Step |
Hyp |
Ref |
Expression |
1 |
|
evls1varsrng.q |
⊢ 𝑄 = ( 𝑆 evalSub1 𝑅 ) |
2 |
|
evls1varsrng.o |
⊢ 𝑂 = ( eval1 ‘ 𝑆 ) |
3 |
|
evls1varsrng.v |
⊢ 𝑉 = ( var1 ‘ 𝑈 ) |
4 |
|
evls1varsrng.u |
⊢ 𝑈 = ( 𝑆 ↾s 𝑅 ) |
5 |
|
evls1varsrng.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
6 |
|
evls1varsrng.s |
⊢ ( 𝜑 → 𝑆 ∈ CRing ) |
7 |
|
evls1varsrng.r |
⊢ ( 𝜑 → 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) |
8 |
1 3 4 5 6 7
|
evls1var |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑉 ) = ( I ↾ 𝐵 ) ) |
9 |
2 5
|
evl1fval1 |
⊢ 𝑂 = ( 𝑆 evalSub1 𝐵 ) |
10 |
9
|
a1i |
⊢ ( 𝜑 → 𝑂 = ( 𝑆 evalSub1 𝐵 ) ) |
11 |
10
|
fveq1d |
⊢ ( 𝜑 → ( 𝑂 ‘ 𝑉 ) = ( ( 𝑆 evalSub1 𝐵 ) ‘ 𝑉 ) ) |
12 |
3
|
a1i |
⊢ ( 𝜑 → 𝑉 = ( var1 ‘ 𝑈 ) ) |
13 |
|
eqid |
⊢ ( var1 ‘ 𝑆 ) = ( var1 ‘ 𝑆 ) |
14 |
13 7 4
|
subrgvr1 |
⊢ ( 𝜑 → ( var1 ‘ 𝑆 ) = ( var1 ‘ 𝑈 ) ) |
15 |
5
|
ressid |
⊢ ( 𝑆 ∈ CRing → ( 𝑆 ↾s 𝐵 ) = 𝑆 ) |
16 |
6 15
|
syl |
⊢ ( 𝜑 → ( 𝑆 ↾s 𝐵 ) = 𝑆 ) |
17 |
16
|
eqcomd |
⊢ ( 𝜑 → 𝑆 = ( 𝑆 ↾s 𝐵 ) ) |
18 |
17
|
fveq2d |
⊢ ( 𝜑 → ( var1 ‘ 𝑆 ) = ( var1 ‘ ( 𝑆 ↾s 𝐵 ) ) ) |
19 |
12 14 18
|
3eqtr2d |
⊢ ( 𝜑 → 𝑉 = ( var1 ‘ ( 𝑆 ↾s 𝐵 ) ) ) |
20 |
19
|
fveq2d |
⊢ ( 𝜑 → ( ( 𝑆 evalSub1 𝐵 ) ‘ 𝑉 ) = ( ( 𝑆 evalSub1 𝐵 ) ‘ ( var1 ‘ ( 𝑆 ↾s 𝐵 ) ) ) ) |
21 |
|
eqid |
⊢ ( 𝑆 evalSub1 𝐵 ) = ( 𝑆 evalSub1 𝐵 ) |
22 |
|
eqid |
⊢ ( var1 ‘ ( 𝑆 ↾s 𝐵 ) ) = ( var1 ‘ ( 𝑆 ↾s 𝐵 ) ) |
23 |
|
eqid |
⊢ ( 𝑆 ↾s 𝐵 ) = ( 𝑆 ↾s 𝐵 ) |
24 |
|
crngring |
⊢ ( 𝑆 ∈ CRing → 𝑆 ∈ Ring ) |
25 |
5
|
subrgid |
⊢ ( 𝑆 ∈ Ring → 𝐵 ∈ ( SubRing ‘ 𝑆 ) ) |
26 |
6 24 25
|
3syl |
⊢ ( 𝜑 → 𝐵 ∈ ( SubRing ‘ 𝑆 ) ) |
27 |
21 22 23 5 6 26
|
evls1var |
⊢ ( 𝜑 → ( ( 𝑆 evalSub1 𝐵 ) ‘ ( var1 ‘ ( 𝑆 ↾s 𝐵 ) ) ) = ( I ↾ 𝐵 ) ) |
28 |
11 20 27
|
3eqtrrd |
⊢ ( 𝜑 → ( I ↾ 𝐵 ) = ( 𝑂 ‘ 𝑉 ) ) |
29 |
8 28
|
eqtrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑉 ) = ( 𝑂 ‘ 𝑉 ) ) |