Theorem pythagtriplem4 16785

Description: Lemma for pythagtrip 16800. Show that 𝐶 − 𝐵 and 𝐶 + 𝐵 are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
pythagtriplem4	⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 1)

Proof of Theorem pythagtriplem4

Step	Hyp	Ref	Expression
1		simp3r 1199	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ 𝐴)
2		nnz 12607	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
3		nnz 12607	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
4		zsubcl 12632	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶 − 𝐵) ∈ ℤ)
5	2, 3, 4	syl2anr 595	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
6	5	3adant1 1127	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
7	6	3ad2ant1 1130	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℤ)
8		simp13 1202	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℕ)
9		simp12 1201	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ)
10	8, 9	nnaddcld 12292	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℕ)
11	10	nnzd 12613	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℤ)
12		gcddvds 16475	. . . . . . . . . 10 ⊢ (((𝐶 − 𝐵) ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ) → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 − 𝐵) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵)))
13	7, 11, 12	syl2anc 582	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 − 𝐵) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵)))
14	13	simprd 494	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵))
15		breq1 5146	. . . . . . . . 9 ⊢ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) ↔ 2 ∥ (𝐶 + 𝐵)))
16	15	biimpd 228	. . . . . . . 8 ⊢ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) → 2 ∥ (𝐶 + 𝐵)))
17	14, 16	mpan9 505	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ (𝐶 + 𝐵))
18		2z 12622	. . . . . . . 8 ⊢ 2 ∈ ℤ
19		simpl13 1247	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐶 ∈ ℕ)
20	19	nnzd 12613	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐶 ∈ ℤ)
21		simpl12 1246	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐵 ∈ ℕ)
22	21	nnzd 12613	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐵 ∈ ℤ)
23	20, 22	zaddcld 12698	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐶 + 𝐵) ∈ ℤ)
24	20, 22	zsubcld 12699	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐶 − 𝐵) ∈ ℤ)
25		dvdsmultr1 16270	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ ∧ (𝐶 − 𝐵) ∈ ℤ) → (2 ∥ (𝐶 + 𝐵) → 2 ∥ ((𝐶 + 𝐵) · (𝐶 − 𝐵))))
26	18, 23, 24, 25	mp3an2i 1462	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (2 ∥ (𝐶 + 𝐵) → 2 ∥ ((𝐶 + 𝐵) · (𝐶 − 𝐵))))
27	17, 26	mpd 15	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
28	19	nncnd 12256	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐶 ∈ ℂ)
29	21	nncnd 12256	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐵 ∈ ℂ)
30		subsq 14203	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
31	28, 29, 30	syl2anc 582	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
32	27, 31	breqtrrd 5171	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ ((𝐶↑2) − (𝐵↑2)))
33		simpl2 1189	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))
34	33	oveq1d 7430	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
35		simpl11 1245	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐴 ∈ ℕ)
36	35	nnsqcld 14236	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐴↑2) ∈ ℕ)
37	36	nncnd 12256	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐴↑2) ∈ ℂ)
38	21	nnsqcld 14236	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐵↑2) ∈ ℕ)
39	38	nncnd 12256	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (𝐵↑2) ∈ ℂ)
40	37, 39	pncand 11600	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = (𝐴↑2))
41	34, 40	eqtr3d 2767	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → ((𝐶↑2) − (𝐵↑2)) = (𝐴↑2))
42	32, 41	breqtrd 5169	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ (𝐴↑2))
43		nnz 12607	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
44	43	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
45	44	3ad2ant1 1130	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℤ)
46	45	adantr 479	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 𝐴 ∈ ℤ)
47		2prm 16660	. . . . . 6 ⊢ 2 ∈ ℙ
48		2nn 12313	. . . . . 6 ⊢ 2 ∈ ℕ
49		prmdvdsexp 16683	. . . . . 6 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 2 ∈ ℕ) → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
