Proof of Theorem dfphi2
Step | Hyp | Ref
| Expression |
1 | | elnn1uz2 12521 |
. 2
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
2 | | phi1 16326 |
. . . . 5
⊢
(ϕ‘1) = 1 |
3 | | 0z 12187 |
. . . . . 6
⊢ 0 ∈
ℤ |
4 | | hashsng 13936 |
. . . . . 6
⊢ (0 ∈
ℤ → (♯‘{0}) = 1) |
5 | 3, 4 | ax-mp 5 |
. . . . 5
⊢
(♯‘{0}) = 1 |
6 | | rabid2 3293 |
. . . . . . 7
⊢ ({0} =
{𝑥 ∈ {0} ∣
(𝑥 gcd 1) = 1} ↔
∀𝑥 ∈ {0} (𝑥 gcd 1) = 1) |
7 | | elsni 4558 |
. . . . . . . . 9
⊢ (𝑥 ∈ {0} → 𝑥 = 0) |
8 | 7 | oveq1d 7228 |
. . . . . . . 8
⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = (0 gcd
1)) |
9 | | gcd1 16087 |
. . . . . . . . 9
⊢ (0 ∈
ℤ → (0 gcd 1) = 1) |
10 | 3, 9 | ax-mp 5 |
. . . . . . . 8
⊢ (0 gcd 1)
= 1 |
11 | 8, 10 | eqtrdi 2794 |
. . . . . . 7
⊢ (𝑥 ∈ {0} → (𝑥 gcd 1) = 1) |
12 | 6, 11 | mprgbir 3076 |
. . . . . 6
⊢ {0} =
{𝑥 ∈ {0} ∣
(𝑥 gcd 1) =
1} |
13 | 12 | fveq2i 6720 |
. . . . 5
⊢
(♯‘{0}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}) |
14 | 2, 5, 13 | 3eqtr2i 2771 |
. . . 4
⊢
(ϕ‘1) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}) |
15 | | fveq2 6717 |
. . . 4
⊢ (𝑁 = 1 → (ϕ‘𝑁) =
(ϕ‘1)) |
16 | | oveq2 7221 |
. . . . . . 7
⊢ (𝑁 = 1 → (0..^𝑁) = (0..^1)) |
17 | | fzo01 13324 |
. . . . . . 7
⊢ (0..^1) =
{0} |
18 | 16, 17 | eqtrdi 2794 |
. . . . . 6
⊢ (𝑁 = 1 → (0..^𝑁) = {0}) |
19 | | oveq2 7221 |
. . . . . . 7
⊢ (𝑁 = 1 → (𝑥 gcd 𝑁) = (𝑥 gcd 1)) |
20 | 19 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑁 = 1 → ((𝑥 gcd 𝑁) = 1 ↔ (𝑥 gcd 1) = 1)) |
21 | 18, 20 | rabeqbidv 3396 |
. . . . 5
⊢ (𝑁 = 1 → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1}) |
22 | 21 | fveq2d 6721 |
. . . 4
⊢ (𝑁 = 1 →
(♯‘{𝑥 ∈
(0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ {0} ∣ (𝑥 gcd 1) = 1})) |
23 | 14, 15, 22 | 3eqtr4a 2804 |
. . 3
⊢ (𝑁 = 1 → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
24 | | eluz2nn 12480 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℕ) |
25 | | phival 16320 |
. . . . 5
⊢ (𝑁 ∈ ℕ →
(ϕ‘𝑁) =
(♯‘{𝑥 ∈
(1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
26 | 24, 25 | syl 17 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
27 | | fzossfz 13261 |
. . . . . . . . . . 11
⊢
(1..^𝑁) ⊆
(1...𝑁) |
28 | 27 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘2) → (1..^𝑁) ⊆ (1...𝑁)) |
29 | | sseqin2 4130 |
. . . . . . . . . 10
⊢
((1..^𝑁) ⊆
(1...𝑁) ↔ ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁)) |
30 | 28, 29 | sylib 221 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = (1..^𝑁)) |
31 | | fzo0ss1 13272 |
. . . . . . . . . 10
⊢
(1..^𝑁) ⊆
(0..^𝑁) |
32 | | sseqin2 4130 |
. . . . . . . . . 10
⊢
((1..^𝑁) ⊆
(0..^𝑁) ↔ ((0..^𝑁) ∩ (1..^𝑁)) = (1..^𝑁)) |
33 | 31, 32 | mpbi 233 |
. . . . . . . . 9
⊢
((0..^𝑁) ∩
(1..^𝑁)) = (1..^𝑁) |
34 | 30, 33 | eqtr4di 2796 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘2) → ((1...𝑁) ∩ (1..^𝑁)) = ((0..^𝑁) ∩ (1..^𝑁))) |
35 | 34 | rabeqdv 3395 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘2) → {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1}) |
36 | | inrab2 4222 |
. . . . . . 7
⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((1...𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1} |
37 | | inrab2 4222 |
. . . . . . 7
⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ ((0..^𝑁) ∩ (1..^𝑁)) ∣ (𝑥 gcd 𝑁) = 1} |
38 | 35, 36, 37 | 3eqtr4g 2803 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁))) |
39 | | phibndlem 16323 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1))) |
40 | | eluzelz 12448 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℤ) |
41 | | fzoval 13244 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℤ →
(1..^𝑁) = (1...(𝑁 − 1))) |
42 | 40, 41 | syl 17 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘2) → (1..^𝑁) = (1...(𝑁 − 1))) |
43 | 39, 42 | sseqtrrd 3942 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁)) |
44 | | df-ss 3883 |
