Step |
Hyp |
Ref |
Expression |
1 |
|
0ex |
⊢ ∅ ∈ V |
2 |
1
|
snid |
⊢ ∅ ∈ { ∅ } |
3 |
|
fvex |
⊢ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∈ V |
4 |
|
map0e |
⊢ ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∈ V → ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m ∅ ) = 1o ) |
5 |
3 4
|
mp1i |
⊢ ( 𝑀 ∈ 𝑋 → ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m ∅ ) = 1o ) |
6 |
|
df1o2 |
⊢ 1o = { ∅ } |
7 |
5 6
|
eqtrdi |
⊢ ( 𝑀 ∈ 𝑋 → ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m ∅ ) = { ∅ } ) |
8 |
2 7
|
eleqtrrid |
⊢ ( 𝑀 ∈ 𝑋 → ∅ ∈ ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m ∅ ) ) |
9 |
|
0elpw |
⊢ ∅ ∈ 𝒫 ( Base ‘ 𝑀 ) |
10 |
9
|
a1i |
⊢ ( 𝑀 ∈ 𝑋 → ∅ ∈ 𝒫 ( Base ‘ 𝑀 ) ) |
11 |
|
lincval |
⊢ ( ( 𝑀 ∈ 𝑋 ∧ ∅ ∈ ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m ∅ ) ∧ ∅ ∈ 𝒫 ( Base ‘ 𝑀 ) ) → ( ∅ ( linC ‘ 𝑀 ) ∅ ) = ( 𝑀 Σg ( 𝑣 ∈ ∅ ↦ ( ( ∅ ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) ) ) |
12 |
8 10 11
|
mpd3an23 |
⊢ ( 𝑀 ∈ 𝑋 → ( ∅ ( linC ‘ 𝑀 ) ∅ ) = ( 𝑀 Σg ( 𝑣 ∈ ∅ ↦ ( ( ∅ ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) ) ) |
13 |
|
mpt0 |
⊢ ( 𝑣 ∈ ∅ ↦ ( ( ∅ ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) = ∅ |
14 |
13
|
a1i |
⊢ ( 𝑀 ∈ 𝑋 → ( 𝑣 ∈ ∅ ↦ ( ( ∅ ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) = ∅ ) |
15 |
14
|
oveq2d |
⊢ ( 𝑀 ∈ 𝑋 → ( 𝑀 Σg ( 𝑣 ∈ ∅ ↦ ( ( ∅ ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) ) = ( 𝑀 Σg ∅ ) ) |
16 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
17 |
16
|
gsum0 |
⊢ ( 𝑀 Σg ∅ ) = ( 0g ‘ 𝑀 ) |
18 |
15 17
|
eqtrdi |
⊢ ( 𝑀 ∈ 𝑋 → ( 𝑀 Σg ( 𝑣 ∈ ∅ ↦ ( ( ∅ ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) ) = ( 0g ‘ 𝑀 ) ) |
19 |
12 18
|
eqtrd |
⊢ ( 𝑀 ∈ 𝑋 → ( ∅ ( linC ‘ 𝑀 ) ∅ ) = ( 0g ‘ 𝑀 ) ) |