Step |
Hyp |
Ref |
Expression |
1 |
|
41prothprm.p |
⊢ 𝑃 = ; 4 1 |
2 |
1
|
41prothprmlem1 |
⊢ ( ( 𝑃 − 1 ) / 2 ) = ; 2 0 |
3 |
2
|
oveq2i |
⊢ ( 3 ↑ ( ( 𝑃 − 1 ) / 2 ) ) = ( 3 ↑ ; 2 0 ) |
4 |
3
|
oveq1i |
⊢ ( ( 3 ↑ ( ( 𝑃 − 1 ) / 2 ) ) mod 𝑃 ) = ( ( 3 ↑ ; 2 0 ) mod 𝑃 ) |
5 |
|
5cn |
⊢ 5 ∈ ℂ |
6 |
|
4cn |
⊢ 4 ∈ ℂ |
7 |
|
5t4e20 |
⊢ ( 5 · 4 ) = ; 2 0 |
8 |
5 6 7
|
mulcomli |
⊢ ( 4 · 5 ) = ; 2 0 |
9 |
8
|
eqcomi |
⊢ ; 2 0 = ( 4 · 5 ) |
10 |
9
|
oveq2i |
⊢ ( 3 ↑ ; 2 0 ) = ( 3 ↑ ( 4 · 5 ) ) |
11 |
|
3cn |
⊢ 3 ∈ ℂ |
12 |
|
4nn0 |
⊢ 4 ∈ ℕ0 |
13 |
|
5nn0 |
⊢ 5 ∈ ℕ0 |
14 |
|
expmul |
⊢ ( ( 3 ∈ ℂ ∧ 4 ∈ ℕ0 ∧ 5 ∈ ℕ0 ) → ( 3 ↑ ( 4 · 5 ) ) = ( ( 3 ↑ 4 ) ↑ 5 ) ) |
15 |
11 12 13 14
|
mp3an |
⊢ ( 3 ↑ ( 4 · 5 ) ) = ( ( 3 ↑ 4 ) ↑ 5 ) |
16 |
10 15
|
eqtri |
⊢ ( 3 ↑ ; 2 0 ) = ( ( 3 ↑ 4 ) ↑ 5 ) |
17 |
16
|
oveq1i |
⊢ ( ( 3 ↑ ; 2 0 ) mod ; 4 1 ) = ( ( ( 3 ↑ 4 ) ↑ 5 ) mod ; 4 1 ) |
18 |
|
3z |
⊢ 3 ∈ ℤ |
19 |
|
zexpcl |
⊢ ( ( 3 ∈ ℤ ∧ 4 ∈ ℕ0 ) → ( 3 ↑ 4 ) ∈ ℤ ) |
20 |
18 12 19
|
mp2an |
⊢ ( 3 ↑ 4 ) ∈ ℤ |
21 |
|
neg1z |
⊢ - 1 ∈ ℤ |
22 |
20 21
|
pm3.2i |
⊢ ( ( 3 ↑ 4 ) ∈ ℤ ∧ - 1 ∈ ℤ ) |
23 |
|
1nn |
⊢ 1 ∈ ℕ |
24 |
12 23
|
decnncl |
⊢ ; 4 1 ∈ ℕ |
25 |
|
nnrp |
⊢ ( ; 4 1 ∈ ℕ → ; 4 1 ∈ ℝ+ ) |
26 |
24 25
|
ax-mp |
⊢ ; 4 1 ∈ ℝ+ |
27 |
13 26
|
pm3.2i |
⊢ ( 5 ∈ ℕ0 ∧ ; 4 1 ∈ ℝ+ ) |
28 |
|
3exp4mod41 |
⊢ ( ( 3 ↑ 4 ) mod ; 4 1 ) = ( - 1 mod ; 4 1 ) |
29 |
|
modexp |
⊢ ( ( ( ( 3 ↑ 4 ) ∈ ℤ ∧ - 1 ∈ ℤ ) ∧ ( 5 ∈ ℕ0 ∧ ; 4 1 ∈ ℝ+ ) ∧ ( ( 3 ↑ 4 ) mod ; 4 1 ) = ( - 1 mod ; 4 1 ) ) → ( ( ( 3 ↑ 4 ) ↑ 5 ) mod ; 4 1 ) = ( ( - 1 ↑ 5 ) mod ; 4 1 ) ) |
30 |
22 27 28 29
|
mp3an |
⊢ ( ( ( 3 ↑ 4 ) ↑ 5 ) mod ; 4 1 ) = ( ( - 1 ↑ 5 ) mod ; 4 1 ) |
31 |
|
3p2e5 |
⊢ ( 3 + 2 ) = 5 |
32 |
31
|
eqcomi |
⊢ 5 = ( 3 + 2 ) |
33 |
32
|
oveq2i |
⊢ ( - 1 ↑ 5 ) = ( - 1 ↑ ( 3 + 2 ) ) |
34 |
|
2z |
⊢ 2 ∈ ℤ |
35 |
|
m1expaddsub |
⊢ ( ( 3 ∈ ℤ ∧ 2 ∈ ℤ ) → ( - 1 ↑ ( 3 − 2 ) ) = ( - 1 ↑ ( 3 + 2 ) ) ) |
36 |
18 34 35
|
mp2an |
⊢ ( - 1 ↑ ( 3 − 2 ) ) = ( - 1 ↑ ( 3 + 2 ) ) |
37 |
36
|
eqcomi |
⊢ ( - 1 ↑ ( 3 + 2 ) ) = ( - 1 ↑ ( 3 − 2 ) ) |
38 |
|
2cn |
⊢ 2 ∈ ℂ |
39 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
40 |
|
2p1e3 |
⊢ ( 2 + 1 ) = 3 |
41 |
11 38 39 40
|
subaddrii |
⊢ ( 3 − 2 ) = 1 |
42 |
41
|
oveq2i |
⊢ ( - 1 ↑ ( 3 − 2 ) ) = ( - 1 ↑ 1 ) |
43 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
44 |
|
exp1 |
⊢ ( - 1 ∈ ℂ → ( - 1 ↑ 1 ) = - 1 ) |
45 |
43 44
|
ax-mp |
⊢ ( - 1 ↑ 1 ) = - 1 |
46 |
42 45
|
eqtri |
⊢ ( - 1 ↑ ( 3 − 2 ) ) = - 1 |
47 |
33 37 46
|
3eqtri |
⊢ ( - 1 ↑ 5 ) = - 1 |
48 |
47
|
oveq1i |
⊢ ( ( - 1 ↑ 5 ) mod ; 4 1 ) = ( - 1 mod ; 4 1 ) |
49 |
17 30 48
|
3eqtri |
⊢ ( ( 3 ↑ ; 2 0 ) mod ; 4 1 ) = ( - 1 mod ; 4 1 ) |
50 |
1
|
oveq2i |
⊢ ( ( 3 ↑ ; 2 0 ) mod 𝑃 ) = ( ( 3 ↑ ; 2 0 ) mod ; 4 1 ) |
51 |
1
|
oveq2i |
⊢ ( - 1 mod 𝑃 ) = ( - 1 mod ; 4 1 ) |
52 |
49 50 51
|
3eqtr4i |
⊢ ( ( 3 ↑ ; 2 0 ) mod 𝑃 ) = ( - 1 mod 𝑃 ) |
53 |
4 52
|
eqtri |
⊢ ( ( 3 ↑ ( ( 𝑃 − 1 ) / 2 ) ) mod 𝑃 ) = ( - 1 mod 𝑃 ) |