Step |
Hyp |
Ref |
Expression |
1 |
|
mplsubg.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
mplsubg.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
3 |
|
mplsubg.u |
⊢ 𝑈 = ( Base ‘ 𝑃 ) |
4 |
|
mplsubg.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
6 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
7 |
2 1 5 6 3
|
mplbas |
⊢ 𝑈 = { 𝑔 ∈ ( Base ‘ 𝑆 ) ∣ 𝑔 finSupp ( 0g ‘ 𝑅 ) } |
8 |
1 5
|
psrelbasfun |
⊢ ( 𝑔 ∈ ( Base ‘ 𝑆 ) → Fun 𝑔 ) |
9 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( Base ‘ 𝑆 ) ) → Fun 𝑔 ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( Base ‘ 𝑆 ) ) → 𝑔 ∈ ( Base ‘ 𝑆 ) ) |
11 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( Base ‘ 𝑆 ) ) → ( 0g ‘ 𝑅 ) ∈ V ) |
12 |
|
funisfsupp |
⊢ ( ( Fun 𝑔 ∧ 𝑔 ∈ ( Base ‘ 𝑆 ) ∧ ( 0g ‘ 𝑅 ) ∈ V ) → ( 𝑔 finSupp ( 0g ‘ 𝑅 ) ↔ ( 𝑔 supp ( 0g ‘ 𝑅 ) ) ∈ Fin ) ) |
13 |
9 10 11 12
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑔 finSupp ( 0g ‘ 𝑅 ) ↔ ( 𝑔 supp ( 0g ‘ 𝑅 ) ) ∈ Fin ) ) |
14 |
13
|
rabbidva |
⊢ ( 𝜑 → { 𝑔 ∈ ( Base ‘ 𝑆 ) ∣ 𝑔 finSupp ( 0g ‘ 𝑅 ) } = { 𝑔 ∈ ( Base ‘ 𝑆 ) ∣ ( 𝑔 supp ( 0g ‘ 𝑅 ) ) ∈ Fin } ) |
15 |
7 14
|
syl5eq |
⊢ ( 𝜑 → 𝑈 = { 𝑔 ∈ ( Base ‘ 𝑆 ) ∣ ( 𝑔 supp ( 0g ‘ 𝑅 ) ) ∈ Fin } ) |