Metamath Proof Explorer
Description: The scalar 1 embedded into an associative algebra corresponds to the 1
of the an associative algebra. (Contributed by AV, 31-Jul-2019)
|
|
Ref |
Expression |
|
Hypotheses |
assaascl0.a |
⊢ 𝐴 = ( algSc ‘ 𝑊 ) |
|
|
assaascl0.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
|
|
assaascl0.w |
⊢ ( 𝜑 → 𝑊 ∈ AssAlg ) |
|
Assertion |
assaascl1 |
⊢ ( 𝜑 → ( 𝐴 ‘ ( 1r ‘ 𝐹 ) ) = ( 1r ‘ 𝑊 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
assaascl0.a |
⊢ 𝐴 = ( algSc ‘ 𝑊 ) |
2 |
|
assaascl0.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
3 |
|
assaascl0.w |
⊢ ( 𝜑 → 𝑊 ∈ AssAlg ) |
4 |
|
assalmod |
⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ LMod ) |
5 |
3 4
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
6 |
|
assaring |
⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ Ring ) |
7 |
3 6
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ Ring ) |
8 |
1 2 5 7
|
ascl1 |
⊢ ( 𝜑 → ( 𝐴 ‘ ( 1r ‘ 𝐹 ) ) = ( 1r ‘ 𝑊 ) ) |