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 |
|
evl1scvarpw.t1 |
⊢ × = ( ·𝑠 ‘ 𝑊 ) |
10 |
|
evl1scvarpw.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) |
11 |
|
evl1scvarpwval.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐵 ) |
12 |
|
evl1scvarpwval.h |
⊢ 𝐻 = ( mulGrp ‘ 𝑅 ) |
13 |
|
evl1scvarpwval.e |
⊢ 𝐸 = ( .g ‘ 𝐻 ) |
14 |
|
evl1scvarpwval.t |
⊢ · = ( .r ‘ 𝑅 ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
16 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
17 |
7 16
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
18 |
2
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑊 ∈ Ring ) |
19 |
17 18
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ Ring ) |
20 |
3
|
ringmgp |
⊢ ( 𝑊 ∈ Ring → 𝐺 ∈ Mnd ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
22 |
4 2 15
|
vr1cl |
⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
23 |
17 22
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
24 |
3 15
|
mgpbas |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝐺 ) |
25 |
24 6
|
mulgnn0cl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ( Base ‘ 𝑊 ) ) → ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) |
26 |
21 8 23 25
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) |
27 |
1 2 3 4 5 6 7 8 11 12 13
|
evl1varpwval |
⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) ) |
28 |
26 27
|
jca |
⊢ ( 𝜑 → ( ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ‘ 𝐶 ) = ( 𝑁 𝐸 𝐶 ) ) ) |
29 |
1 2 5 15 7 11 28 10 9 14
|
evl1vsd |
⊢ ( 𝜑 → ( ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) ‘ 𝐶 ) = ( 𝐴 · ( 𝑁 𝐸 𝐶 ) ) ) ) |
30 |
29
|
simprd |
⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) ‘ 𝐶 ) = ( 𝐴 · ( 𝑁 𝐸 𝐶 ) ) ) |