Step |
Hyp |
Ref |
Expression |
1 |
|
subrgmvr.v |
⊢ 𝑉 = ( 𝐼 mVar 𝑅 ) |
2 |
|
subrgmvr.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
3 |
|
subrgmvr.r |
⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) ) |
4 |
|
subrgmvr.h |
⊢ 𝐻 = ( 𝑅 ↾s 𝑇 ) |
5 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
6 |
4 5
|
subrg1 |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝐻 ) ) |
7 |
3 6
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝐻 ) ) |
8 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
9 |
4 8
|
subrg0 |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝐻 ) ) |
10 |
3 9
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝐻 ) ) |
11 |
7 10
|
ifeq12d |
⊢ ( 𝜑 → if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) = if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝐻 ) , ( 0g ‘ 𝐻 ) ) ) |
12 |
11
|
mpteq2dv |
⊢ ( 𝜑 → ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) = ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝐻 ) , ( 0g ‘ 𝐻 ) ) ) ) |
13 |
12
|
mpteq2dv |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝐻 ) , ( 0g ‘ 𝐻 ) ) ) ) ) |
14 |
|
eqid |
⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
15 |
|
subrgrcl |
⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑅 ∈ Ring ) |
16 |
3 15
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
17 |
1 14 8 5 2 16
|
mvrfval |
⊢ ( 𝜑 → 𝑉 = ( 𝑥 ∈ 𝐼 ↦ ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) ) |
18 |
|
eqid |
⊢ ( 𝐼 mVar 𝐻 ) = ( 𝐼 mVar 𝐻 ) |
19 |
|
eqid |
⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 ) |
20 |
|
eqid |
⊢ ( 1r ‘ 𝐻 ) = ( 1r ‘ 𝐻 ) |
21 |
4
|
ovexi |
⊢ 𝐻 ∈ V |
22 |
21
|
a1i |
⊢ ( 𝜑 → 𝐻 ∈ V ) |
23 |
18 14 19 20 2 22
|
mvrfval |
⊢ ( 𝜑 → ( 𝐼 mVar 𝐻 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ if ( 𝑦 = ( 𝑧 ∈ 𝐼 ↦ if ( 𝑧 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝐻 ) , ( 0g ‘ 𝐻 ) ) ) ) ) |
24 |
13 17 23
|
3eqtr4d |
⊢ ( 𝜑 → 𝑉 = ( 𝐼 mVar 𝐻 ) ) |