| Step |
Hyp |
Ref |
Expression |
| 1 |
|
0exp |
⊢ ( 𝑁 ∈ ℕ → ( 0 ↑ 𝑁 ) = 0 ) |
| 2 |
1
|
oveq1d |
⊢ ( 𝑁 ∈ ℕ → ( ( 0 ↑ 𝑁 ) gcd 0 ) = ( 0 gcd 0 ) ) |
| 3 |
2
|
eqeq1d |
⊢ ( 𝑁 ∈ ℕ → ( ( ( 0 ↑ 𝑁 ) gcd 0 ) = 1 ↔ ( 0 gcd 0 ) = 1 ) ) |
| 4 |
|
oveq1 |
⊢ ( 𝐴 = 0 → ( 𝐴 ↑ 𝑁 ) = ( 0 ↑ 𝑁 ) ) |
| 5 |
|
oveq12 |
⊢ ( ( ( 𝐴 ↑ 𝑁 ) = ( 0 ↑ 𝑁 ) ∧ 𝐵 = 0 ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = ( ( 0 ↑ 𝑁 ) gcd 0 ) ) |
| 6 |
4 5
|
sylan |
⊢ ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = ( ( 0 ↑ 𝑁 ) gcd 0 ) ) |
| 7 |
6
|
eqeq1d |
⊢ ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 ↔ ( ( 0 ↑ 𝑁 ) gcd 0 ) = 1 ) ) |
| 8 |
|
oveq12 |
⊢ ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( 𝐴 gcd 𝐵 ) = ( 0 gcd 0 ) ) |
| 9 |
8
|
eqeq1d |
⊢ ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( ( 𝐴 gcd 𝐵 ) = 1 ↔ ( 0 gcd 0 ) = 1 ) ) |
| 10 |
7 9
|
bibi12d |
⊢ ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 ↔ ( 𝐴 gcd 𝐵 ) = 1 ) ↔ ( ( ( 0 ↑ 𝑁 ) gcd 0 ) = 1 ↔ ( 0 gcd 0 ) = 1 ) ) ) |
| 11 |
3 10
|
syl5ibrcom |
⊢ ( 𝑁 ∈ ℕ → ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 ↔ ( 𝐴 gcd 𝐵 ) = 1 ) ) ) |
| 12 |
11
|
3ad2ant3 |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 ↔ ( 𝐴 gcd 𝐵 ) = 1 ) ) ) |
| 13 |
|
exprmfct |
⊢ ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) → ∃ 𝑝 ∈ ℙ 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ) |
| 14 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → 𝐴 ∈ ℤ ) |
| 15 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → 𝑁 ∈ ℕ ) |
| 16 |
15
|
nnnn0d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → 𝑁 ∈ ℕ0 ) |
| 17 |
|
zexpcl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℤ ) |
| 18 |
14 16 17
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( 𝐴 ↑ 𝑁 ) ∈ ℤ ) |
| 19 |
18
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℤ ) |
| 20 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → 𝐵 ∈ ℤ ) |
| 21 |
20
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → 𝐵 ∈ ℤ ) |
| 22 |
|
gcddvds |
⊢ ( ( ( 𝐴 ↑ 𝑁 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ 𝐵 ) ) |
| 23 |
19 21 22
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ 𝐵 ) ) |
| 24 |
23
|
simpld |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ) |
| 25 |
|
prmz |
⊢ ( 𝑝 ∈ ℙ → 𝑝 ∈ ℤ ) |
| 26 |
25
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑝 ∈ ℤ ) |
| 27 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) |
| 28 |
14
|
zcnd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → 𝐴 ∈ ℂ ) |
| 29 |
|
expeq0 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐴 ↑ 𝑁 ) = 0 ↔ 𝐴 = 0 ) ) |
| 30 |
28 15 29
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( 𝐴 ↑ 𝑁 ) = 0 ↔ 𝐴 = 0 ) ) |
| 31 |
30
|
anbi1d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( ( 𝐴 ↑ 𝑁 ) = 0 ∧ 𝐵 = 0 ) ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ) |
| 32 |
27 31
|
mtbird |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ¬ ( ( 𝐴 ↑ 𝑁 ) = 0 ∧ 𝐵 = 0 ) ) |
| 33 |
|
gcdn0cl |
⊢ ( ( ( ( 𝐴 ↑ 𝑁 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ¬ ( ( 𝐴 ↑ 𝑁 ) = 0 ∧ 𝐵 = 0 ) ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ℕ ) |
| 34 |
18 20 32 33
|
syl21anc |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ℕ ) |
| 35 |
34
|
nnzd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ℤ ) |
| 36 |
35
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ℤ ) |
| 37 |
|
dvdstr |
⊢ ( ( 𝑝 ∈ ℤ ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ℤ ∧ ( 𝐴 ↑ 𝑁 ) ∈ ℤ ) → ( ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ) → 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ) ) |
| 38 |
26 36 19 37
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ) → 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ) ) |
| 39 |
24 38
|
mpan2d |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ) ) |
| 40 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑝 ∈ ℙ ) |
| 41 |
|
simpll1 |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → 𝐴 ∈ ℤ ) |
| 42 |
15
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑁 ∈ ℕ ) |
