Step |
Hyp |
Ref |
Expression |
1 |
|
asclmulg.a |
⊢ 𝐴 = ( algSc ‘ 𝑊 ) |
2 |
|
asclmulg.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
3 |
|
asclmulg.k |
⊢ 𝐾 = ( Base ‘ 𝐹 ) |
4 |
|
asclmulg.m |
⊢ ↑ = ( .g ‘ 𝑊 ) |
5 |
|
asclmulg.t |
⊢ ∗ = ( .g ‘ 𝐹 ) |
6 |
|
assalmod |
⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ LMod ) |
7 |
6
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → 𝑊 ∈ LMod ) |
8 |
|
simp3 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → 𝑋 ∈ 𝐾 ) |
9 |
|
simp2 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → 𝑁 ∈ ℕ0 ) |
10 |
|
assaring |
⊢ ( 𝑊 ∈ AssAlg → 𝑊 ∈ Ring ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
12 |
|
eqid |
⊢ ( 1r ‘ 𝑊 ) = ( 1r ‘ 𝑊 ) |
13 |
11 12
|
ringidcl |
⊢ ( 𝑊 ∈ Ring → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
14 |
10 13
|
syl |
⊢ ( 𝑊 ∈ AssAlg → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
16 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
17 |
11 2 16 3 4 5
|
lmodvsmmulgdi |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) ) → ( 𝑁 ↑ ( 𝑋 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) = ( ( 𝑁 ∗ 𝑋 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
18 |
7 8 9 15 17
|
syl13anc |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → ( 𝑁 ↑ ( 𝑋 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) = ( ( 𝑁 ∗ 𝑋 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
19 |
1 2 3 16 12
|
asclval |
⊢ ( 𝑋 ∈ 𝐾 → ( 𝐴 ‘ 𝑋 ) = ( 𝑋 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
20 |
8 19
|
syl |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → ( 𝐴 ‘ 𝑋 ) = ( 𝑋 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
21 |
20
|
oveq2d |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → ( 𝑁 ↑ ( 𝐴 ‘ 𝑋 ) ) = ( 𝑁 ↑ ( 𝑋 ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) ) |
22 |
2
|
assasca |
⊢ ( 𝑊 ∈ AssAlg → 𝐹 ∈ CRing ) |
23 |
22
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → 𝐹 ∈ CRing ) |
24 |
23
|
crnggrpd |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → 𝐹 ∈ Grp ) |
25 |
9
|
nn0zd |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → 𝑁 ∈ ℤ ) |
26 |
3 5 24 25 8
|
mulgcld |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → ( 𝑁 ∗ 𝑋 ) ∈ 𝐾 ) |
27 |
1 2 3 16 12
|
asclval |
⊢ ( ( 𝑁 ∗ 𝑋 ) ∈ 𝐾 → ( 𝐴 ‘ ( 𝑁 ∗ 𝑋 ) ) = ( ( 𝑁 ∗ 𝑋 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
28 |
26 27
|
syl |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → ( 𝐴 ‘ ( 𝑁 ∗ 𝑋 ) ) = ( ( 𝑁 ∗ 𝑋 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
29 |
18 21 28
|
3eqtr4rd |
⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾 ) → ( 𝐴 ‘ ( 𝑁 ∗ 𝑋 ) ) = ( 𝑁 ↑ ( 𝐴 ‘ 𝑋 ) ) ) |