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Theorem pythagtriplem4 12786

Description: Lemma for pythagtrip 12801. 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 1050	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ 𝐴)
2		nnz 9461	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
3		nnz 9461	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
4		zsubcl 9483	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶 − 𝐵) ∈ ℤ)
5	2, 3, 4	syl2anr 290	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
6	5	3adant1 1039	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
7	6	3ad2ant1 1042	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℤ)
8		simp13 1053	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℕ)
9		simp12 1052	. . . . . . . . . . . 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 9154	. . . . . . . . . . 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 9564	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℤ)
12		gcddvds 12479	. . . . . . . . . 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 411	. . . . . . . . 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 114	. . . . . . . 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 4085	. . . . . . . . 9 ⊢ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) ↔ 2 ∥ (𝐶 + 𝐵)))
16	15	biimpd 144	. . . . . . . 8 ⊢ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) → 2 ∥ (𝐶 + 𝐵)))
17	14, 16	mpan9 281	. . . . . . 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 9470	. . . . . . . 8 ⊢ 2 ∈ ℤ
19		simpl13 1098	. . . . . . . . . 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 9564	. . . . . . . . 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 1097	. . . . . . . . . 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 9564	. . . . . . . . 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 9569	. . . . . . . 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 9570	. . . . . . . 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 12337	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ ∧ (𝐶 − 𝐵) ∈ ℤ) → (2 ∥ (𝐶 + 𝐵) → 2 ∥ ((𝐶 + 𝐵) · (𝐶 − 𝐵))))
26	18, 23, 24, 25	mp3an2i 1376	. . . . . . 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 13	. . . . . 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 9120	. . . . . . 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 9120	. . . . . . 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 10863	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
31	28, 29, 30	syl2anc 411	. . . . . 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 4110	. . . . 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 1025	. . . . . . 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 6015	. . . . . 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 1096	. . . . . . . . 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 10911	. . . . . . . 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 9120	. . . . . . 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 10911	. . . . . . . 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 9120	. . . . . . 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 8454	. . . . . 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 2264	. . . . 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 4108	. . . 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 9461	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
44	43	3ad2ant1 1042	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
45	44	3ad2ant1 1042	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℤ)
46	45	adantr 276	. . . . 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 12644	. . . . . 6 ⊢ 2 ∈ ℙ
48		2nn 9268	. . . . . 6 ⊢ 2 ∈ ℕ
49		prmdvdsexp 12665	. . . . . 6 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 2 ∈ ℕ) → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
50	47, 48, 49	mp3an13 1362	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
51	46, 50	syl 14	. . . 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 147	. . 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 669	. 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 9474	. . . . . . . 8 ⊢ -1 ∈ ℤ
55		gcdaddm 12500	. . . . . . . 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 1376	. . . . . . 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 9120	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
58	9	nncnd 9120	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
59		pnncan 8383	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶 − 𝐵)) = (𝐵 + 𝐵))
60	59	3anidm23 1331	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶 − 𝐵)) = (𝐵 + 𝐵))
61		subcl 8341	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ)
62	61	mulm1d 8552	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-1 · (𝐶 − 𝐵)) = -(𝐶 − 𝐵))
63	62	oveq2d 6016	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) + -(𝐶 − 𝐵)))
64		addcl 8120	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
65	64, 61	negsubd 8459	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + -(𝐶 − 𝐵)) = ((𝐶 + 𝐵) − (𝐶 − 𝐵)))
66	63, 65	eqtrd 2262	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) − (𝐶 − 𝐵)))
67		2times 9234	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℂ → (2 · 𝐵) = (𝐵 + 𝐵))
68	67	adantl 277	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐵) = (𝐵 + 𝐵))
69	60, 66, 68	3eqtr4d 2272	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = (2 · 𝐵))
70	69	oveq2d 6016	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵)))) = ((𝐶 − 𝐵) gcd (2 · 𝐵)))
71	57, 58, 70	syl2anc 411	. . . . . . 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 2262	. . . . . 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 9564	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℤ)
74		zmulcl 9496	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐵) ∈ ℤ)
75	18, 73, 74	sylancr 414	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐵) ∈ ℤ)
76		gcddvds 12479	. . . . . . . 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 411	. . . . . . 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 114	. . . . . 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 4104	. . . . 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 9468	. . . . . . . 8 ⊢ 1 ∈ ℤ
81		gcdaddm 12500	. . . . . . . 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 1376	. . . . . . 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 8384	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
84	83	3anidm13 1330	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
85	61	mulid2d 8161	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · (𝐶 − 𝐵)) = (𝐶 − 𝐵))
86	85	oveq2d 6016	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) + (𝐶 − 𝐵)))
87		2times 9234	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℂ → (2 · 𝐶) = (𝐶 + 𝐶))
88	87	adantr 276	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐶) = (𝐶 + 𝐶))
89	84, 86, 88	3eqtr4d 2272	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = (2 · 𝐶))
90	57, 58, 89	syl2anc 411	. . . . . . . 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 6016	. . . . . . 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 2262	. . . . . 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 9564	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℤ)
94		zmulcl 9496	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (2 · 𝐶) ∈ ℤ)
95	18, 93, 94	sylancr 414	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) ∈ ℤ)
96		gcddvds 12479	. . . . . . . 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 411	. . . . . . 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 114	. . . . . 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 4104	. . . . 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 9126	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℕ)
101	100	nnne0d 9151	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
102	101	ancoms 268	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
103	102	3adant1 1039	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
104	103	3ad2ant1 1042	. . . . . . . . . 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 2421	. . . . . . . . 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 936	. . . . . . . 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 12478	. . . . . . . 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 1270	. . . . . . 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 9564	. . . . . 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 12528	. . . . . 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 1271	. . . . 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 433	. . . 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 9382	. . . . . 6 ⊢ 2 ∈ ℕ₀
114		mulgcd 12532	. . . . . 6 ⊢ ((2 ∈ ℕ₀ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 · 𝐵) gcd (2 · 𝐶)) = (2 · (𝐵 gcd 𝐶)))
115	113, 73, 93, 114	mp3an2i 1376	. . . . 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 12785	. . . . . . 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 6016	. . . . . 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 9260	. . . . . 6 ⊢ (2 · 1) = 2
119	117, 118	eqtrdi 2278	. . . . 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 2262	. . . 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 4108	. . 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 12639	. . . 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 414	. . 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 147	. 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 730	. 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 62	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)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 713 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 class class class wbr 4082 (class class class)co 6000 ℂcc 7993 0cc0 7995 1c1 7996 + caddc 7998 · cmul 8000 − cmin 8313 -cneg 8314 ℕcn 9106 2c2 9157 ℕ₀cn0 9365 ℤcz 9442 ↑cexp 10755 ∥ cdvds 12293 gcd cgcd 12469 ℙcprime 12624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-1o 6560 df-2o 6561 df-er 6678 df-en 6886 df-sup 7147 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-q 9811 df-rp 9846 df-fz 10201 df-fzo 10335 df-fl 10485 df-mod 10540 df-seqfrec 10665 df-exp 10756 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-dvds 12294 df-gcd 12470 df-prm 12625

This theorem is referenced by: pythagtriplem6 12788 pythagtriplem7 12789

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