Step |
Hyp |
Ref |
Expression |
1 |
|
lfladd0l.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lfladd0l.r |
⊢ 𝑅 = ( Scalar ‘ 𝑊 ) |
3 |
|
lfladd0l.p |
⊢ + = ( +g ‘ 𝑅 ) |
4 |
|
lfladd0l.o |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
lfladd0l.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
6 |
|
lfladd0l.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
7 |
|
lfladd0l.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
8 |
1
|
fvexi |
⊢ 𝑉 ∈ V |
9 |
8
|
a1i |
⊢ ( 𝜑 → 𝑉 ∈ V ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
11 |
2 10 1 5
|
lflf |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → 𝐺 : 𝑉 ⟶ ( Base ‘ 𝑅 ) ) |
12 |
6 7 11
|
syl2anc |
⊢ ( 𝜑 → 𝐺 : 𝑉 ⟶ ( Base ‘ 𝑅 ) ) |
13 |
4
|
fvexi |
⊢ 0 ∈ V |
14 |
13
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
15 |
2
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝑅 ∈ Ring ) |
16 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
17 |
6 15 16
|
3syl |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
18 |
10 3 4
|
grplid |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑘 ∈ ( Base ‘ 𝑅 ) ) → ( 0 + 𝑘 ) = 𝑘 ) |
19 |
17 18
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑅 ) ) → ( 0 + 𝑘 ) = 𝑘 ) |
20 |
9 12 14 19
|
caofid0l |
⊢ ( 𝜑 → ( ( 𝑉 × { 0 } ) ∘f + 𝐺 ) = 𝐺 ) |