Step |
Hyp |
Ref |
Expression |
1 |
|
rspsnel.1 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
rspsnel.2 |
⊢ · = ( .r ‘ 𝑅 ) |
3 |
|
rspsnel.3 |
⊢ 𝐾 = ( RSpan ‘ 𝑅 ) |
4 |
|
rlmlmod |
⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
5 |
|
simpr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
6 |
5 1
|
eleqtrdi |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ ( Base ‘ 𝑅 ) ) |
7 |
|
eqid |
⊢ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) |
8 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) = ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
9 |
|
rlmbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) |
10 |
|
rlmvsca |
⊢ ( .r ‘ 𝑅 ) = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) ) |
11 |
2 10
|
eqtri |
⊢ · = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) ) |
12 |
|
rspval |
⊢ ( RSpan ‘ 𝑅 ) = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) ) |
13 |
3 12
|
eqtri |
⊢ 𝐾 = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) ) |
14 |
7 8 9 11 13
|
lspsnel |
⊢ ( ( ( ringLMod ‘ 𝑅 ) ∈ LMod ∧ 𝑋 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐼 ∈ ( 𝐾 ‘ { 𝑋 } ) ↔ ∃ 𝑥 ∈ ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) 𝐼 = ( 𝑥 · 𝑋 ) ) ) |
15 |
4 6 14
|
syl2an2r |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ∈ ( 𝐾 ‘ { 𝑋 } ) ↔ ∃ 𝑥 ∈ ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) 𝐼 = ( 𝑥 · 𝑋 ) ) ) |
16 |
|
rlmsca |
⊢ ( 𝑅 ∈ Ring → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
18 |
17
|
fveq2d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) |
19 |
1 18
|
eqtr2id |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) = 𝐵 ) |
20 |
19
|
rexeqdv |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) 𝐼 = ( 𝑥 · 𝑋 ) ↔ ∃ 𝑥 ∈ 𝐵 𝐼 = ( 𝑥 · 𝑋 ) ) ) |
21 |
15 20
|
bitrd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ∈ ( 𝐾 ‘ { 𝑋 } ) ↔ ∃ 𝑥 ∈ 𝐵 𝐼 = ( 𝑥 · 𝑋 ) ) ) |