Proof of Theorem fpprel2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 2nn 12339 | . . . . 5
⊢ 2 ∈
ℕ | 
| 2 |  | fpprel 47715 | . . . . 5
⊢ (2 ∈
ℕ → (𝑋 ∈ (
FPPr ‘2) ↔ (𝑋
∈ (ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1))) | 
| 3 | 1, 2 | mp1i 13 | . . . 4
⊢ (𝑋 ∈ ( FPPr ‘2) →
(𝑋 ∈ ( FPPr ‘2)
↔ (𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1))) | 
| 4 |  | eluz4eluz2 12925 | . . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 𝑋 ∈
(ℤ≥‘2)) | 
| 5 | 4 | 3ad2ant1 1134 | . . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1) → 𝑋 ∈
(ℤ≥‘2)) | 
| 6 | 5 | adantl 481 | . . . . . . 7
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → 𝑋 ∈
(ℤ≥‘2)) | 
| 7 |  | fppr2odd 47718 | . . . . . . . 8
⊢ (𝑋 ∈ ( FPPr ‘2) →
𝑋 ∈ Odd
) | 
| 8 | 7 | adantr 480 | . . . . . . 7
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → 𝑋 ∈ Odd ) | 
| 9 |  | simpr2 1196 | . . . . . . 7
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → 𝑋 ∉ ℙ) | 
| 10 | 6, 8, 9 | 3jca 1129 | . . . . . 6
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → (𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ)) | 
| 11 |  | fpprwppr 47726 | . . . . . . 7
⊢ (𝑋 ∈ ( FPPr ‘2) →
((2↑𝑋) mod 𝑋) = (2 mod 𝑋)) | 
| 12 |  | 2re 12340 | . . . . . . . . . 10
⊢ 2 ∈
ℝ | 
| 13 | 12 | a1i 11 | . . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 2 ∈ ℝ) | 
| 14 |  | eluz4nn 12928 | . . . . . . . . . 10
⊢ (𝑋 ∈
(ℤ≥‘4) → 𝑋 ∈ ℕ) | 
| 15 | 14 | nnrpd 13075 | . . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 𝑋 ∈
ℝ+) | 
| 16 |  | 0le2 12368 | . . . . . . . . . 10
⊢ 0 ≤
2 | 
| 17 | 16 | a1i 11 | . . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 0 ≤ 2) | 
| 18 |  | eluz2 12884 | . . . . . . . . . 10
⊢ (𝑋 ∈
(ℤ≥‘4) ↔ (4 ∈ ℤ ∧ 𝑋 ∈ ℤ ∧ 4 ≤
𝑋)) | 
| 19 |  | 4z 12651 | . . . . . . . . . . . . . 14
⊢ 4 ∈
ℤ | 
| 20 |  | zlem1lt 12669 | . . . . . . . . . . . . . 14
⊢ ((4
∈ ℤ ∧ 𝑋
∈ ℤ) → (4 ≤ 𝑋 ↔ (4 − 1) < 𝑋)) | 
| 21 | 19, 20 | mpan 690 | . . . . . . . . . . . . 13
⊢ (𝑋 ∈ ℤ → (4 ≤
𝑋 ↔ (4 − 1) <
𝑋)) | 
| 22 |  | 4m1e3 12395 | . . . . . . . . . . . . . . 15
⊢ (4
− 1) = 3 | 
| 23 | 22 | breq1i 5150 | . . . . . . . . . . . . . 14
⊢ ((4
− 1) < 𝑋 ↔ 3
< 𝑋) | 
| 24 | 12 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 2 ∈
ℝ) | 
| 25 |  | 3re 12346 | . . . . . . . . . . . . . . . . 17
⊢ 3 ∈
ℝ | 
| 26 | 25 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 3 ∈
ℝ) | 
| 27 |  | zre 12617 | . . . . . . . . . . . . . . . . 17
⊢ (𝑋 ∈ ℤ → 𝑋 ∈
ℝ) | 
| 28 | 27 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 𝑋 ∈ ℝ) | 
| 29 |  | 2lt3 12438 | . . . . . . . . . . . . . . . . 17
⊢ 2 <
3 | 
| 30 | 29 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 2 <
3) | 
| 31 |  | simpr 484 | . . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 3 < 𝑋) | 
| 32 | 24, 26, 28, 30, 31 | lttrd 11422 | . . . . . . . . . . . . . . 15
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 2 < 𝑋) | 
| 33 | 32 | ex 412 | . . . . . . . . . . . . . 14
⊢ (𝑋 ∈ ℤ → (3 <
𝑋 → 2 < 𝑋)) | 
| 34 | 23, 33 | biimtrid 242 | . . . . . . . . . . . . 13
⊢ (𝑋 ∈ ℤ → ((4
− 1) < 𝑋 → 2
< 𝑋)) | 
| 35 | 21, 34 | sylbid 240 | . . . . . . . . . . . 12
⊢ (𝑋 ∈ ℤ → (4 ≤
𝑋 → 2 < 𝑋)) | 
| 36 | 35 | a1i 11 | . . . . . . . . . . 11
⊢ (4 ∈
ℤ → (𝑋 ∈
ℤ → (4 ≤ 𝑋
→ 2 < 𝑋))) | 
| 37 | 36 | 3imp 1111 | . . . . . . . . . 10
⊢ ((4
∈ ℤ ∧ 𝑋
∈ ℤ ∧ 4 ≤ 𝑋) → 2 < 𝑋) | 
| 38 | 18, 37 | sylbi 217 | . . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 2 < 𝑋) | 
| 39 |  | modid 13936 | . . . . . . . . 9
⊢ (((2
∈ ℝ ∧ 𝑋
∈ ℝ+) ∧ (0 ≤ 2 ∧ 2 < 𝑋)) → (2 mod 𝑋) = 2) | 
