Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑗 = 0 → ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 · 𝐵 ) ↑ 0 ) ) |
2 |
|
oveq2 |
⊢ ( 𝑗 = 0 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 0 ) ) |
3 |
|
oveq2 |
⊢ ( 𝑗 = 0 → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ 0 ) ) |
4 |
2 3
|
oveq12d |
⊢ ( 𝑗 = 0 → ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) = ( ( 𝐴 ↑ 0 ) · ( 𝐵 ↑ 0 ) ) ) |
5 |
1 4
|
eqeq12d |
⊢ ( 𝑗 = 0 → ( ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) ↔ ( ( 𝐴 · 𝐵 ) ↑ 0 ) = ( ( 𝐴 ↑ 0 ) · ( 𝐵 ↑ 0 ) ) ) ) |
6 |
5
|
imbi2d |
⊢ ( 𝑗 = 0 → ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) ) ↔ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 0 ) = ( ( 𝐴 ↑ 0 ) · ( 𝐵 ↑ 0 ) ) ) ) ) |
7 |
|
oveq2 |
⊢ ( 𝑗 = 𝑘 → ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) ) |
8 |
|
oveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 𝑘 ) ) |
9 |
|
oveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ 𝑘 ) ) |
10 |
8 9
|
oveq12d |
⊢ ( 𝑗 = 𝑘 → ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) ) |
11 |
7 10
|
eqeq12d |
⊢ ( 𝑗 = 𝑘 → ( ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) ↔ ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑗 = 𝑘 → ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) ) ↔ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) ) |
14 |
|
oveq2 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ ( 𝑘 + 1 ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) |
16 |
14 15
|
oveq12d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) |
17 |
13 16
|
eqeq12d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) ↔ ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) ) |
18 |
17
|
imbi2d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) ) ↔ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
19 |
|
oveq2 |
⊢ ( 𝑗 = 𝑁 → ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) ) |
20 |
|
oveq2 |
⊢ ( 𝑗 = 𝑁 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 𝑁 ) ) |
21 |
|
oveq2 |
⊢ ( 𝑗 = 𝑁 → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ 𝑁 ) ) |
22 |
20 21
|
oveq12d |
⊢ ( 𝑗 = 𝑁 → ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) |
23 |
19 22
|
eqeq12d |
⊢ ( 𝑗 = 𝑁 → ( ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) ↔ ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) ) |
24 |
23
|
imbi2d |
⊢ ( 𝑗 = 𝑁 → ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑗 ) = ( ( 𝐴 ↑ 𝑗 ) · ( 𝐵 ↑ 𝑗 ) ) ) ↔ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) ) ) |
25 |
|
mulcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ ) |
26 |
|
exp0 |
⊢ ( ( 𝐴 · 𝐵 ) ∈ ℂ → ( ( 𝐴 · 𝐵 ) ↑ 0 ) = 1 ) |
27 |
25 26
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 0 ) = 1 ) |
28 |
|
exp0 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 0 ) = 1 ) |
29 |
|
exp0 |
⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ↑ 0 ) = 1 ) |
30 |
28 29
|
oveqan12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑ 0 ) · ( 𝐵 ↑ 0 ) ) = ( 1 · 1 ) ) |
31 |
|
1t1e1 |
⊢ ( 1 · 1 ) = 1 |
32 |
30 31
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑ 0 ) · ( 𝐵 ↑ 0 ) ) = 1 ) |
33 |
27 32
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 0 ) = ( ( 𝐴 ↑ 0 ) · ( 𝐵 ↑ 0 ) ) ) |
34 |
|
expp1 |
⊢ ( ( ( 𝐴 · 𝐵 ) ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) · ( 𝐴 · 𝐵 ) ) ) |
35 |
25 34
|
sylan |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) · ( 𝐴 · 𝐵 ) ) ) |
36 |
35
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) ) → ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) · ( 𝐴 · 𝐵 ) ) ) |
37 |
|
oveq1 |
⊢ ( ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) → ( ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) · ( 𝐴 · 𝐵 ) ) = ( ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) · ( 𝐴 · 𝐵 ) ) ) |
38 |
|
expcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℂ ) |
39 |
|
expcl |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ 𝑘 ) ∈ ℂ ) |
40 |
38 39
|
anim12i |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) ) → ( ( 𝐴 ↑ 𝑘 ) ∈ ℂ ∧ ( 𝐵 ↑ 𝑘 ) ∈ ℂ ) ) |
41 |
40
|
anandirs |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ↑ 𝑘 ) ∈ ℂ ∧ ( 𝐵 ↑ 𝑘 ) ∈ ℂ ) ) |
42 |
|
simpl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ) |
43 |
|
mul4 |
⊢ ( ( ( ( 𝐴 ↑ 𝑘 ) ∈ ℂ ∧ ( 𝐵 ↑ 𝑘 ) ∈ ℂ ) ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ) → ( ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) · ( 𝐴 · 𝐵 ) ) = ( ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) · ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) ) |
44 |
41 42 43
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) · ( 𝐴 · 𝐵 ) ) = ( ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) · ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) ) |
45 |
|
expp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
46 |
45
|
adantlr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
47 |
|
expp1 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) |
48 |
47
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) |
49 |
46 48
|
oveq12d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) = ( ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) · ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) ) |
50 |
44 49
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) · ( 𝐴 · 𝐵 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) |
51 |
37 50
|
sylan9eqr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) ) → ( ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) · ( 𝐴 · 𝐵 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) |
52 |
36 51
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) ) → ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) |
53 |
52
|
exp31 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑘 ∈ ℕ0 → ( ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) → ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
54 |
53
|
com12 |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) → ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
55 |
54
|
a2d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · ( 𝐵 ↑ 𝑘 ) ) ) → ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) · ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) ) ) |
56 |
6 12 18 24 33 55
|
nn0ind |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) ) |
57 |
56
|
expdcom |
⊢ ( 𝐴 ∈ ℂ → ( 𝐵 ∈ ℂ → ( 𝑁 ∈ ℕ0 → ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) ) ) |
58 |
57
|
3imp |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐴 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) |