Step |
Hyp |
Ref |
Expression |
1 |
|
elrsp.n |
⊢ 𝑁 = ( RSpan ‘ 𝑅 ) |
2 |
|
elrsp.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
elrsp.1 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
elrsp.x |
⊢ · = ( .r ‘ 𝑅 ) |
5 |
|
elrsp.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
6 |
|
elrsp.i |
⊢ ( 𝜑 → 𝐼 ⊆ 𝐵 ) |
7 |
|
rspval |
⊢ ( RSpan ‘ 𝑅 ) = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) ) |
8 |
1 7
|
eqtri |
⊢ 𝑁 = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) ) |
9 |
|
rlmbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) |
10 |
2 9
|
eqtri |
⊢ 𝐵 = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) |
11 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) = ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
12 |
|
eqid |
⊢ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) |
13 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) = ( 0g ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
14 |
|
rlmvsca |
⊢ ( .r ‘ 𝑅 ) = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) ) |
15 |
4 14
|
eqtri |
⊢ · = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) ) |
16 |
|
rlmlmod |
⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
17 |
5 16
|
syl |
⊢ ( 𝜑 → ( ringLMod ‘ 𝑅 ) ∈ LMod ) |
18 |
8 10 11 12 13 15 17 6
|
ellspds |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ 𝐼 ) ↔ ∃ 𝑎 ∈ ( ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ↑m 𝐼 ) ( 𝑎 finSupp ( 0g ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ∧ 𝑋 = ( ( ringLMod ‘ 𝑅 ) Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ) ) ) |
19 |
|
rlmsca |
⊢ ( 𝑅 ∈ Ring → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
20 |
5 19
|
syl |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
21 |
20
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) |
22 |
2 21
|
eqtrid |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) |
23 |
22
|
oveq1d |
⊢ ( 𝜑 → ( 𝐵 ↑m 𝐼 ) = ( ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ↑m 𝐼 ) ) |
24 |
20
|
fveq2d |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) = ( 0g ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) |
25 |
3 24
|
eqtrid |
⊢ ( 𝜑 → 0 = ( 0g ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) |
26 |
25
|
breq2d |
⊢ ( 𝜑 → ( 𝑎 finSupp 0 ↔ 𝑎 finSupp ( 0g ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) ) |
27 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
28 |
27
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
29 |
28 6
|
ssexd |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
30 |
29
|
mptexd |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ∈ V ) |
31 |
9
|
a1i |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) ) |
32 |
|
rlmplusg |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ ( ringLMod ‘ 𝑅 ) ) |
33 |
32
|
a1i |
⊢ ( 𝜑 → ( +g ‘ 𝑅 ) = ( +g ‘ ( ringLMod ‘ 𝑅 ) ) ) |
34 |
30 5 17 31 33
|
gsumpropd |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) = ( ( ringLMod ‘ 𝑅 ) Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ) |
35 |
34
|
eqeq2d |
⊢ ( 𝜑 → ( 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ↔ 𝑋 = ( ( ringLMod ‘ 𝑅 ) Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ) ) |
36 |
26 35
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ) ↔ ( 𝑎 finSupp ( 0g ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ∧ 𝑋 = ( ( ringLMod ‘ 𝑅 ) Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ) ) ) |
37 |
23 36
|
rexeqbidv |
⊢ ( 𝜑 → ( ∃ 𝑎 ∈ ( 𝐵 ↑m 𝐼 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ) ↔ ∃ 𝑎 ∈ ( ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ↑m 𝐼 ) ( 𝑎 finSupp ( 0g ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ∧ 𝑋 = ( ( ringLMod ‘ 𝑅 ) Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ) ) ) |
38 |
18 37
|
bitr4d |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ 𝐼 ) ↔ ∃ 𝑎 ∈ ( 𝐵 ↑m 𝐼 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝐼 ↦ ( ( 𝑎 ‘ 𝑖 ) · 𝑖 ) ) ) ) ) ) |