Metamath Proof Explorer


Theorem flt4lem2

Description: If A is even, B is odd. (Contributed by SN, 22-Aug-2024)

Ref Expression
Hypotheses flt4lem2.a ( 𝜑𝐴 ∈ ℕ )
flt4lem2.b ( 𝜑𝐵 ∈ ℕ )
flt4lem2.c ( 𝜑𝐶 ∈ ℕ )
flt4lem2.1 ( 𝜑 → 2 ∥ 𝐴 )
flt4lem2.2 ( 𝜑 → ( 𝐴 gcd 𝐶 ) = 1 )
flt4lem2.3 ( 𝜑 → ( ( 𝐴 ↑ 2 ) + ( 𝐵 ↑ 2 ) ) = ( 𝐶 ↑ 2 ) )
Assertion flt4lem2 ( 𝜑 → ¬ 2 ∥ 𝐵 )

Proof

Step Hyp Ref Expression
1 flt4lem2.a ( 𝜑𝐴 ∈ ℕ )
2 flt4lem2.b ( 𝜑𝐵 ∈ ℕ )
3 flt4lem2.c ( 𝜑𝐶 ∈ 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 → ( 𝑖𝐴 ↔ 2 ∥ 𝐴 ) )
8 breq1 ( 𝑖 = 2 → ( 𝑖𝐶 ↔ 2 ∥ 𝐶 ) )
9 7 8 anbi12d ( 𝑖 = 2 → ( ( 𝑖𝐴𝑖𝐶 ) ↔ ( 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 ( 𝜑𝐵 ∈ 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 ) ( 𝑖𝐴𝑖𝐶 ) )
37 ncoprmgcdne1b ( ( 𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ) → ( ∃ 𝑖 ∈ ( ℤ ‘ 2 ) ( 𝑖𝐴𝑖𝐶 ) ↔ ( 𝐴 gcd 𝐶 ) ≠ 1 ) )
38 23 20 37 syl2anc ( ( 𝜑 ∧ 2 ∥ 𝐵 ) → ( ∃ 𝑖 ∈ ( ℤ ‘ 2 ) ( 𝑖𝐴𝑖𝐶 ) ↔ ( 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 )