| Step |
Hyp |
Ref |
Expression |
| 1 |
|
r19.26 |
⊢ ( ∀ 𝑝 ∈ ℙ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) |
| 2 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → 𝑝 ∈ ℙ ) |
| 3 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → 𝐴 ∈ ℕ ) |
| 4 |
2 3
|
pccld |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt 𝐴 ) ∈ ℕ0 ) |
| 5 |
4
|
nn0red |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt 𝐴 ) ∈ ℝ ) |
| 6 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → 𝐵 ∈ ℕ ) |
| 7 |
2 6
|
pccld |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt 𝐵 ) ∈ ℕ0 ) |
| 8 |
7
|
nn0red |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt 𝐵 ) ∈ ℝ ) |
| 9 |
|
1red |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → 1 ∈ ℝ ) |
| 10 |
|
le2add |
⊢ ( ( ( ( 𝑝 pCnt 𝐴 ) ∈ ℝ ∧ ( 𝑝 pCnt 𝐵 ) ∈ ℝ ) ∧ ( 1 ∈ ℝ ∧ 1 ∈ ℝ ) ) → ( ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) → ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ ( 1 + 1 ) ) ) |
| 11 |
5 8 9 9 10
|
syl22anc |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) → ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ ( 1 + 1 ) ) ) |
| 12 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
| 13 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝐴 gcd 𝐵 ) = 1 ) |
| 14 |
13
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt ( 𝐴 gcd 𝐵 ) ) = ( 𝑝 pCnt 1 ) ) |
| 15 |
3
|
nnzd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → 𝐴 ∈ ℤ ) |
| 16 |
6
|
nnzd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → 𝐵 ∈ ℤ ) |
| 17 |
|
pcgcd |
⊢ ( ( 𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝑝 pCnt ( 𝐴 gcd 𝐵 ) ) = if ( ( 𝑝 pCnt 𝐴 ) ≤ ( 𝑝 pCnt 𝐵 ) , ( 𝑝 pCnt 𝐴 ) , ( 𝑝 pCnt 𝐵 ) ) ) |
| 18 |
2 15 16 17
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt ( 𝐴 gcd 𝐵 ) ) = if ( ( 𝑝 pCnt 𝐴 ) ≤ ( 𝑝 pCnt 𝐵 ) , ( 𝑝 pCnt 𝐴 ) , ( 𝑝 pCnt 𝐵 ) ) ) |
| 19 |
|
pc1 |
⊢ ( 𝑝 ∈ ℙ → ( 𝑝 pCnt 1 ) = 0 ) |
| 20 |
19
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt 1 ) = 0 ) |
| 21 |
14 18 20
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → if ( ( 𝑝 pCnt 𝐴 ) ≤ ( 𝑝 pCnt 𝐵 ) , ( 𝑝 pCnt 𝐴 ) , ( 𝑝 pCnt 𝐵 ) ) = 0 ) |
| 22 |
|
ifid |
⊢ if ( ( 𝑝 pCnt 𝐴 ) ≤ ( 𝑝 pCnt 𝐵 ) , 1 , 1 ) = 1 |
| 23 |
|
ifeq12 |
⊢ ( ( 1 = ( 𝑝 pCnt 𝐴 ) ∧ 1 = ( 𝑝 pCnt 𝐵 ) ) → if ( ( 𝑝 pCnt 𝐴 ) ≤ ( 𝑝 pCnt 𝐵 ) , 1 , 1 ) = if ( ( 𝑝 pCnt 𝐴 ) ≤ ( 𝑝 pCnt 𝐵 ) , ( 𝑝 pCnt 𝐴 ) , ( 𝑝 pCnt 𝐵 ) ) ) |
| 24 |
22 23
|
eqtr3id |
⊢ ( ( 1 = ( 𝑝 pCnt 𝐴 ) ∧ 1 = ( 𝑝 pCnt 𝐵 ) ) → 1 = if ( ( 𝑝 pCnt 𝐴 ) ≤ ( 𝑝 pCnt 𝐵 ) , ( 𝑝 pCnt 𝐴 ) , ( 𝑝 pCnt 𝐵 ) ) ) |
| 25 |
24
|
eqeq1d |
⊢ ( ( 1 = ( 𝑝 pCnt 𝐴 ) ∧ 1 = ( 𝑝 pCnt 𝐵 ) ) → ( 1 = 0 ↔ if ( ( 𝑝 pCnt 𝐴 ) ≤ ( 𝑝 pCnt 𝐵 ) , ( 𝑝 pCnt 𝐴 ) , ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) |
| 26 |
21 25
|
syl5ibrcom |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 1 = ( 𝑝 pCnt 𝐴 ) ∧ 1 = ( 𝑝 pCnt 𝐵 ) ) → 1 = 0 ) ) |
| 27 |
26
|
necon3ad |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 1 ≠ 0 → ¬ ( 1 = ( 𝑝 pCnt 𝐴 ) ∧ 1 = ( 𝑝 pCnt 𝐵 ) ) ) ) |
| 28 |
12 27
|
mpi |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ¬ ( 1 = ( 𝑝 pCnt 𝐴 ) ∧ 1 = ( 𝑝 pCnt 𝐵 ) ) ) |
| 29 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 30 |
5
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt 𝐴 ) ∈ ℂ ) |
| 31 |
|
subeq0 |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝑝 pCnt 𝐴 ) ∈ ℂ ) → ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ↔ 1 = ( 𝑝 pCnt 𝐴 ) ) ) |
| 32 |
29 30 31
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ↔ 1 = ( 𝑝 pCnt 𝐴 ) ) ) |
| 33 |
8
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt 𝐵 ) ∈ ℂ ) |
| 34 |
|