| 43 |
|
prmdvdsexp |
⊢ ( ( 𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ) → ( 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ↔ 𝑝 ∥ 𝐴 ) ) |
| 44 |
40 41 42 43
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ↔ 𝑝 ∥ 𝐴 ) ) |
| 45 |
39 44
|
sylibd |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → 𝑝 ∥ 𝐴 ) ) |
| 46 |
23
|
simprd |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ 𝐵 ) |
| 47 |
|
dvdstr |
⊢ ( ( 𝑝 ∈ ℤ ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ 𝐵 ) → 𝑝 ∥ 𝐵 ) ) |
| 48 |
26 36 21 47
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∥ 𝐵 ) → 𝑝 ∥ 𝐵 ) ) |
| 49 |
46 48
|
mpan2d |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → 𝑝 ∥ 𝐵 ) ) |
| 50 |
45 49
|
jcad |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → ( 𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵 ) ) ) |
| 51 |
|
dvdsgcd |
⊢ ( ( 𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵 ) → 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ) ) |
| 52 |
26 41 21 51
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵 ) → 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ) ) |
| 53 |
|
nprmdvds1 |
⊢ ( 𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1 ) |
| 54 |
|
breq2 |
⊢ ( ( 𝐴 gcd 𝐵 ) = 1 → ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ↔ 𝑝 ∥ 1 ) ) |
| 55 |
54
|
notbid |
⊢ ( ( 𝐴 gcd 𝐵 ) = 1 → ( ¬ 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ↔ ¬ 𝑝 ∥ 1 ) ) |
| 56 |
53 55
|
syl5ibrcom |
⊢ ( 𝑝 ∈ ℙ → ( ( 𝐴 gcd 𝐵 ) = 1 → ¬ 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ) ) |
| 57 |
56
|
necon2ad |
⊢ ( 𝑝 ∈ ℙ → ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) → ( 𝐴 gcd 𝐵 ) ≠ 1 ) ) |
| 58 |
57
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) → ( 𝐴 gcd 𝐵 ) ≠ 1 ) ) |
| 59 |
50 52 58
|
3syld |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → ( 𝐴 gcd 𝐵 ) ≠ 1 ) ) |
| 60 |
59
|
rexlimdva |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ∃ 𝑝 ∈ ℙ 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → ( 𝐴 gcd 𝐵 ) ≠ 1 ) ) |
| 61 |
|
gcdn0cl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( 𝐴 gcd 𝐵 ) ∈ ℕ ) |
| 62 |
61
|
3adantl3 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( 𝐴 gcd 𝐵 ) ∈ ℕ ) |
| 63 |
|
eluz2b3 |
⊢ ( ( 𝐴 gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( ( 𝐴 gcd 𝐵 ) ∈ ℕ ∧ ( 𝐴 gcd 𝐵 ) ≠ 1 ) ) |
| 64 |
63
|
baib |
⊢ ( ( 𝐴 gcd 𝐵 ) ∈ ℕ → ( ( 𝐴 gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝐴 gcd 𝐵 ) ≠ 1 ) ) |
| 65 |
62 64
|
syl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( 𝐴 gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝐴 gcd 𝐵 ) ≠ 1 ) ) |
| 66 |
60 65
|
sylibrd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ∃ 𝑝 ∈ ℙ 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → ( 𝐴 gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
| 67 |
13 66
|
syl5 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
| 68 |
|
exprmfct |
⊢ ( ( 𝐴 gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) → ∃ 𝑝 ∈ ℙ 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ) |
| 69 |
|
gcddvds |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ∧ ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) ) |
| 70 |
41 21 69
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ∧ ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) ) |
| 71 |
70
|
simpld |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ) |
| 72 |
|
iddvdsexp |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ) → 𝐴 ∥ ( 𝐴 ↑ 𝑁 ) ) |
| 73 |
41 42 72
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → 𝐴 ∥ ( 𝐴 ↑ 𝑁 ) ) |
| 74 |
62
|
nnzd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( 𝐴 gcd 𝐵 ) ∈ ℤ ) |
| 75 |
74
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝐴 gcd 𝐵 ) ∈ ℤ ) |
| 76 |
|
dvdstr |
⊢ ( ( ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ ( 𝐴 ↑ 𝑁 ) ∈ ℤ ) → ( ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ∧ 𝐴 ∥ ( 𝐴 ↑ 𝑁 ) ) → ( 𝐴 gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ) ) |
| 77 |
75 41 19 76
