Step |
Hyp |
Ref |
Expression |
1 |
|
frlmval.f |
⊢ 𝐹 = ( 𝑅 freeLMod 𝐼 ) |
2 |
|
frlmpws.b |
⊢ 𝐵 = ( Base ‘ 𝐹 ) |
3 |
|
frlmlss.u |
⊢ 𝑈 = ( LSubSp ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) |
4 |
1
|
frlmval |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → 𝐹 = ( 𝑅 ⊕m ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) |
5 |
4
|
fveq2d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( Base ‘ 𝐹 ) = ( Base ‘ ( 𝑅 ⊕m ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) ) |
6 |
2 5
|
syl5eq |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → 𝐵 = ( Base ‘ ( 𝑅 ⊕m ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) ) |
7 |
|
simpr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → 𝐼 ∈ 𝑊 ) |
8 |
|
simpl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → 𝑅 ∈ Ring ) |
9 |
|
rlmlmod |
⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
10 |
9
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
11 |
|
fconst6g |
⊢ ( ( ringLMod ‘ 𝑅 ) ∈ LMod → ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) : 𝐼 ⟶ LMod ) |
12 |
10 11
|
syl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) : 𝐼 ⟶ LMod ) |
13 |
|
fvex |
⊢ ( ringLMod ‘ 𝑅 ) ∈ V |
14 |
13
|
fvconst2 |
⊢ ( 𝑖 ∈ 𝐼 → ( ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ‘ 𝑖 ) = ( ringLMod ‘ 𝑅 ) ) |
15 |
14
|
adantl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ‘ 𝑖 ) = ( ringLMod ‘ 𝑅 ) ) |
16 |
15
|
fveq2d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) ∧ 𝑖 ∈ 𝐼 ) → ( Scalar ‘ ( ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ‘ 𝑖 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
17 |
|
rlmsca |
⊢ ( 𝑅 ∈ Ring → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
18 |
17
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) ∧ 𝑖 ∈ 𝐼 ) → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
19 |
16 18
|
eqtr4d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) ∧ 𝑖 ∈ 𝐼 ) → ( Scalar ‘ ( ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ‘ 𝑖 ) ) = 𝑅 ) |
20 |
|
eqid |
⊢ ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) = ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) |
21 |
|
eqid |
⊢ ( LSubSp ‘ ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) = ( LSubSp ‘ ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) |
22 |
|
eqid |
⊢ ( Base ‘ ( 𝑅 ⊕m ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) = ( Base ‘ ( 𝑅 ⊕m ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) |
23 |
7 8 12 19 20 21 22
|
dsmmlss |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( Base ‘ ( 𝑅 ⊕m ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) ∈ ( LSubSp ‘ ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) ) |
24 |
|
eqid |
⊢ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) = ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) |
25 |
|
eqid |
⊢ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) |
26 |
24 25
|
pwsval |
⊢ ( ( ( ringLMod ‘ 𝑅 ) ∈ V ∧ 𝐼 ∈ 𝑊 ) → ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) = ( ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) |
27 |
13 26
|
mpan |
⊢ ( 𝐼 ∈ 𝑊 → ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) = ( ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) |
28 |
27
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) = ( ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) |
29 |
17
|
eqcomd |
⊢ ( 𝑅 ∈ Ring → ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) = 𝑅 ) |
30 |
29
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) = 𝑅 ) |
31 |
30
|
oveq1d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) = ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) |
32 |
28 31
|
eqtr2d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) = ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) |
33 |
32
|
fveq2d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( LSubSp ‘ ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) = ( LSubSp ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) ) |
34 |
33 3
|
eqtr4di |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( LSubSp ‘ ( 𝑅 Xs ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) = 𝑈 ) |
35 |
23 34
|
eleqtrd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → ( Base ‘ ( 𝑅 ⊕m ( 𝐼 × { ( ringLMod ‘ 𝑅 ) } ) ) ) ∈ 𝑈 ) |
36 |
6 35
|
eqeltrd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → 𝐵 ∈ 𝑈 ) |