Description: Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | srapart.a | |- ( ph -> A = ( ( subringAlg ` W ) ` S ) ) |
|
srapart.s | |- ( ph -> S C_ ( Base ` W ) ) |
||
Assertion | srabase | |- ( ph -> ( Base ` W ) = ( Base ` A ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srapart.a | |- ( ph -> A = ( ( subringAlg ` W ) ` S ) ) |
|
2 | srapart.s | |- ( ph -> S C_ ( Base ` W ) ) |
|
3 | df-base | |- Base = Slot 1 |
|
4 | 1nn | |- 1 e. NN |
|
5 | 1lt5 | |- 1 < 5 |
|
6 | 5 | orci | |- ( 1 < 5 \/ 8 < 1 ) |
7 | 1 2 3 4 6 | sralem | |- ( ph -> ( Base ` W ) = ( Base ` A ) ) |