Step |
Hyp |
Ref |
Expression |
1 |
|
1nprm |
⊢ ¬ 1 ∈ ℙ |
2 |
|
eleq1 |
⊢ ( 𝑃 = 1 → ( 𝑃 ∈ ℙ ↔ 1 ∈ ℙ ) ) |
3 |
2
|
biimpcd |
⊢ ( 𝑃 ∈ ℙ → ( 𝑃 = 1 → 1 ∈ ℙ ) ) |
4 |
1 3
|
mtoi |
⊢ ( 𝑃 ∈ ℙ → ¬ 𝑃 = 1 ) |
5 |
4
|
neqned |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ≠ 1 ) |
6 |
5
|
pm4.71i |
⊢ ( 𝑃 ∈ ℙ ↔ ( 𝑃 ∈ ℙ ∧ 𝑃 ≠ 1 ) ) |
7 |
|
isprm |
⊢ ( 𝑃 ∈ ℙ ↔ ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ≈ 2o ) ) |
8 |
|
isprm2lem |
⊢ ( ( 𝑃 ∈ ℕ ∧ 𝑃 ≠ 1 ) → ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ≈ 2o ↔ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } = { 1 , 𝑃 } ) ) |
9 |
|
eqss |
⊢ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } = { 1 , 𝑃 } ↔ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ∧ { 1 , 𝑃 } ⊆ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ) ) |
10 |
9
|
imbi2i |
⊢ ( ( 𝑃 ∈ ℕ → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } = { 1 , 𝑃 } ) ↔ ( 𝑃 ∈ ℕ → ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ∧ { 1 , 𝑃 } ⊆ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ) ) ) |
11 |
|
1idssfct |
⊢ ( 𝑃 ∈ ℕ → { 1 , 𝑃 } ⊆ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ) |
12 |
|
jcab |
⊢ ( ( 𝑃 ∈ ℕ → ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ∧ { 1 , 𝑃 } ⊆ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ) ) ↔ ( ( 𝑃 ∈ ℕ → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ∧ ( 𝑃 ∈ ℕ → { 1 , 𝑃 } ⊆ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ) ) ) |
13 |
11 12
|
mpbiran2 |
⊢ ( ( 𝑃 ∈ ℕ → ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ∧ { 1 , 𝑃 } ⊆ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ) ) ↔ ( 𝑃 ∈ ℕ → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) |
14 |
10 13
|
bitri |
⊢ ( ( 𝑃 ∈ ℕ → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } = { 1 , 𝑃 } ) ↔ ( 𝑃 ∈ ℕ → { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) |
15 |
14
|
pm5.74ri |
⊢ ( 𝑃 ∈ ℕ → ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } = { 1 , 𝑃 } ↔ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝑃 ∈ ℕ ∧ 𝑃 ≠ 1 ) → ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } = { 1 , 𝑃 } ↔ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) |
17 |
8 16
|
bitrd |
⊢ ( ( 𝑃 ∈ ℕ ∧ 𝑃 ≠ 1 ) → ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ≈ 2o ↔ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) |
18 |
17
|
expcom |
⊢ ( 𝑃 ≠ 1 → ( 𝑃 ∈ ℕ → ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ≈ 2o ↔ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) ) |
19 |
18
|
pm5.32d |
⊢ ( 𝑃 ≠ 1 → ( ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ≈ 2o ) ↔ ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) ) |
20 |
7 19
|
syl5bb |
⊢ ( 𝑃 ≠ 1 → ( 𝑃 ∈ ℙ ↔ ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) ) |
21 |
20
|
pm5.32ri |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑃 ≠ 1 ) ↔ ( ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ∧ 𝑃 ≠ 1 ) ) |
22 |
|
ancom |
⊢ ( ( ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ∧ 𝑃 ≠ 1 ) ↔ ( 𝑃 ≠ 1 ∧ ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) ) |