subeq0 |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝑝 pCnt 𝐵 ) ∈ ℂ ) → ( ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ↔ 1 = ( 𝑝 pCnt 𝐵 ) ) ) |
| 35 |
29 33 34
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ↔ 1 = ( 𝑝 pCnt 𝐵 ) ) ) |
| 36 |
32 35
|
anbi12d |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ↔ ( 1 = ( 𝑝 pCnt 𝐴 ) ∧ 1 = ( 𝑝 pCnt 𝐵 ) ) ) ) |
| 37 |
28 36
|
mtbird |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ¬ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) |
| 38 |
37
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) ∧ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) → ¬ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) |
| 39 |
|
eqcom |
⊢ ( ( 1 + 1 ) = ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ↔ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) = ( 1 + 1 ) ) |
| 40 |
|
1re |
⊢ 1 ∈ ℝ |
| 41 |
40 40
|
readdcli |
⊢ ( 1 + 1 ) ∈ ℝ |
| 42 |
41
|
recni |
⊢ ( 1 + 1 ) ∈ ℂ |
| 43 |
4 7
|
nn0addcld |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ∈ ℕ0 ) |
| 44 |
43
|
nn0red |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ∈ ℝ ) |
| 45 |
44
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ∈ ℂ ) |
| 46 |
|
subeq0 |
⊢ ( ( ( 1 + 1 ) ∈ ℂ ∧ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ∈ ℂ ) → ( ( ( 1 + 1 ) − ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( 1 + 1 ) = ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) ) |
| 47 |
42 45 46
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 1 + 1 ) − ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( 1 + 1 ) = ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) ) |
| 48 |
47 39
|
bitrdi |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 1 + 1 ) − ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) = ( 1 + 1 ) ) ) |
| 49 |
9
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → 1 ∈ ℂ ) |
| 50 |
49 49 30 33
|
addsub4d |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 1 + 1 ) − ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) = ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) ) |
| 51 |
50
|
eqeq1d |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 1 + 1 ) − ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ) ) |
| 52 |
48 51
|
bitr3d |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) = ( 1 + 1 ) ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ) ) |
| 53 |
52
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) ∧ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) → ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) = ( 1 + 1 ) ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ) ) |
| 54 |
|
subge0 |
⊢ ( ( 1 ∈ ℝ ∧ ( 𝑝 pCnt 𝐴 ) ∈ ℝ ) → ( 0 ≤ ( 1 − ( 𝑝 pCnt 𝐴 ) ) ↔ ( 𝑝 pCnt 𝐴 ) ≤ 1 ) ) |
| 55 |
40 5 54
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 0 ≤ ( 1 − ( 𝑝 pCnt 𝐴 ) ) ↔ ( 𝑝 pCnt 𝐴 ) ≤ 1 ) ) |
| 56 |
|
subge0 |
⊢ ( ( 1 ∈ ℝ ∧ ( 𝑝 pCnt 𝐵 ) ∈ ℝ ) → ( 0 ≤ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ↔ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) |
| 57 |
40 8 56
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 0 ≤ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ↔ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) |
| 58 |
55 57
|
anbi12d |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 0 ≤ ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∧ 0 ≤ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) ↔ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) ) |
| 59 |
|
resubcl |
⊢ ( ( 1 ∈ ℝ ∧ ( 𝑝 pCnt 𝐴 ) ∈ ℝ ) → ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∈ ℝ ) |
| 60 |
40 5 59
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∈ ℝ ) |
| 61 |
|
resubcl |
