Description: The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014) (Revised by Thierry Arnoux, 1-Apr-2018)
Ref | Expression | ||
---|---|---|---|
Hypotheses | srg0cl.b | |- B = ( Base ` R ) |
|
srg0cl.z | |- .0. = ( 0g ` R ) |
||
Assertion | srg0cl | |- ( R e. SRing -> .0. e. B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srg0cl.b | |- B = ( Base ` R ) |
|
2 | srg0cl.z | |- .0. = ( 0g ` R ) |
|
3 | srgmnd | |- ( R e. SRing -> R e. Mnd ) |
|
4 | 1 2 | mndidcl | |- ( R e. Mnd -> .0. e. B ) |
5 | 3 4 | syl | |- ( R e. SRing -> .0. e. B ) |