23 |
|
anass |
⊢ ( ( ( 𝑃 ≠ 1 ∧ 𝑃 ∈ ℕ ) ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ↔ ( 𝑃 ≠ 1 ∧ ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) ) |
24 |
22 23
|
bitr4i |
⊢ ( ( ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ∧ 𝑃 ≠ 1 ) ↔ ( ( 𝑃 ≠ 1 ∧ 𝑃 ∈ ℕ ) ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) |
25 |
|
ancom |
⊢ ( ( 𝑃 ≠ 1 ∧ 𝑃 ∈ ℕ ) ↔ ( 𝑃 ∈ ℕ ∧ 𝑃 ≠ 1 ) ) |
26 |
|
eluz2b3 |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝑃 ∈ ℕ ∧ 𝑃 ≠ 1 ) ) |
27 |
25 26
|
bitr4i |
⊢ ( ( 𝑃 ≠ 1 ∧ 𝑃 ∈ ℕ ) ↔ 𝑃 ∈ ( ℤ≥ ‘ 2 ) ) |
28 |
27
|
anbi1i |
⊢ ( ( ( 𝑃 ≠ 1 ∧ 𝑃 ∈ ℕ ) ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ↔ ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ) |
29 |
|
dfss2 |
⊢ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ↔ ∀ 𝑧 ( 𝑧 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } → 𝑧 ∈ { 1 , 𝑃 } ) ) |
30 |
|
breq1 |
⊢ ( 𝑛 = 𝑧 → ( 𝑛 ∥ 𝑃 ↔ 𝑧 ∥ 𝑃 ) ) |
31 |
30
|
elrab |
⊢ ( 𝑧 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ↔ ( 𝑧 ∈ ℕ ∧ 𝑧 ∥ 𝑃 ) ) |
32 |
|
vex |
⊢ 𝑧 ∈ V |
33 |
32
|
elpr |
⊢ ( 𝑧 ∈ { 1 , 𝑃 } ↔ ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) |
34 |
31 33
|
imbi12i |
⊢ ( ( 𝑧 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } → 𝑧 ∈ { 1 , 𝑃 } ) ↔ ( ( 𝑧 ∈ ℕ ∧ 𝑧 ∥ 𝑃 ) → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) |
35 |
|
impexp |
⊢ ( ( ( 𝑧 ∈ ℕ ∧ 𝑧 ∥ 𝑃 ) → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ↔ ( 𝑧 ∈ ℕ → ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) ) |
36 |
34 35
|
bitri |
⊢ ( ( 𝑧 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } → 𝑧 ∈ { 1 , 𝑃 } ) ↔ ( 𝑧 ∈ ℕ → ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) ) |
37 |
36
|
albii |
⊢ ( ∀ 𝑧 ( 𝑧 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } → 𝑧 ∈ { 1 , 𝑃 } ) ↔ ∀ 𝑧 ( 𝑧 ∈ ℕ → ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) ) |
38 |
|
df-ral |
⊢ ( ∀ 𝑧 ∈ ℕ ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ↔ ∀ 𝑧 ( 𝑧 ∈ ℕ → ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) ) |
39 |
37 38
|
bitr4i |
⊢ ( ∀ 𝑧 ( 𝑧 ∈ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } → 𝑧 ∈ { 1 , 𝑃 } ) ↔ ∀ 𝑧 ∈ ℕ ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) |
40 |
29 39
|
bitri |
⊢ ( { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ↔ ∀ 𝑧 ∈ ℕ ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) |
41 |
40
|
anbi2i |
⊢ ( ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ↔ ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) ∧ ∀ 𝑧 ∈ ℕ ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) ) |
42 |
24 28 41
|
3bitri |
⊢ ( ( ( 𝑃 ∈ ℕ ∧ { 𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃 } ⊆ { 1 , 𝑃 } ) ∧ 𝑃 ≠ 1 ) ↔ ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) ∧ ∀ 𝑧 ∈ ℕ ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) ) |
43 |
6 21 42
|
3bitri |
⊢ ( 𝑃 ∈ ℙ ↔ ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) ∧ ∀ 𝑧 ∈ ℕ ( 𝑧 ∥ 𝑃 → ( 𝑧 = 1 ∨ 𝑧 = 𝑃 ) ) ) ) |