Step |
Hyp |
Ref |
Expression |
1 |
|
lsmidl.1 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
lsmidl.3 |
⊢ ⊕ = ( LSSum ‘ 𝑅 ) |
3 |
|
lsmidl.4 |
⊢ 𝐾 = ( RSpan ‘ 𝑅 ) |
4 |
|
lsmidl.5 |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
5 |
|
lsmidl.6 |
⊢ ( 𝜑 → 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) |
6 |
|
lsmidl.7 |
⊢ ( 𝜑 → 𝐽 ∈ ( LIdeal ‘ 𝑅 ) ) |
7 |
|
rlmlsm |
⊢ ( 𝑅 ∈ Ring → ( LSSum ‘ 𝑅 ) = ( LSSum ‘ ( ringLMod ‘ 𝑅 ) ) ) |
8 |
4 7
|
syl |
⊢ ( 𝜑 → ( LSSum ‘ 𝑅 ) = ( LSSum ‘ ( ringLMod ‘ 𝑅 ) ) ) |
9 |
2 8
|
syl5eq |
⊢ ( 𝜑 → ⊕ = ( LSSum ‘ ( ringLMod ‘ 𝑅 ) ) ) |
10 |
9
|
oveqd |
⊢ ( 𝜑 → ( 𝐼 ⊕ 𝐽 ) = ( 𝐼 ( LSSum ‘ ( ringLMod ‘ 𝑅 ) ) 𝐽 ) ) |
11 |
|
rlmlmod |
⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
12 |
4 11
|
syl |
⊢ ( 𝜑 → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
13 |
|
lidlval |
⊢ ( LIdeal ‘ 𝑅 ) = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) ) |
14 |
|
rspval |
⊢ ( RSpan ‘ 𝑅 ) = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) ) |
15 |
3 14
|
eqtri |
⊢ 𝐾 = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) ) |
16 |
|
eqid |
⊢ ( LSSum ‘ ( ringLMod ‘ 𝑅 ) ) = ( LSSum ‘ ( ringLMod ‘ 𝑅 ) ) |
17 |
13 15 16
|
lsmsp |
⊢ ( ( ( ringLMod ‘ 𝑅 ) ∈ LMod ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝐽 ∈ ( LIdeal ‘ 𝑅 ) ) → ( 𝐼 ( LSSum ‘ ( ringLMod ‘ 𝑅 ) ) 𝐽 ) = ( 𝐾 ‘ ( 𝐼 ∪ 𝐽 ) ) ) |
18 |
12 5 6 17
|
syl3anc |
⊢ ( 𝜑 → ( 𝐼 ( LSSum ‘ ( ringLMod ‘ 𝑅 ) ) 𝐽 ) = ( 𝐾 ‘ ( 𝐼 ∪ 𝐽 ) ) ) |
19 |
10 18
|
eqtrd |
⊢ ( 𝜑 → ( 𝐼 ⊕ 𝐽 ) = ( 𝐾 ‘ ( 𝐼 ∪ 𝐽 ) ) ) |