pythagtriplem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pythagtriplem4		Structured version Visualization version GIF version

Theorem pythagtriplem4 16903

Description: Lemma for pythagtrip 16918. 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 1221	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ 𝐴)
2		nnz 12629	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
3		nnz 12629	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
4		zsubcl 12653	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶 − 𝐵) ∈ ℤ)
5	2, 3, 4	syl2anr 609	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
6	5	3adant1 1148	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
7	6	3ad2ant1 1151	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℤ)
8		simp13 1224	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℕ)
9		simp12 1223	. . . . . . . . . . . 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 12305	. . . . . . . . . . 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 12634	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℤ)
12		gcddvds 16585	. . . . . . . . . 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 596	. . . . . . . . 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 501	. . . . . . . 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 5114	. . . . . . . . 9 ⊢ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) ↔ 2 ∥ (𝐶 + 𝐵)))
16	15	biimpd 232	. . . . . . . 8 ⊢ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) → 2 ∥ (𝐶 + 𝐵)))
17	14, 16	mpan9 516	. . . . . . 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 12643	. . . . . . . 8 ⊢ 2 ∈ ℤ
19		simpl13 1269	. . . . . . . . . 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 12634	. . . . . . . . 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 1268	. . . . . . . . . 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 12634	. . . . . . . . 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 12722	. . . . . . . 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 12723	. . . . . . . 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 16378	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ ∧ (𝐶 − 𝐵) ∈ ℤ) → (2 ∥ (𝐶 + 𝐵) → 2 ∥ ((𝐶 + 𝐵) · (𝐶 − 𝐵))))
26	18, 23, 24, 25	mp3an2i 1495	. . . . . . 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 16	. . . . . 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 12266	. . . . . . 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 12266	. . . . . . 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 14266	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
31	28, 29, 30	syl2anc 596	. . . . . 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 5141	. . . . 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 1211	. . . . . . 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 7434	. . . . . 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 1267	. . . . . . . . 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 14300	. . . . . . . 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 12266	. . . . . . 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 14300	. . . . . . . 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 12266	. . . . . . 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 11587	. . . . . 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 2802	. . . . 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 5139	. . . 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 12629	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
44	43	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
45	44	3ad2ant1 1151	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℤ)
46	45	adantr 486	. . . . 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 16774	. . . . . 6 ⊢ 2 ∈ ℙ
48		2nn 12331	. . . . . 6 ⊢ 2 ∈ ℕ
49		prmdvdsexp 16798	. . . . . 6 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 2 ∈ ℕ) → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
50	47, 48, 49	mp3an13 1481	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
51	46, 50	syl 18	. . . 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 235	. . 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 828	. 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 12647	. . . . . . . 8 ⊢ -1 ∈ ℤ
55		gcdaddm 16607	. . . . . . . 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 1495	. . . . . . 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 12266	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
58	9	nncnd 12266	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
59		pnncan 11516	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶 − 𝐵)) = (𝐵 + 𝐵))
60	59	3anidm23 1448	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶 − 𝐵)) = (𝐵 + 𝐵))
61		subcl 11473	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ)
62	61	mulm1d 11683	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-1 · (𝐶 − 𝐵)) = -(𝐶 − 𝐵))
63	62	oveq2d 7435	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) + -(𝐶 − 𝐵)))
64		addcl 11199	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
65	64, 61	negsubd 11592	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + -(𝐶 − 𝐵)) = ((𝐶 + 𝐵) − (𝐶 − 𝐵)))
66	63, 65	eqtrd 2800	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) − (𝐶 − 𝐵)))
67		2times 12393	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℂ → (2 · 𝐵) = (𝐵 + 𝐵))
68	67	adantl 487	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐵) = (𝐵 + 𝐵))
69	60, 66, 68	3eqtr4d 2810	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = (2 · 𝐵))
70	69	oveq2d 7435	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵)))) = ((𝐶 − 𝐵) gcd (2 · 𝐵)))
71	57, 58, 70	syl2anc 596	. . . . . . 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 2800	. . . . . 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 12634	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℤ)
74		zmulcl 12660	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐵) ∈ ℤ)
75	18, 73, 74	sylancr 599	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐵) ∈ ℤ)
76		gcddvds 16585	. . . . . . . 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 596	. . . . . . 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 501	. . . . . 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 5135	. . . . 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 12641	. . . . . . . 8 ⊢ 1 ∈ ℤ
81		gcdaddm 16607	. . . . . . . 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 1495	. . . . . . 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 11517	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
84	83	3anidm13 1447	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
85	61	mullidd 11244	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · (𝐶 − 𝐵)) = (𝐶 − 𝐵))
86	85	oveq2d 7435	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) + (𝐶 − 𝐵)))
87		2times 12393	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℂ → (2 · 𝐶) = (𝐶 + 𝐶))
88	87	adantr 486	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐶) = (𝐶 + 𝐶))
89	84, 86, 88	3eqtr4d 2810	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = (2 · 𝐶))
90	57, 58, 89	syl2anc 596	. . . . . . . 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 7435	. . . . . . 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 2800	. . . . . 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 12634	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℤ)
94		zmulcl 12660	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (2 · 𝐶) ∈ ℤ)
95	18, 93, 94	sylancr 599	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) ∈ ℤ)
96		gcddvds 16585	. . . . . . . 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 596	. . . . . . 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 501	. . . . . 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 5135	. . . . 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 12273	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℕ)
101	100	nnne0d 12303	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
102	101	ancoms 464	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
103	102	3adant1 1148	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
104	103	3ad2ant1 1151	. . . . . . . . . 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 2965	. . . . . . . . 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 494	. . . . . . . 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 16584	. . . . . . . 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 851	. . . . . . 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 12634	. . . . . 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 16626	. . . . . 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 1398	. . . . 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 712	. . . 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 12538	. . . . . 6 ⊢ 2 ∈ ℕ₀
114		mulgcd 16630	. . . . . 6 ⊢ ((2 ∈ ℕ₀ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 · 𝐵) gcd (2 · 𝐶)) = (2 · (𝐵 gcd 𝐶)))
115	113, 73, 93, 114	mp3an2i 1495	. . . . 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 16902	. . . . . . 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 7435	. . . . . 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 12420	. . . . . 6 ⊢ (2 · 1) = 2
119	117, 118	eqtrdi 2816	. . . . 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 2800	. . . 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 5139	. . 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 16769	. . . 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 599	. . 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 235	. 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 902	. 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 66	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 setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 class class class wbr 5111 (class class class)co 7419 ℂcc 11115 0cc0 11117 1c1 11118 + caddc 11120 · cmul 11122 − cmin 11458 -cneg 11459 ℕcn 12250 2c2 12312 ℕ₀cn0 12521 ℤcz 12608 ↑cexp 14117 ∥ cdvds 16334 gcd cgcd 16576 ℙcprime 16753

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-n0 12522 df-z 12609 df-uz 12881 df-rp 13035 df-fz 13554 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-dvds 16335 df-gcd 16577 df-prm 16754

This theorem is used by: pythagtriplem6 16905 pythagtriplem7 16906 flt4lem3 43440

W3C validator