | va | . . . . . . 7 setvar π | |
7 | 6 | cv 1532 | . . . . . 6 class π |
8 | 5, 7 | wss 3943 | . . . . 5 wff π β π |
9 | 2 | cv 1532 | . . . . . 6 class π |
10 | csdrg 20637 | . . . . . 6 class SubDRing | |
11 | 9, 10 | cfv 6537 | . . . . 5 class (SubDRingβπ) |
12 | 8, 6, 11 | crab 3426 | . . . 4 class {π β (SubDRingβπ) β£ π β π} |
13 | 12 | cint 4943 | . . 3 class β© {π β (SubDRingβπ) β£ π β π} |
14 | 2, 3, 4, 4, 13 | cmpo 7407 | . 2 class (π β V, π β V β¦ β© {π β (SubDRingβπ) β£ π β π}) |
15 | 1, 14 | wceq 1533 | 1 wff fldGen = (π β V, π β V β¦ β© {π β (SubDRingβπ) β£ π β π}) |
Colors of variables: wff setvar class |
This definition is referenced by: fldgenval 32905 |
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