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 |
|
evl1scvarpw.s |
⊢ 𝑆 = ( 𝑅 ↑s 𝐵 ) |
12 |
|
evl1scvarpw.t2 |
⊢ ∙ = ( .r ‘ 𝑆 ) |
13 |
|
evl1scvarpw.m |
⊢ 𝑀 = ( mulGrp ‘ 𝑆 ) |
14 |
|
evl1scvarpw.f |
⊢ 𝐹 = ( .g ‘ 𝑀 ) |
15 |
2
|
ply1assa |
⊢ ( 𝑅 ∈ CRing → 𝑊 ∈ AssAlg ) |
16 |
7 15
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ AssAlg ) |
17 |
10 5
|
eleqtrdi |
⊢ ( 𝜑 → 𝐴 ∈ ( Base ‘ 𝑅 ) ) |
18 |
2
|
ply1sca |
⊢ ( 𝑅 ∈ CRing → 𝑅 = ( Scalar ‘ 𝑊 ) ) |
19 |
18
|
eqcomd |
⊢ ( 𝑅 ∈ CRing → ( Scalar ‘ 𝑊 ) = 𝑅 ) |
20 |
7 19
|
syl |
⊢ ( 𝜑 → ( Scalar ‘ 𝑊 ) = 𝑅 ) |
21 |
20
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ 𝑅 ) ) |
22 |
17 21
|
eleqtrrd |
⊢ ( 𝜑 → 𝐴 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
23 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
24 |
3 23
|
mgpbas |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝐺 ) |
25 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
26 |
7 25
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
27 |
2
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑊 ∈ Ring ) |
28 |
26 27
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ Ring ) |
29 |
3
|
ringmgp |
⊢ ( 𝑊 ∈ Ring → 𝐺 ∈ Mnd ) |
30 |
28 29
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
31 |
4 2 23
|
vr1cl |
⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
32 |
26 31
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑊 ) ) |
33 |
24 6 30 8 32
|
mulgnn0cld |
⊢ ( 𝜑 → ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) |
34 |
|
eqid |
⊢ ( algSc ‘ 𝑊 ) = ( algSc ‘ 𝑊 ) |
35 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
36 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
37 |
|
eqid |
⊢ ( .r ‘ 𝑊 ) = ( .r ‘ 𝑊 ) |
38 |
34 35 36 23 37 9
|
asclmul1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝐴 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) → ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) = ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) |
39 |
16 22 33 38
|
syl3anc |
⊢ ( 𝜑 → ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) = ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) |
40 |
39
|
eqcomd |
⊢ ( 𝜑 → ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) = ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) ) |
41 |
40
|
fveq2d |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) = ( 𝑄 ‘ ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) ) ) |
42 |
1 2 11 5
|
evl1rhm |
⊢ ( 𝑅 ∈ CRing → 𝑄 ∈ ( 𝑊 RingHom 𝑆 ) ) |
43 |
7 42
|
syl |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑊 RingHom 𝑆 ) ) |
44 |
2
|
ply1lmod |
⊢ ( 𝑅 ∈ Ring → 𝑊 ∈ LMod ) |
45 |
26 44
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
46 |
34 35 28 45 36 23
|
asclf |
⊢ ( 𝜑 → ( algSc ‘ 𝑊 ) : ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⟶ ( Base ‘ 𝑊 ) ) |
47 |
46 22
|
ffvelcdmd |
⊢ ( 𝜑 → ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ∈ ( Base ‘ 𝑊 ) ) |
48 |
23 37 12
|
rhmmul |
⊢ ( ( 𝑄 ∈ ( 𝑊 RingHom 𝑆 ) ∧ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑁 ↑ 𝑋 ) ∈ ( Base ‘ 𝑊 ) ) → ( 𝑄 ‘ ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) ∙ ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ) ) |
49 |
43 47 33 48
|
syl3anc |
⊢ ( 𝜑 → ( 𝑄 ‘ ( ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ( .r ‘ 𝑊 ) ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) ∙ ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ) ) |
50 |
1 2 5 34
|
evl1sca |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝐴 ∈ 𝐵 ) → ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) = ( 𝐵 × { 𝐴 } ) ) |
51 |
7 10 50
|
syl2anc |
⊢ ( 𝜑 → ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) = ( 𝐵 × { 𝐴 } ) ) |
52 |
1 2 3 4 5 6 7 8
|
evl1varpw |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) = ( 𝑁 ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) ( 𝑄 ‘ 𝑋 ) ) ) |
53 |
11
|
fveq2i |
⊢ ( mulGrp ‘ 𝑆 ) = ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) |
54 |
13 53
|
eqtri |
⊢ 𝑀 = ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) |
55 |
54
|
fveq2i |
⊢ ( .g ‘ 𝑀 ) = ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) |
56 |
14 55
|
eqtri |
⊢ 𝐹 = ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) |
57 |
56
|
a1i |
⊢ ( 𝜑 → 𝐹 = ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) ) |
58 |
57
|
eqcomd |
⊢ ( 𝜑 → ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) = 𝐹 ) |
59 |
58
|
oveqd |
⊢ ( 𝜑 → ( 𝑁 ( .g ‘ ( mulGrp ‘ ( 𝑅 ↑s 𝐵 ) ) ) ( 𝑄 ‘ 𝑋 ) ) = ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) ) |
60 |
52 59
|
eqtrd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) = ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) ) |
61 |
51 60
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑄 ‘ ( ( algSc ‘ 𝑊 ) ‘ 𝐴 ) ) ∙ ( 𝑄 ‘ ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝐵 × { 𝐴 } ) ∙ ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) ) ) |
62 |
41 49 61
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐴 × ( 𝑁 ↑ 𝑋 ) ) ) = ( ( 𝐵 × { 𝐴 } ) ∙ ( 𝑁 𝐹 ( 𝑄 ‘ 𝑋 ) ) ) ) |