Step |
Hyp |
Ref |
Expression |
1 |
|
simp2l |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( 𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ+ ) ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → 𝐶 ∈ ℕ0 ) |
2 |
|
id |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ) |
3 |
2
|
3adant2l |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( 𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ+ ) ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ) |
4 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 𝐴 ↑ 𝑥 ) = ( 𝐴 ↑ 0 ) ) |
5 |
4
|
oveq1d |
⊢ ( 𝑥 = 0 → ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐴 ↑ 0 ) mod 𝐷 ) ) |
6 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 𝐵 ↑ 𝑥 ) = ( 𝐵 ↑ 0 ) ) |
7 |
6
|
oveq1d |
⊢ ( 𝑥 = 0 → ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 0 ) mod 𝐷 ) ) |
8 |
5 7
|
eqeq12d |
⊢ ( 𝑥 = 0 → ( ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) ↔ ( ( 𝐴 ↑ 0 ) mod 𝐷 ) = ( ( 𝐵 ↑ 0 ) mod 𝐷 ) ) ) |
9 |
8
|
imbi2d |
⊢ ( 𝑥 = 0 → ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) ) ↔ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 0 ) mod 𝐷 ) = ( ( 𝐵 ↑ 0 ) mod 𝐷 ) ) ) ) |
10 |
|
oveq2 |
⊢ ( 𝑥 = 𝑘 → ( 𝐴 ↑ 𝑥 ) = ( 𝐴 ↑ 𝑘 ) ) |
11 |
10
|
oveq1d |
⊢ ( 𝑥 = 𝑘 → ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) ) |
12 |
|
oveq2 |
⊢ ( 𝑥 = 𝑘 → ( 𝐵 ↑ 𝑥 ) = ( 𝐵 ↑ 𝑘 ) ) |
13 |
12
|
oveq1d |
⊢ ( 𝑥 = 𝑘 → ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) |
14 |
11 13
|
eqeq12d |
⊢ ( 𝑥 = 𝑘 → ( ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) ↔ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) ) |
15 |
14
|
imbi2d |
⊢ ( 𝑥 = 𝑘 → ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) ) ↔ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) ) ) |
16 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( 𝐴 ↑ 𝑥 ) = ( 𝐴 ↑ ( 𝑘 + 1 ) ) ) |
17 |
16
|
oveq1d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) ) |
18 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( 𝐵 ↑ 𝑥 ) = ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) |
19 |
18
|
oveq1d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) ) |
20 |
17 19
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) ↔ ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) = ( ( 𝐵 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) ) ) |
21 |
20
|
imbi2d |
⊢ ( 𝑥 = ( 𝑘 + 1 ) → ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) ) ↔ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) = ( ( 𝐵 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) ) ) ) |
22 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐴 ↑ 𝑥 ) = ( 𝐴 ↑ 𝐶 ) ) |
23 |
22
|
oveq1d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐴 ↑ 𝐶 ) mod 𝐷 ) ) |
24 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐵 ↑ 𝑥 ) = ( 𝐵 ↑ 𝐶 ) ) |
25 |
24
|
oveq1d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝐶 ) mod 𝐷 ) ) |
26 |
23 25
|
eqeq12d |
⊢ ( 𝑥 = 𝐶 → ( ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) ↔ ( ( 𝐴 ↑ 𝐶 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝐶 ) mod 𝐷 ) ) ) |
27 |
26
|
imbi2d |
⊢ ( 𝑥 = 𝐶 → ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝑥 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑥 ) mod 𝐷 ) ) ↔ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝐶 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝐶 ) mod 𝐷 ) ) ) ) |
28 |
|
zcn |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℂ ) |
29 |
|
exp0 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 0 ) = 1 ) |
30 |
28 29
|
syl |
⊢ ( 𝐴 ∈ ℤ → ( 𝐴 ↑ 0 ) = 1 ) |
31 |
|
zcn |
⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℂ ) |
32 |
|
exp0 |
⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ↑ 0 ) = 1 ) |
33 |
31 32
|
syl |
⊢ ( 𝐵 ∈ ℤ → ( 𝐵 ↑ 0 ) = 1 ) |
34 |
33
|
eqcomd |
⊢ ( 𝐵 ∈ ℤ → 1 = ( 𝐵 ↑ 0 ) ) |
35 |
30 34
|
sylan9eq |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ↑ 0 ) = ( 𝐵 ↑ 0 ) ) |
36 |
35
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 ↑ 0 ) mod 𝐷 ) = ( ( 𝐵 ↑ 0 ) mod 𝐷 ) ) |
37 |
36
|
3ad2ant1 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 0 ) mod 𝐷 ) = ( ( 𝐵 ↑ 0 ) mod 𝐷 ) ) |
38 |
|
simp21l |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → 𝐴 ∈ ℤ ) |
39 |
|
simp1 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → 𝑘 ∈ ℕ0 ) |
40 |
|
zexpcl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℤ ) |
41 |
38 39 40
|
syl2anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( 𝐴 ↑ 𝑘 ) ∈ ℤ ) |
42 |
|
simp21r |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → 𝐵 ∈ ℤ ) |
43 |
|
zexpcl |
⊢ ( ( 𝐵 ∈ ℤ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ 𝑘 ) ∈ ℤ ) |
44 |
42 39 43
|
syl2anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( 𝐵 ↑ 𝑘 ) ∈ ℤ ) |
45 |
|
simp22 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → 𝐷 ∈ ℝ+ ) |
46 |
|
simp3 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) |
47 |
|
simp23 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) |
48 |
41 44 38 42 45 46 47
|
modmul12d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) mod 𝐷 ) = ( ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) mod 𝐷 ) ) |
49 |
38
|
zcnd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → 𝐴 ∈ ℂ ) |
50 |
|
expp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
51 |
49 39 50
|
syl2anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
52 |
51
|
oveq1d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) = ( ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) mod 𝐷 ) ) |
53 |
42
|
zcnd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → 𝐵 ∈ ℂ ) |
54 |
|
expp1 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) |
55 |
53 39 54
|
syl2anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( 𝐵 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) |
56 |
55
|
oveq1d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( ( 𝐵 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) = ( ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) mod 𝐷 ) ) |
57 |
48 52 56
|
3eqtr4d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) ∧ ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) = ( ( 𝐵 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) ) |
58 |
57
|
3exp |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) → ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) = ( ( 𝐵 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) ) ) ) |
59 |
58
|
a2d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝑘 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝑘 ) mod 𝐷 ) ) → ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) = ( ( 𝐵 ↑ ( 𝑘 + 1 ) ) mod 𝐷 ) ) ) ) |
60 |
9 15 21 27 37 59
|
nn0ind |
⊢ ( 𝐶 ∈ ℕ0 → ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ 𝐷 ∈ ℝ+ ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝐶 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝐶 ) mod 𝐷 ) ) ) |
61 |
1 3 60
|
sylc |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ( 𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ+ ) ∧ ( 𝐴 mod 𝐷 ) = ( 𝐵 mod 𝐷 ) ) → ( ( 𝐴 ↑ 𝐶 ) mod 𝐷 ) = ( ( 𝐵 ↑ 𝐶 ) mod 𝐷 ) ) |