50	47, 48, 49	mp3an13 1448	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
51	46, 50	syl 17	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
52	42, 51	mpbid 231	. . 3 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2) → 2 ∥ 𝐴)
53	1, 52	mtand 814	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2)
54		neg1z 12626	. . . . . . . 8 ⊢ -1 ∈ ℤ
55		gcdaddm 16497	. . . . . . . 8 ⊢ ((-1 ∈ ℤ ∧ (𝐶 − 𝐵) ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵)))))
56	54, 7, 11, 55	mp3an2i 1462	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵)))))
57	8	nncnd 12256	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
58	9	nncnd 12256	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
59		pnncan 11529	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶 − 𝐵)) = (𝐵 + 𝐵))
60	59	3anidm23 1418	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶 − 𝐵)) = (𝐵 + 𝐵))
61		subcl 11487	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ)
62	61	mulm1d 11694	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-1 · (𝐶 − 𝐵)) = -(𝐶 − 𝐵))
63	62	oveq2d 7431	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) + -(𝐶 − 𝐵)))
64		addcl 11218	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
65	64, 61	negsubd 11605	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + -(𝐶 − 𝐵)) = ((𝐶 + 𝐵) − (𝐶 − 𝐵)))
66	63, 65	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) − (𝐶 − 𝐵)))
67		2times 12376	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℂ → (2 · 𝐵) = (𝐵 + 𝐵))
68	67	adantl 480	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐵) = (𝐵 + 𝐵))
69	60, 66, 68	3eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = (2 · 𝐵))
70	69	oveq2d 7431	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵)))) = ((𝐶 − 𝐵) gcd (2 · 𝐵)))
71	57, 58, 70	syl2anc 582	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵)))) = ((𝐶 − 𝐵) gcd (2 · 𝐵)))
72	56, 71	eqtrd 2765	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = ((𝐶 − 𝐵) gcd (2 · 𝐵)))
73	9	nnzd 12613	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℤ)
74		zmulcl 12639	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐵) ∈ ℤ)
75	18, 73, 74	sylancr 585	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐵) ∈ ℤ)
76		gcddvds 16475	. . . . . . . 8 ⊢ (((𝐶 − 𝐵) ∈ ℤ ∧ (2 · 𝐵) ∈ ℤ) → (((𝐶 − 𝐵) gcd (2 · 𝐵)) ∥ (𝐶 − 𝐵) ∧ ((𝐶 − 𝐵) gcd (2 · 𝐵)) ∥ (2 · 𝐵)))
77	7, 75, 76	syl2anc 582	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 − 𝐵) gcd (2 · 𝐵)) ∥ (𝐶 − 𝐵) ∧ ((𝐶 − 𝐵) gcd (2 · 𝐵)) ∥ (2 · 𝐵)))
78	77	simprd 494	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (2 · 𝐵)) ∥ (2 · 𝐵))
79	72, 78	eqbrtrd 5165	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐵))
80		1z 12620	. . . . . . . 8 ⊢ 1 ∈ ℤ
81		gcdaddm 16497	. . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ (𝐶 − 𝐵) ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵)))))
82	80, 7, 11, 81	mp3an2i 1462	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵)))))
83		ppncan 11530	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
84	83	3anidm13 1417	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
85	61	mullidd 11260	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · (𝐶 − 𝐵)) = (𝐶 − 𝐵))
86	85	oveq2d 7431	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) + (𝐶 − 𝐵)))
87		2times 12376	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℂ → (2 · 𝐶) = (𝐶 + 𝐶))
88	87	adantr 479	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐶) = (𝐶 + 𝐶))
89	84, 86, 88	3eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = (2 · 𝐶))
90	57, 58, 89	syl2anc 582	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = (2 · 𝐶))
91	90	oveq2d 7431	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵)))) = ((𝐶 − 𝐵) gcd (2 · 𝐶)))
92	82, 91	eqtrd 2765	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = ((𝐶 − 𝐵) gcd (2 · 𝐶)))
93	8	nnzd 12613	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℤ)
94		zmulcl 12639	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (2 · 𝐶) ∈ ℤ)
95	18, 93, 94	sylancr 585	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) ∈ ℤ)
96		gcddvds 16475	. . . . . . . 8 ⊢ (((𝐶 − 𝐵) ∈ ℤ ∧ (2 · 𝐶) ∈ ℤ) → (((𝐶 − 𝐵) gcd (2 · 𝐶)) ∥ (𝐶 − 𝐵) ∧ ((𝐶 − 𝐵) gcd (2 · 𝐶)) ∥ (2 · 𝐶)))
97	7, 95, 96	syl2anc 582	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 − 𝐵) gcd (2 · 𝐶)) ∥ (𝐶 − 𝐵) ∧ ((𝐶 − 𝐵) gcd (2 · 𝐶)) ∥ (2 · 𝐶)))
