Description: The unit element of a semiring belongs to the base set of the semiring. (Contributed by NM, 27-Aug-2011) (Revised by Mario Carneiro, 27-Dec-2014) (Revised by Thierry Arnoux, 1-Apr-2018)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | srgidcl.b | |- B = ( Base ` R ) |
|
| srgidcl.u | |- .1. = ( 1r ` R ) |
||
| Assertion | srgidcl | |- ( R e. SRing -> .1. e. B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | srgidcl.b | |- B = ( Base ` R ) |
|
| 2 | srgidcl.u | |- .1. = ( 1r ` R ) |
|
| 3 | eqid | |- ( mulGrp ` R ) = ( mulGrp ` R ) |
|
| 4 | 3 | srgmgp | |- ( R e. SRing -> ( mulGrp ` R ) e. Mnd ) |
| 5 | 3 1 | mgpbas | |- B = ( Base ` ( mulGrp ` R ) ) |
| 6 | 3 2 | ringidval | |- .1. = ( 0g ` ( mulGrp ` R ) ) |
| 7 | 5 6 | mndidcl | |- ( ( mulGrp ` R ) e. Mnd -> .1. e. B ) |
| 8 | 4 7 | syl | |- ( R e. SRing -> .1. e. B ) |