Step |
Hyp |
Ref |
Expression |
1 |
|
evl1varpw.q |
⊢ 𝑄 = ( eval1 ‘ 𝑅 ) |
2 |
|
evl1varpw.w |
⊢ 𝑊 = ( Poly1 ‘ 𝑅 ) |
3 |
|
evl1varpw.g |
⊢ 𝐺 = ( mulGrp ‘ 𝑊 ) |
4 |
|
evl1varpw.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
5 |
|
evl1varpw.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
6 |
|
evl1varpw.e |
⊢ ↑ = ( .g ‘ 𝐺 ) |
7 |
|
evl1varpw.r |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
8 |
|
evl1varpw.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
9 |
|
evl1varpwval.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐵 ) |
10 |
|
evl1varpwval.h |
⊢ 𝐻 = ( mulGrp ‘ 𝑅 ) |
11 |
|
evl1varpwval.e |
⊢ 𝐸 = ( .g ‘ 𝐻 ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
13 |
1 4 5 2 12 7 9
|
evl1vard |
⊢ ( 𝜑 → ( 𝑋 ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝑄 ‘ 𝑋 ) ‘ 𝐶 ) = 𝐶 ) ) |
14 |
3
|
fveq2i |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ ( mulGrp ‘ 𝑊 ) ) |
15 |
6 14
|
eqtri |
⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑊 ) ) |
16 |
10
|
fveq2i |
⊢ ( .g ‘ 𝐻 ) = ( .g ‘ ( mulGrp ‘ 𝑅 ) ) |
17 |
11 16
|
eqtri |
⊢ 𝐸 = ( .g ‘ ( mulGrp ‘ 𝑅 ) ) |
18 |
1 2 5 12 7 9 13 15 17 8
|
evl1expd |
⊢ ( 𝜑 → ( ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) ) ) |
19 |
18
|
simprd |
⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) ) |