Description: The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | drhmsubc.c | |- C = ( U i^i DivRing ) |
|
| drhmsubc.j | |- J = ( r e. C , s e. C |-> ( r RingHom s ) ) |
||
| fldhmsubc.d | |- D = ( U i^i Field ) |
||
| fldhmsubc.f | |- F = ( r e. D , s e. D |-> ( r RingHom s ) ) |
||
| Assertion | fldcat | |- ( U e. V -> ( ( RingCat ` U ) |`cat F ) e. Cat ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | drhmsubc.c | |- C = ( U i^i DivRing ) |
|
| 2 | drhmsubc.j | |- J = ( r e. C , s e. C |-> ( r RingHom s ) ) |
|
| 3 | fldhmsubc.d | |- D = ( U i^i Field ) |
|
| 4 | fldhmsubc.f | |- F = ( r e. D , s e. D |-> ( r RingHom s ) ) |
|
| 5 | isfld | |- ( r e. Field <-> ( r e. DivRing /\ r e. CRing ) ) |
|
| 6 | crngring | |- ( r e. CRing -> r e. Ring ) |
|
| 7 | 6 | adantl | |- ( ( r e. DivRing /\ r e. CRing ) -> r e. Ring ) |
| 8 | 5 7 | sylbi | |- ( r e. Field -> r e. Ring ) |
| 9 | 8 | rgen | |- A. r e. Field r e. Ring |
| 10 | 9 3 4 | sringcat | |- ( U e. V -> ( ( RingCat ` U ) |`cat F ) e. Cat ) |