Description: Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Revised by Thierry Arnoux, 1-Apr-2018)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | srgcl.b | |- B = ( Base ` R ) |
|
| srgcl.t | |- .x. = ( .r ` R ) |
||
| Assertion | srgcl | |- ( ( R e. SRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | srgcl.b | |- B = ( Base ` R ) |
|
| 2 | srgcl.t | |- .x. = ( .r ` 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 | mgpplusg | |- .x. = ( +g ` ( mulGrp ` R ) ) |
| 7 | 5 6 | mndcl | |- ( ( ( mulGrp ` R ) e. Mnd /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B ) |
| 8 | 4 7 | syl3an1 | |- ( ( R e. SRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B ) |