Description: A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lkrcl.v | ⊢ 𝑉 = ( Base ‘ 𝑊 ) | |
lkrcl.f | ⊢ 𝐹 = ( LFnl ‘ 𝑊 ) | ||
lkrcl.k | ⊢ 𝐾 = ( LKer ‘ 𝑊 ) | ||
Assertion | lkrcl | ⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → 𝑋 ∈ 𝑉 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrcl.v | ⊢ 𝑉 = ( Base ‘ 𝑊 ) | |
2 | lkrcl.f | ⊢ 𝐹 = ( LFnl ‘ 𝑊 ) | |
3 | lkrcl.k | ⊢ 𝐾 = ( LKer ‘ 𝑊 ) | |
4 | eqid | ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) | |
5 | eqid | ⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) | |
6 | 1 4 5 2 3 | ellkr | ⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ) → ( 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ↔ ( 𝑋 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑋 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) ) |
7 | 6 | simprbda | ⊢ ( ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ) ∧ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → 𝑋 ∈ 𝑉 ) |
8 | 7 | 3impa | ⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ) → 𝑋 ∈ 𝑉 ) |