⊢ ( ( 1 ∈ ℝ ∧ ( 𝑝 pCnt 𝐵 ) ∈ ℝ ) → ( 1 − ( 𝑝 pCnt 𝐵 ) ) ∈ ℝ ) |
| 62 |
40 8 61
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 1 − ( 𝑝 pCnt 𝐵 ) ) ∈ ℝ ) |
| 63 |
|
add20 |
⊢ ( ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∈ ℝ ∧ 0 ≤ ( 1 − ( 𝑝 pCnt 𝐴 ) ) ) ∧ ( ( 1 − ( 𝑝 pCnt 𝐵 ) ) ∈ ℝ ∧ 0 ≤ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) ) → ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) |
| 64 |
63
|
an4s |
⊢ ( ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∈ ℝ ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ∈ ℝ ) ∧ ( 0 ≤ ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∧ 0 ≤ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) ) → ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) |
| 65 |
64
|
ex |
⊢ ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∈ ℝ ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ∈ ℝ ) → ( ( 0 ≤ ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∧ 0 ≤ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) → ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) ) |
| 66 |
60 62 65
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 0 ≤ ( 1 − ( 𝑝 pCnt 𝐴 ) ) ∧ 0 ≤ ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) → ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) ) |
| 67 |
58 66
|
sylbird |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) → ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) ) |
| 68 |
67
|
imp |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) ∧ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) → ( ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) + ( 1 − ( 𝑝 pCnt 𝐵 ) ) ) = 0 ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) |
| 69 |
53 68
|
bitrd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) ∧ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) → ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) = ( 1 + 1 ) ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) |
| 70 |
39 69
|
syl5bb |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) ∧ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) → ( ( 1 + 1 ) = ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ↔ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) |
| 71 |
70
|
necon3abid |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) ∧ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) → ( ( 1 + 1 ) ≠ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ↔ ¬ ( ( 1 − ( 𝑝 pCnt 𝐴 ) ) = 0 ∧ ( 1 − ( 𝑝 pCnt 𝐵 ) ) = 0 ) ) ) |
| 72 |
38 71
|
mpbird |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) ∧ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) → ( 1 + 1 ) ≠ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) |
| 73 |
72
|
ex |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) → ( 1 + 1 ) ≠ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) ) |
| 74 |
11 73
|
jcad |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) → ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ ( 1 + 1 ) ∧ ( 1 + 1 ) ≠ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) ) ) |
| 75 |
|
nnz |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ ) |
| 76 |
|
nnne0 |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ≠ 0 ) |
| 77 |
75 76
|
jca |
⊢ ( 𝐴 ∈ ℕ → ( 𝐴 ∈ ℤ ∧ 𝐴 ≠ 0 ) ) |
| 78 |
3 77
|
syl |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝐴 ∈ ℤ ∧ 𝐴 ≠ 0 ) ) |
| 79 |
|
nnz |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℤ ) |
| 80 |
|
nnne0 |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ≠ 0 ) |
| 81 |
79 80
|
jca |
⊢ ( 𝐵 ∈ ℕ → ( 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) ) |
| 82 |
6 81
|