| 40 | 13, 15, 17, 38, 39 | syl22anc 839 | . . . . . . . 8
⊢ (𝑋 ∈
(ℤ≥‘4) → (2 mod 𝑋) = 2) | 
| 41 | 40 | 3ad2ant1 1134 | . . . . . . 7
⊢ ((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1) → (2 mod 𝑋) = 2) | 
| 42 | 11, 41 | sylan9eq 2797 | . . . . . 6
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → ((2↑𝑋) mod 𝑋) = 2) | 
| 43 | 10, 42 | jca 511 | . . . . 5
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2)) | 
| 44 | 43 | ex 412 | . . . 4
⊢ (𝑋 ∈ ( FPPr ‘2) →
((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1) → ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2))) | 
| 45 | 3, 44 | sylbid 240 | . . 3
⊢ (𝑋 ∈ ( FPPr ‘2) →
(𝑋 ∈ ( FPPr ‘2)
→ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2))) | 
| 46 | 45 | pm2.43i 52 | . 2
⊢ (𝑋 ∈ ( FPPr ‘2) →
((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2)) | 
| 47 |  | ge2nprmge4 16738 | . . . . . 6
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∉ ℙ) → 𝑋 ∈
(ℤ≥‘4)) | 
| 48 | 47 | 3adant2 1132 | . . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 𝑋 ∈
(ℤ≥‘4)) | 
| 49 |  | simp3 1139 | . . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 𝑋 ∉ ℙ) | 
| 50 | 48, 49 | jca 511 | . . . 4
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → (𝑋 ∈ (ℤ≥‘4)
∧ 𝑋 ∉
ℙ)) | 
| 51 | 50 | adantr 480 | . . 3
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → (𝑋 ∈ (ℤ≥‘4)
∧ 𝑋 ∉
ℙ)) | 
| 52 | 1 | a1i 11 | . . . 4
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → 2 ∈
ℕ) | 
| 53 | 12 | a1i 11 | . . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 2 ∈
ℝ) | 
| 54 |  | eluz2nn 12924 | . . . . . . . . . 10
⊢ (𝑋 ∈
(ℤ≥‘2) → 𝑋 ∈ ℕ) | 
| 55 | 54 | nnrpd 13075 | . . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘2) → 𝑋 ∈
ℝ+) | 
| 56 | 55 | 3ad2ant1 1134 | . . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 𝑋 ∈
ℝ+) | 
| 57 | 16 | a1i 11 | . . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 0 ≤
2) | 
| 58 | 48, 38 | syl 17 | . . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 2 < 𝑋) | 
| 59 | 53, 56, 57, 58, 39 | syl22anc 839 | . . . . . . 7
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → (2 mod 𝑋) = 2) | 
| 60 | 59 | eqcomd 2743 | . . . . . 6
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 2 = (2 mod 𝑋)) | 
| 61 | 60 | eqeq2d 2748 | . . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → (((2↑𝑋) mod 𝑋) = 2 ↔ ((2↑𝑋) mod 𝑋) = (2 mod 𝑋))) | 
| 62 | 61 | biimpa 476 | . . . 4
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → ((2↑𝑋) mod 𝑋) = (2 mod 𝑋)) | 
| 63 | 52, 62 | jca 511 | . . 3
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → (2 ∈ ℕ ∧
((2↑𝑋) mod 𝑋) = (2 mod 𝑋))) | 
| 64 |  | gcd2odd1 47655 | . . . . . 6
⊢ (𝑋 ∈ Odd → (𝑋 gcd 2) = 1) | 
| 65 | 64 | 3ad2ant2 1135 | . . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → (𝑋 gcd 2) = 1) | 
| 66 | 65 | adantr 480 | . . . 4
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → (𝑋 gcd 2) = 1) | 
| 67 |  | fpprwpprb 47727 | . . . 4
⊢ ((𝑋 gcd 2) = 1 → (𝑋 ∈ ( FPPr ‘2) ↔
((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ) ∧ (2 ∈ ℕ
∧ ((2↑𝑋) mod 𝑋) = (2 mod 𝑋))))) | 
| 68 | 66, 67 | syl 17 | . . 3
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → (𝑋 ∈ ( FPPr ‘2) ↔ ((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ) ∧ (2 ∈ ℕ
∧ ((2↑𝑋) mod 𝑋) = (2 mod 𝑋))))) | 
| 69 | 51, 63, 68 | mpbir2and 713 | . 2
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → 𝑋 ∈ ( FPPr ‘2)) | 
| 70 | 46, 69 | impbii 209 | 1
⊢ (𝑋 ∈ ( FPPr ‘2) ↔
((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2)) |