Step |
Hyp |
Ref |
Expression |
1 |
|
flt4lem2.a |
⊢ ( 𝜑 → 𝐴 ∈ ℕ ) |
2 |
|
flt4lem2.b |
⊢ ( 𝜑 → 𝐵 ∈ ℕ ) |
3 |
|
flt4lem2.c |
⊢ ( 𝜑 → 𝐶 ∈ ℕ ) |
4 |
|
flt4lem2.1 |
⊢ ( 𝜑 → 2 ∥ 𝐴 ) |
5 |
|
flt4lem2.2 |
⊢ ( 𝜑 → ( 𝐴 gcd 𝐶 ) = 1 ) |
6 |
|
flt4lem2.3 |
⊢ ( 𝜑 → ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) = ( 𝐶 ↑ 2 ) ) |
7 |
|
breq1 |
⊢ ( 𝑖 = 2 → ( 𝑖 ∥ 𝐴 ↔ 2 ∥ 𝐴 ) ) |
8 |
|
breq1 |
⊢ ( 𝑖 = 2 → ( 𝑖 ∥ 𝐶 ↔ 2 ∥ 𝐶 ) ) |
9 |
7 8
|
anbi12d |
⊢ ( 𝑖 = 2 → ( ( 𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶 ) ↔ ( 2 ∥ 𝐴 ∧ 2 ∥ 𝐶 ) ) ) |
10 |
|
2z |
⊢ 2 ∈ ℤ |
11 |
|
uzid |
⊢ ( 2 ∈ ℤ → 2 ∈ ( ℤ≥ ‘ 2 ) ) |
12 |
10 11
|
ax-mp |
⊢ 2 ∈ ( ℤ≥ ‘ 2 ) |
13 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 2 ∈ ( ℤ≥ ‘ 2 ) ) |
14 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 2 ∥ 𝐴 ) |
15 |
10
|
a1i |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 2 ∈ ℤ ) |
16 |
|
gcdnncl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 gcd 𝐵 ) ∈ ℕ ) |
17 |
1 2 16
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 gcd 𝐵 ) ∈ ℕ ) |
18 |
17
|
nnzd |
⊢ ( 𝜑 → ( 𝐴 gcd 𝐵 ) ∈ ℤ ) |
19 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → ( 𝐴 gcd 𝐵 ) ∈ ℤ ) |
20 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 𝐶 ∈ ℕ ) |
21 |
20
|
nnzd |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 𝐶 ∈ ℤ ) |
22 |
|
simpr |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 2 ∥ 𝐵 ) |
23 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 𝐴 ∈ ℕ ) |
24 |
23
|
nnzd |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 𝐴 ∈ ℤ ) |
25 |
2
|
nnzd |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
26 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 𝐵 ∈ ℤ ) |
27 |
|
dvdsgcd |
⊢ ( ( 2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 2 ∥ 𝐴 ∧ 2 ∥ 𝐵 ) → 2 ∥ ( 𝐴 gcd 𝐵 ) ) ) |
28 |
15 24 26 27
|
syl3anc |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → ( ( 2 ∥ 𝐴 ∧ 2 ∥ 𝐵 ) → 2 ∥ ( 𝐴 gcd 𝐵 ) ) ) |
29 |
14 22 28
|
mp2and |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 2 ∥ ( 𝐴 gcd 𝐵 ) ) |
30 |
|
2nn |
⊢ 2 ∈ ℕ |
31 |
30
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℕ ) |
32 |
1 2 3 31 6
|
fltdvdsabdvdsc |
⊢ ( 𝜑 → ( 𝐴 gcd 𝐵 ) ∥ 𝐶 ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → ( 𝐴 gcd 𝐵 ) ∥ 𝐶 ) |
34 |
15 19 21 29 33
|
dvdstrd |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → 2 ∥ 𝐶 ) |
35 |
14 34
|
jca |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → ( 2 ∥ 𝐴 ∧ 2 ∥ 𝐶 ) ) |
36 |
9 13 35
|
rspcedvdw |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → ∃ 𝑖 ∈ ( ℤ≥ ‘ 2 ) ( 𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶 ) ) |
37 |
|
ncoprmgcdne1b |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ) → ( ∃ 𝑖 ∈ ( ℤ≥ ‘ 2 ) ( 𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶 ) ↔ ( 𝐴 gcd 𝐶 ) ≠ 1 ) ) |
38 |
23 20 37
|
syl2anc |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → ( ∃ 𝑖 ∈ ( ℤ≥ ‘ 2 ) ( 𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶 ) ↔ ( 𝐴 gcd 𝐶 ) ≠ 1 ) ) |
39 |
36 38
|
mpbid |
⊢ ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → ( 𝐴 gcd 𝐶 ) ≠ 1 ) |
40 |
39
|
ex |
⊢ ( 𝜑 → ( 2 ∥ 𝐵 → ( 𝐴 gcd 𝐶 ) ≠ 1 ) ) |
41 |
40
|
necon2bd |
⊢ ( 𝜑 → ( ( 𝐴 gcd 𝐶 ) = 1 → ¬ 2 ∥ 𝐵 ) ) |
42 |
5 41
|
mpd |
⊢ ( 𝜑 → ¬ 2 ∥ 𝐵 ) |