Step |
Hyp |
Ref |
Expression |
1 |
|
baerlem3.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
baerlem3.m |
⊢ − = ( -g ‘ 𝑊 ) |
3 |
|
baerlem3.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
4 |
|
baerlem3.s |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
5 |
|
baerlem3.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
6 |
|
baerlem3.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
7 |
|
baerlem3.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
8 |
|
baerlem3.c |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) |
9 |
|
baerlem3.d |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) |
10 |
|
baerlem3.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
11 |
|
baerlem3.z |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) ) |
12 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
13 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
14 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
15 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
16 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑊 ) ) = ( +g ‘ ( Scalar ‘ 𝑊 ) ) |
17 |
|
eqid |
⊢ ( -g ‘ ( Scalar ‘ 𝑊 ) ) = ( -g ‘ ( Scalar ‘ 𝑊 ) ) |
18 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
19 |
|
eqid |
⊢ ( invg ‘ ( Scalar ‘ 𝑊 ) ) = ( invg ‘ ( Scalar ‘ 𝑊 ) ) |
20 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
baerlem3lem2 |
⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 − 𝑍 ) } ) = ( ( ( 𝑁 ‘ { 𝑌 } ) ⊕ ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊕ ( 𝑁 ‘ { ( 𝑋 − 𝑍 ) } ) ) ) ) |