Step |
Hyp |
Ref |
Expression |
1 |
|
lfladdcl.r |
⊢ 𝑅 = ( Scalar ‘ 𝑊 ) |
2 |
|
lfladdcl.p |
⊢ + = ( +g ‘ 𝑅 ) |
3 |
|
lfladdcl.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
4 |
|
lfladdcl.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
5 |
|
lfladdcl.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
6 |
|
lfladdcl.h |
⊢ ( 𝜑 → 𝐻 ∈ 𝐹 ) |
7 |
|
fvexd |
⊢ ( 𝜑 → ( Base ‘ 𝑊 ) ∈ V ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
10 |
1 8 9 3
|
lflf |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → 𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ 𝑅 ) ) |
11 |
4 5 10
|
syl2anc |
⊢ ( 𝜑 → 𝐺 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ 𝑅 ) ) |
12 |
1 8 9 3
|
lflf |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ) → 𝐻 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ 𝑅 ) ) |
13 |
4 6 12
|
syl2anc |
⊢ ( 𝜑 → 𝐻 : ( Base ‘ 𝑊 ) ⟶ ( Base ‘ 𝑅 ) ) |
14 |
1
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝑅 ∈ Ring ) |
15 |
|
ringabl |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Abel ) |
16 |
4 14 15
|
3syl |
⊢ ( 𝜑 → 𝑅 ∈ Abel ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑅 ∈ Abel ) |
18 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) ) |
19 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑅 ) ) |
20 |
8 2
|
ablcom |
⊢ ( ( 𝑅 ∈ Abel ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) |
21 |
17 18 19 20
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) |
22 |
7 11 13 21
|
caofcom |
⊢ ( 𝜑 → ( 𝐺 ∘f + 𝐻 ) = ( 𝐻 ∘f + 𝐺 ) ) |