Step |
Hyp |
Ref |
Expression |
1 |
|
lkrfval2.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lkrfval2.d |
⊢ 𝐷 = ( Scalar ‘ 𝑊 ) |
3 |
|
lkrfval2.o |
⊢ 0 = ( 0g ‘ 𝐷 ) |
4 |
|
lkrfval2.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
5 |
|
lkrfval2.k |
⊢ 𝐾 = ( LKer ‘ 𝑊 ) |
6 |
2 3 4 5
|
lkrval |
⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ) → ( 𝐾 ‘ 𝐺 ) = ( ◡ 𝐺 “ { 0 } ) ) |
7 |
6
|
eleq2d |
⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ) → ( 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ↔ 𝑋 ∈ ( ◡ 𝐺 “ { 0 } ) ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
9 |
2 8 1 4
|
lflf |
⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ) → 𝐺 : 𝑉 ⟶ ( Base ‘ 𝐷 ) ) |
10 |
|
ffn |
⊢ ( 𝐺 : 𝑉 ⟶ ( Base ‘ 𝐷 ) → 𝐺 Fn 𝑉 ) |
11 |
|
elpreima |
⊢ ( 𝐺 Fn 𝑉 → ( 𝑋 ∈ ( ◡ 𝐺 “ { 0 } ) ↔ ( 𝑋 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑋 ) ∈ { 0 } ) ) ) |
12 |
9 10 11
|
3syl |
⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ) → ( 𝑋 ∈ ( ◡ 𝐺 “ { 0 } ) ↔ ( 𝑋 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑋 ) ∈ { 0 } ) ) ) |
13 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑋 ) ∈ V |
14 |
13
|
elsn |
⊢ ( ( 𝐺 ‘ 𝑋 ) ∈ { 0 } ↔ ( 𝐺 ‘ 𝑋 ) = 0 ) |
15 |
14
|
anbi2i |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑋 ) ∈ { 0 } ) ↔ ( 𝑋 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑋 ) = 0 ) ) |
16 |
12 15
|
bitrdi |
⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ) → ( 𝑋 ∈ ( ◡ 𝐺 “ { 0 } ) ↔ ( 𝑋 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑋 ) = 0 ) ) ) |
17 |
7 16
|
bitrd |
⊢ ( ( 𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ) → ( 𝑋 ∈ ( 𝐾 ‘ 𝐺 ) ↔ ( 𝑋 ∈ 𝑉 ∧ ( 𝐺 ‘ 𝑋 ) = 0 ) ) ) |