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ∧ 𝐴 ∥ ( 𝐴 ↑ 𝑁 ) ) → ( 𝐴 gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ) ) |
| 78 |
71 73 77
|
mp2and |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝐴 gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ) |
| 79 |
|
dvdstr |
⊢ ( ( 𝑝 ∈ ℤ ∧ ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ ( 𝐴 ↑ 𝑁 ) ∈ ℤ ) → ( ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ∧ ( 𝐴 gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ) → 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ) ) |
| 80 |
26 75 19 79
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ∧ ( 𝐴 gcd 𝐵 ) ∥ ( 𝐴 ↑ 𝑁 ) ) → 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ) ) |
| 81 |
78 80
|
mpan2d |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) → 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ) ) |
| 82 |
70
|
simprd |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) |
| 83 |
|
dvdstr |
⊢ ( ( 𝑝 ∈ ℤ ∧ ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ∧ ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) → 𝑝 ∥ 𝐵 ) ) |
| 84 |
26 75 21 83
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) ∧ ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) → 𝑝 ∥ 𝐵 ) ) |
| 85 |
82 84
|
mpan2d |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) → 𝑝 ∥ 𝐵 ) ) |
| 86 |
81 85
|
jcad |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) → ( 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ∧ 𝑝 ∥ 𝐵 ) ) ) |
| 87 |
|
dvdsgcd |
⊢ ( ( 𝑝 ∈ ℤ ∧ ( 𝐴 ↑ 𝑁 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ∧ 𝑝 ∥ 𝐵 ) → 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ) ) |
| 88 |
26 19 21 87
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( 𝐴 ↑ 𝑁 ) ∧ 𝑝 ∥ 𝐵 ) → 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ) ) |
| 89 |
|
breq2 |
⊢ ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 → ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ↔ 𝑝 ∥ 1 ) ) |
| 90 |
89
|
notbid |
⊢ ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 → ( ¬ 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ↔ ¬ 𝑝 ∥ 1 ) ) |
| 91 |
53 90
|
syl5ibrcom |
⊢ ( 𝑝 ∈ ℙ → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 → ¬ 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ) ) |
| 92 |
91
|
necon2ad |
⊢ ( 𝑝 ∈ ℙ → ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ≠ 1 ) ) |
| 93 |
92
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ≠ 1 ) ) |
| 94 |
86 88 93
|
3syld |
⊢ ( ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( 𝐴 gcd 𝐵 ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ≠ 1 ) ) |
| 95 |
94
|
rexlimdva |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ∃ 𝑝 ∈ ℙ 𝑝 ∥ ( 𝐴 gcd 𝐵 ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ≠ 1 ) ) |
| 96 |
|
eluz2b3 |
⊢ ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ℕ ∧ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ≠ 1 ) ) |
| 97 |
96
|
baib |
⊢ ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ℕ → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ≠ 1 ) ) |
| 98 |
34 97
|
syl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ≠ 1 ) ) |
| 99 |
95 98
|
sylibrd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ∃ 𝑝 ∈ ℙ 𝑝 ∥ ( 𝐴 gcd 𝐵 ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
| 100 |
68 99
|
syl5 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( 𝐴 gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
| 101 |
67 100
|
impbid |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝐴 gcd 𝐵 ) ∈ ( ℤ≥ ‘ 2 ) ) ) |
| 102 |
101 98 65
|
3bitr3d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) ≠ 1 ↔ ( 𝐴 gcd 𝐵 ) ≠ 1 ) ) |
| 103 |
102
|
necon4bid |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 ↔ ( 𝐴 gcd 𝐵 ) = 1 ) ) |
| 104 |
103
|
ex |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) → ( ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 ↔ ( 𝐴 gcd 𝐵 ) = 1 ) ) ) |
| 105 |
12 104
|
pm2.61d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ ) → ( ( ( 𝐴 ↑ 𝑁 ) gcd 𝐵 ) = 1 ↔ ( 𝐴 gcd 𝐵 ) = 1 ) ) |