98	97	simprd 494	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (2 · 𝐶)) ∥ (2 · 𝐶))
99	92, 98	eqbrtrd 5165	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐶))
100		nnaddcl 12263	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℕ)
101	100	nnne0d 12290	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
102	101	ancoms 457	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
103	102	3adant1 1127	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
104	103	3ad2ant1 1130	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ≠ 0)
105	104	neneqd 2935	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ (𝐶 + 𝐵) = 0)
106	105	intnand 487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ ((𝐶 − 𝐵) = 0 ∧ (𝐶 + 𝐵) = 0))
107		gcdn0cl 16474	. . . . . . . 8 ⊢ ((((𝐶 − 𝐵) ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ) ∧ ¬ ((𝐶 − 𝐵) = 0 ∧ (𝐶 + 𝐵) = 0)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∈ ℕ)
108	7, 11, 106, 107	syl21anc 836	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∈ ℕ)
109	108	nnzd 12613	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∈ ℤ)
110		dvdsgcd 16517	. . . . . 6 ⊢ ((((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∈ ℤ ∧ (2 · 𝐵) ∈ ℤ ∧ (2 · 𝐶) ∈ ℤ) → ((((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐵) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐶)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ ((2 · 𝐵) gcd (2 · 𝐶))))
111	109, 75, 95, 110	syl3anc 1368	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐵) ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (2 · 𝐶)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ ((2 · 𝐵) gcd (2 · 𝐶))))
112	79, 99, 111	mp2and 697	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ ((2 · 𝐵) gcd (2 · 𝐶)))
113		2nn0 12517	. . . . . 6 ⊢ 2 ∈ ℕ0
114		mulgcd 16521	. . . . . 6 ⊢ ((2 ∈ ℕ0 ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 · 𝐵) gcd (2 · 𝐶)) = (2 · (𝐵 gcd 𝐶)))
115	113, 73, 93, 114	mp3an2i 1462	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 · 𝐵) gcd (2 · 𝐶)) = (2 · (𝐵 gcd 𝐶)))
116		pythagtriplem3 16784	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵 gcd 𝐶) = 1)
117	116	oveq2d 7431	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐵 gcd 𝐶)) = (2 · 1))
118		2t1e2 12403	. . . . . 6 ⊢ (2 · 1) = 2
119	117, 118	eqtrdi 2781	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐵 gcd 𝐶)) = 2)
120	115, 119	eqtrd 2765	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 · 𝐵) gcd (2 · 𝐶)) = 2)
121	112, 120	breqtrd 5169	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ 2)
122		dvdsprime 16655	. . . 4 ⊢ ((2 ∈ ℙ ∧ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∈ ℕ) → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ 2 ↔ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 ∨ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 1)))
123	47, 108, 122	sylancr 585	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ 2 ↔ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 ∨ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 1)))
124	121, 123	mpbid 231	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 ∨ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 1))
125		orel1 886	. 2 ⊢ (¬ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → ((((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 ∨ ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 1) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 1))
126	53, 124, 125	sylc 65	1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2930   class class class wbr 5143  (class class class)co 7415  ℂcc 11134  0cc0 11136  1c1 11137   + caddc 11139   · cmul 11141   − cmin 11472  -cneg 11473  ℕcn 12240  2c2 12295  ℕ₀cn0 12500  ℤcz 12586  ↑cexp 14056   ∥ cdvds 16228   gcd cgcd 16466  ℙcprime 16639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-2o 8484  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-sup 9463  df-inf 9464  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-n0 12501  df-z 12587  df-uz 12851  df-rp 13005  df-fz 13515  df-fl 13787  df-mod 13865  df-seq 13997  df-exp 14057  df-cj 15076  df-re 15077  df-im 15078  df-sqrt 15212  df-abs 15213  df-dvds 16229  df-gcd 16467  df-prm 16640

This theorem is referenced by:  pythagtriplem6  16787  pythagtriplem7  16788  flt4lem3  42136