. . . . . . 7
⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) |
45 | 43, 44 | sylib 221 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘2) → ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) |
46 | | gcd0id 16078 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℤ → (0 gcd
𝑁) = (abs‘𝑁)) |
47 | 40, 46 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) → (0 gcd 𝑁) = (abs‘𝑁)) |
48 | | eluzelre 12449 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℝ) |
49 | | eluzge2nn0 12483 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈
ℕ0) |
50 | 49 | nn0ge0d 12153 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → 0 ≤ 𝑁) |
51 | 48, 50 | absidd 14986 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) → (abs‘𝑁) = 𝑁) |
52 | 47, 51 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘2) → (0 gcd 𝑁) = 𝑁) |
53 | | eluz2b3 12518 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) |
54 | 53 | simprbi 500 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ≠ 1) |
55 | 52, 54 | eqnetrd 3008 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈
(ℤ≥‘2) → (0 gcd 𝑁) ≠ 1) |
56 | 55 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (0 gcd 𝑁) ≠ 1) |
57 | 7 | oveq1d 7228 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ {0} → (𝑥 gcd 𝑁) = (0 gcd 𝑁)) |
58 | 57, 17 | eleq2s 2856 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) = (0 gcd 𝑁)) |
59 | 58 | neeq1d 3000 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (0..^1) → ((𝑥 gcd 𝑁) ≠ 1 ↔ (0 gcd 𝑁) ≠ 1)) |
60 | 56, 59 | syl5ibrcom 250 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) → (𝑥 gcd 𝑁) ≠ 1)) |
61 | 60 | necon2bd 2956 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → ¬ 𝑥 ∈ (0..^1))) |
62 | | simpr 488 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁)) |
63 | | 1z 12207 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℤ |
64 | | fzospliti 13274 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ (0..^𝑁) ∧ 1 ∈ ℤ) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁))) |
65 | 62, 63, 64 | sylancl 589 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 ∈ (0..^1) ∨ 𝑥 ∈ (1..^𝑁))) |
66 | 65 | ord 864 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → (¬ 𝑥 ∈ (0..^1) → 𝑥 ∈ (1..^𝑁))) |
67 | 61, 66 | syld 47 |
. . . . . . . . 9
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁))) |
68 | 67 | ralrimiva 3105 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘2) → ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁))) |
69 | | rabss 3985 |
. . . . . . . 8
⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ∀𝑥 ∈ (0..^𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1..^𝑁))) |
70 | 68, 69 | sylibr 237 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘2) → {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁)) |
71 | | df-ss 3883 |
. . . . . . 7
⊢ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1..^𝑁) ↔ ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) |
72 | 70, 71 | sylib 221 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘2) → ({𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∩ (1..^𝑁)) = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) |
73 | 38, 45, 72 | 3eqtr3d 2785 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} = {𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1}) |
74 | 73 | fveq2d 6721 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘2) → (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
75 | 26, 74 | eqtrd 2777 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘2) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
76 | 23, 75 | jaoi 857 |
. 2
⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))
→ (ϕ‘𝑁) =
(♯‘{𝑥 ∈
(0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
77 | 1, 76 | sylbi 220 |
1
⊢ (𝑁 ∈ ℕ →
(ϕ‘𝑁) =
(♯‘{𝑥 ∈
(0..^𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |