Description: Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | rlmbas | ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmval | ⊢ ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) ) | |
2 | 1 | a1i | ⊢ ( ⊤ → ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) ) ) |
3 | ssidd | ⊢ ( ⊤ → ( Base ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) ) | |
4 | 2 3 | srabase | ⊢ ( ⊤ → ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) ) |
5 | 4 | mptru | ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) |