Step |
Hyp |
Ref |
Expression |
1 |
|
idlsrgmulrss2.1 |
⊢ 𝑆 = ( IDLsrg ‘ 𝑅 ) |
2 |
|
idlsrgmulrss2.2 |
⊢ 𝐵 = ( LIdeal ‘ 𝑅 ) |
3 |
|
idlsrgmulrss2.3 |
⊢ ⊗ = ( .r ‘ 𝑆 ) |
4 |
|
idlsrgmulrss2.5 |
⊢ · = ( .r ‘ 𝑅 ) |
5 |
|
idlsrgmulrss2.6 |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
6 |
|
idlsrgmulrss2.7 |
⊢ ( 𝜑 → 𝐼 ∈ 𝐵 ) |
7 |
|
idlsrgmulrss2.8 |
⊢ ( 𝜑 → 𝐽 ∈ 𝐵 ) |
8 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
9 |
|
eqid |
⊢ ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) = ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) |
10 |
1 2 3 8 9 5 6 7
|
idlsrgmulrval |
⊢ ( 𝜑 → ( 𝐼 ⊗ 𝐽 ) = ( ( RSpan ‘ 𝑅 ) ‘ ( 𝐼 ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) 𝐽 ) ) ) |
11 |
|
rlmlmod |
⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
12 |
5 11
|
syl |
⊢ ( 𝜑 → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
14 |
13 2
|
lidlss |
⊢ ( 𝐽 ∈ 𝐵 → 𝐽 ⊆ ( Base ‘ 𝑅 ) ) |
15 |
7 14
|
syl |
⊢ ( 𝜑 → 𝐽 ⊆ ( Base ‘ 𝑅 ) ) |
16 |
13 2
|
lidlss |
⊢ ( 𝐼 ∈ 𝐵 → 𝐼 ⊆ ( Base ‘ 𝑅 ) ) |
17 |
6 16
|
syl |
⊢ ( 𝜑 → 𝐼 ⊆ ( Base ‘ 𝑅 ) ) |
18 |
7 2
|
eleqtrdi |
⊢ ( 𝜑 → 𝐽 ∈ ( LIdeal ‘ 𝑅 ) ) |
19 |
13 8 9 5 17 18
|
ringlsmss2 |
⊢ ( 𝜑 → ( 𝐼 ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) 𝐽 ) ⊆ 𝐽 ) |
20 |
|
rlmbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) |
21 |
|
rspval |
⊢ ( RSpan ‘ 𝑅 ) = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) ) |
22 |
20 21
|
lspss |
⊢ ( ( ( ringLMod ‘ 𝑅 ) ∈ LMod ∧ 𝐽 ⊆ ( Base ‘ 𝑅 ) ∧ ( 𝐼 ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) 𝐽 ) ⊆ 𝐽 ) → ( ( RSpan ‘ 𝑅 ) ‘ ( 𝐼 ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) 𝐽 ) ) ⊆ ( ( RSpan ‘ 𝑅 ) ‘ 𝐽 ) ) |
23 |
12 15 19 22
|
syl3anc |
⊢ ( 𝜑 → ( ( RSpan ‘ 𝑅 ) ‘ ( 𝐼 ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) 𝐽 ) ) ⊆ ( ( RSpan ‘ 𝑅 ) ‘ 𝐽 ) ) |
24 |
|
eqid |
⊢ ( RSpan ‘ 𝑅 ) = ( RSpan ‘ 𝑅 ) |
25 |
24 2
|
rspidlid |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐽 ∈ 𝐵 ) → ( ( RSpan ‘ 𝑅 ) ‘ 𝐽 ) = 𝐽 ) |
26 |
5 7 25
|
syl2anc |
⊢ ( 𝜑 → ( ( RSpan ‘ 𝑅 ) ‘ 𝐽 ) = 𝐽 ) |
27 |
23 26
|
sseqtrd |
⊢ ( 𝜑 → ( ( RSpan ‘ 𝑅 ) ‘ ( 𝐼 ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) 𝐽 ) ) ⊆ 𝐽 ) |
28 |
10 27
|
eqsstrd |
⊢ ( 𝜑 → ( 𝐼 ⊗ 𝐽 ) ⊆ 𝐽 ) |