syl |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) ) |
| 83 |
|
pcmul |
⊢ ( ( 𝑝 ∈ ℙ ∧ ( 𝐴 ∈ ℤ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) ) → ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) = ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) |
| 84 |
2 78 82 83
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) = ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) |
| 85 |
84
|
breq1d |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) ≤ 1 ↔ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ 1 ) ) |
| 86 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
| 87 |
|
nn0leltp1 |
⊢ ( ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ∈ ℕ0 ∧ 1 ∈ ℕ0 ) → ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ 1 ↔ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) < ( 1 + 1 ) ) ) |
| 88 |
43 86 87
|
sylancl |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ 1 ↔ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) < ( 1 + 1 ) ) ) |
| 89 |
|
ltlen |
⊢ ( ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ∈ ℝ ∧ ( 1 + 1 ) ∈ ℝ ) → ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) < ( 1 + 1 ) ↔ ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ ( 1 + 1 ) ∧ ( 1 + 1 ) ≠ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) ) ) |
| 90 |
44 41 89
|
sylancl |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) < ( 1 + 1 ) ↔ ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ ( 1 + 1 ) ∧ ( 1 + 1 ) ≠ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) ) ) |
| 91 |
85 88 90
|
3bitrd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) ≤ 1 ↔ ( ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ≤ ( 1 + 1 ) ∧ ( 1 + 1 ) ≠ ( ( 𝑝 pCnt 𝐴 ) + ( 𝑝 pCnt 𝐵 ) ) ) ) ) |
| 92 |
74 91
|
sylibrd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) → ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) ≤ 1 ) ) |
| 93 |
92
|
ralimdva |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) → ( ∀ 𝑝 ∈ ℙ ( ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) → ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) ≤ 1 ) ) |
| 94 |
1 93
|
syl5bir |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) → ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) ≤ 1 ) ) |
| 95 |
|
issqf |
⊢ ( 𝐴 ∈ ℕ → ( ( μ ‘ 𝐴 ) ≠ 0 ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐴 ) ≤ 1 ) ) |
| 96 |
|
issqf |
⊢ ( 𝐵 ∈ ℕ → ( ( μ ‘ 𝐵 ) ≠ 0 ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) |
| 97 |
95 96
|
bi2anan9 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( ( μ ‘ 𝐴 ) ≠ 0 ∧ ( μ ‘ 𝐵 ) ≠ 0 ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) ) |
| 98 |
97
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) → ( ( ( μ ‘ 𝐴 ) ≠ 0 ∧ ( μ ‘ 𝐵 ) ≠ 0 ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐴 ) ≤ 1 ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt 𝐵 ) ≤ 1 ) ) ) |
| 99 |
|
nnmulcl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 · 𝐵 ) ∈ ℕ ) |
| 100 |
99
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) → ( 𝐴 · 𝐵 ) ∈ ℕ ) |
| 101 |
|
issqf |
⊢ ( ( 𝐴 · 𝐵 ) ∈ ℕ → ( ( μ ‘ ( 𝐴 · 𝐵 ) ) ≠ 0 ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) ≤ 1 ) ) |
| 102 |
100 101
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) → ( ( μ ‘ ( 𝐴 · 𝐵 ) ) ≠ 0 ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 pCnt ( 𝐴 · 𝐵 ) ) ≤ 1 ) ) |
| 103 |
94 98 102
|
3imtr4d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) → ( ( ( μ ‘ 𝐴 ) ≠ 0 ∧ ( μ ‘ 𝐵 ) ≠ 0 ) → ( μ ‘ ( 𝐴 · 𝐵 ) ) ≠ 0 ) ) |
| 104 |
103
|
imp |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) = 1 ) ∧ ( ( μ ‘ 𝐴 ) ≠ 0 ∧ ( μ ‘ 𝐵 ) ≠ 0 ) ) → ( μ ‘ ( 𝐴 · 𝐵 ) ) ≠ 0 ) |