Proof of Theorem fermltl
| Step | Hyp | Ref
| Expression |
| 1 | | prmnn 16711 |
. . . 4
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
| 2 | | dvdsmodexp 16298 |
. . . . 5
⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ∈ ℕ ∧ 𝑃 ∥ 𝐴) → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃)) |
| 3 | 2 | 3exp 1120 |
. . . 4
⊢ (𝑃 ∈ ℕ → (𝑃 ∈ ℕ → (𝑃 ∥ 𝐴 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃)))) |
| 4 | 1, 1, 3 | sylc 65 |
. . 3
⊢ (𝑃 ∈ ℙ → (𝑃 ∥ 𝐴 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))) |
| 5 | 4 | adantr 480 |
. 2
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 ∥ 𝐴 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))) |
| 6 | | coprm 16748 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬
𝑃 ∥ 𝐴 ↔ (𝑃 gcd 𝐴) = 1)) |
| 7 | | prmz 16712 |
. . . . . 6
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℤ) |
| 8 | | gcdcom 16550 |
. . . . . 6
⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑃 gcd 𝐴) = (𝐴 gcd 𝑃)) |
| 9 | 7, 8 | sylan 580 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 gcd 𝐴) = (𝐴 gcd 𝑃)) |
| 10 | 9 | eqeq1d 2739 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝑃 gcd 𝐴) = 1 ↔ (𝐴 gcd 𝑃) = 1)) |
| 11 | 6, 10 | bitrd 279 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬
𝑃 ∥ 𝐴 ↔ (𝐴 gcd 𝑃) = 1)) |
| 12 | | simp2 1138 |
. . . . . . . 8
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 𝐴 ∈ ℤ) |
| 13 | 1 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 𝑃 ∈ ℕ) |
| 14 | 13 | phicld 16809 |
. . . . . . . . 9
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (ϕ‘𝑃) ∈
ℕ) |
| 15 | 14 | nnnn0d 12587 |
. . . . . . . 8
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (ϕ‘𝑃) ∈
ℕ0) |
| 16 | | zexpcl 14117 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧
(ϕ‘𝑃) ∈
ℕ0) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ) |
| 17 | 12, 15, 16 | syl2anc 584 |
. . . . . . 7
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ) |
| 18 | 17 | zred 12722 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (𝐴↑(ϕ‘𝑃)) ∈ ℝ) |
| 19 | | 1red 11262 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 1 ∈
ℝ) |
| 20 | 13 | nnrpd 13075 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 𝑃 ∈
ℝ+) |
| 21 | | eulerth 16820 |
. . . . . . 7
⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃)) |
| 22 | 1, 21 | syl3an1 1164 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃)) |
| 23 | | modmul1 13965 |
. . . . . 6
⊢ ((((𝐴↑(ϕ‘𝑃)) ∈ ℝ ∧ 1 ∈
ℝ) ∧ (𝐴 ∈
ℤ ∧ 𝑃 ∈
ℝ+) ∧ ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃)) → (((𝐴↑(ϕ‘𝑃)) · 𝐴) mod 𝑃) = ((1 · 𝐴) mod 𝑃)) |
| 24 | 18, 19, 12, 20, 22, 23 | syl221anc 1383 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (((𝐴↑(ϕ‘𝑃)) · 𝐴) mod 𝑃) = ((1 · 𝐴) mod 𝑃)) |
| 25 | | phiprm 16814 |
. . . . . . . . . 10
⊢ (𝑃 ∈ ℙ →
(ϕ‘𝑃) = (𝑃 − 1)) |
| 26 | 25 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (ϕ‘𝑃) = (𝑃 − 1)) |
| 27 | 26 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (𝐴↑(ϕ‘𝑃)) = (𝐴↑(𝑃 − 1))) |
| 28 | 27 | oveq1d 7446 |
. . . . . . 7
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) · 𝐴) = ((𝐴↑(𝑃 − 1)) · 𝐴)) |
| 29 | 12 | zcnd 12723 |
. . . . . . . 8
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 𝐴 ∈ ℂ) |
| 30 | | expm1t 14131 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℕ) → (𝐴↑𝑃) = ((𝐴↑(𝑃 − 1)) · 𝐴)) |
| 31 | 29, 13, 30 | syl2anc 584 |
. . . . . . 7
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (𝐴↑𝑃) = ((𝐴↑(𝑃 − 1)) · 𝐴)) |
| 32 | 28, 31 | eqtr4d 2780 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) · 𝐴) = (𝐴↑𝑃)) |
| 33 | 32 | oveq1d 7446 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (((𝐴↑(ϕ‘𝑃)) · 𝐴) mod 𝑃) = ((𝐴↑𝑃) mod 𝑃)) |
| 34 | 29 | mullidd 11279 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (1 · 𝐴) = 𝐴) |
| 35 | 34 | oveq1d 7446 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((1 · 𝐴) mod 𝑃) = (𝐴 mod 𝑃)) |
| 36 | 24, 33, 35 | 3eqtr3d 2785 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃)) |
| 37 | 36 | 3expia 1122 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝑃) = 1 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))) |
| 38 | 11, 37 | sylbid 240 |
. 2
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬
𝑃 ∥ 𝐴 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))) |
| 39 | 5, 38 | pm2.61d 179 |
1
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃)) |