Metamath Proof Explorer
Description: A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014)
|
|
Ref |
Expression |
|
Hypotheses |
mvrf.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
|
|
mvrf.v |
⊢ 𝑉 = ( 𝐼 mVar 𝑅 ) |
|
|
mvrf.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
|
|
mvrf.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
|
|
mvrf.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
|
|
mvrcl2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐼 ) |
|
Assertion |
mvrcl2 |
⊢ ( 𝜑 → ( 𝑉 ‘ 𝑋 ) ∈ 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mvrf.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
mvrf.v |
⊢ 𝑉 = ( 𝐼 mVar 𝑅 ) |
3 |
|
mvrf.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
4 |
|
mvrf.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
mvrf.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
6 |
|
mvrcl2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐼 ) |
7 |
1 2 3 4 5
|
mvrf |
⊢ ( 𝜑 → 𝑉 : 𝐼 ⟶ 𝐵 ) |
8 |
7 6
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑉 ‘ 𝑋 ) ∈ 𝐵 ) |