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 16741

Description: Lemma for pythagtrip 16756. 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 1203	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ 𝐴)
2		nnz 12499	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
3		nnz 12499	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
4		zsubcl 12524	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐶 − 𝐵) ∈ ℤ)
5	2, 3, 4	syl2anr 597	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
6	5	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
7	6	3ad2ant1 1133	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℤ)
8		simp13 1206	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℕ)
9		simp12 1205	. . . . . . . . . . . 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 12187	. . . . . . . . . . 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 12505	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℤ)
12		gcddvds 16424	. . . . . . . . . 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 584	. . . . . . . . 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 495	. . . . . . . 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 5098	. . . . . . . . 9 ⊢ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) ↔ 2 ∥ (𝐶 + 𝐵)))
16	15	biimpd 229	. . . . . . . 8 ⊢ (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) = 2 → (((𝐶 − 𝐵) gcd (𝐶 + 𝐵)) ∥ (𝐶 + 𝐵) → 2 ∥ (𝐶 + 𝐵)))
17	14, 16	mpan9 506	. . . . . . 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 12514	. . . . . . . 8 ⊢ 2 ∈ ℤ
19		simpl13 1251	. . . . . . . . . 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 12505	. . . . . . . . 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 1250	. . . . . . . . . 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 12505	. . . . . . . . 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 12591	. . . . . . . 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 12592	. . . . . . . 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 16217	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ ∧ (𝐶 − 𝐵) ∈ ℤ) → (2 ∥ (𝐶 + 𝐵) → 2 ∥ ((𝐶 + 𝐵) · (𝐶 − 𝐵))))
26	18, 23, 24, 25	mp3an2i 1468	. . . . . . 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 12151	. . . . . . 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 12151	. . . . . . 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 14127	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
31	28, 29, 30	syl2anc 584	. . . . . 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 5123	. . . . 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 1193	. . . . . . 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 7370	. . . . . 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 1249	. . . . . . . . 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 14161	. . . . . . . 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 12151	. . . . . . 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 14161	. . . . . . . 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 12151	. . . . . . 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 11483	. . . . . 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 2770	. . . . 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 5121	. . . 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 12499	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
44	43	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
45	44	3ad2ant1 1133	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℤ)
46	45	adantr 480	. . . . 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 16613	. . . . . 6 ⊢ 2 ∈ ℙ
48		2nn 12208	. . . . . 6 ⊢ 2 ∈ ℕ
49		prmdvdsexp 16636	. . . . . 6 ⊢ ((2 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 2 ∈ ℕ) → (2 ∥ (𝐴↑2) ↔ 2 ∥ 𝐴))
50	47, 48, 49	mp3an13 1454	. . . . 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 232	. . 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 815	. 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 12518	. . . . . . . 8 ⊢ -1 ∈ ℤ
55		gcdaddm 16446	. . . . . . . 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 1468	. . . . . . 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 12151	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
58	9	nncnd 12151	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
59		pnncan 11412	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶 − 𝐵)) = (𝐵 + 𝐵))
60	59	3anidm23 1423	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) − (𝐶 − 𝐵)) = (𝐵 + 𝐵))
61		subcl 11369	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ)
62	61	mulm1d 11579	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-1 · (𝐶 − 𝐵)) = -(𝐶 − 𝐵))
63	62	oveq2d 7371	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) + -(𝐶 − 𝐵)))
64		addcl 11098	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
65	64, 61	negsubd 11488	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + -(𝐶 − 𝐵)) = ((𝐶 + 𝐵) − (𝐶 − 𝐵)))
66	63, 65	eqtrd 2768	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) − (𝐶 − 𝐵)))
67		2times 12266	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℂ → (2 · 𝐵) = (𝐵 + 𝐵))
68	67	adantl 481	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐵) = (𝐵 + 𝐵))
69	60, 66, 68	3eqtr4d 2778	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵))) = (2 · 𝐵))
70	69	oveq2d 7371	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 − 𝐵) gcd ((𝐶 + 𝐵) + (-1 · (𝐶 − 𝐵)))) = ((𝐶 − 𝐵) gcd (2 · 𝐵)))
71	57, 58, 70	syl2anc 584	. . . . . . 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 2768	. . . . . 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 12505	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℤ)
74		zmulcl 12531	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐵) ∈ ℤ)
75	18, 73, 74	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐵) ∈ ℤ)
76		gcddvds 16424	. . . . . . . 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 584	. . . . . . 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 495	. . . . . 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 5117	. . . . 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 12512	. . . . . . . 8 ⊢ 1 ∈ ℤ
81		gcdaddm 16446	. . . . . . . 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 1468	. . . . . . 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 11413	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
84	83	3anidm13 1422	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
85	61	mullidd 11140	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 · (𝐶 − 𝐵)) = (𝐶 − 𝐵))
86	85	oveq2d 7371	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = ((𝐶 + 𝐵) + (𝐶 − 𝐵)))
87		2times 12266	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℂ → (2 · 𝐶) = (𝐶 + 𝐶))
88	87	adantr 480	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐶) = (𝐶 + 𝐶))
89	84, 86, 88	3eqtr4d 2778	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (1 · (𝐶 − 𝐵))) = (2 · 𝐶))
90	57, 58, 89	syl2anc 584	. . . . . . . 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 7371	. . . . . . 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 2768	. . . . . 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 12505	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℤ)
94		zmulcl 12531	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (2 · 𝐶) ∈ ℤ)
95	18, 93, 94	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) ∈ ℤ)
96		gcddvds 16424	. . . . . . . 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 584	. . . . . . 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 495	. . . . . 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 5117	. . . . 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 12158	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℕ)
101	100	nnne0d 12185	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
102	101	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
103	102	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ≠ 0)
104	103	3ad2ant1 1133	. . . . . . . . . 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 488	. . . . . . . 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 16423	. . . . . . . 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 837	. . . . . . 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 12505	. . . . . 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 16465	. . . . . 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 1373	. . . . 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 699	. . . 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 12408	. . . . . 6 ⊢ 2 ∈ ℕ₀
114		mulgcd 16469	. . . . . 6 ⊢ ((2 ∈ ℕ₀ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 · 𝐵) gcd (2 · 𝐶)) = (2 · (𝐵 gcd 𝐶)))
115	113, 73, 93, 114	mp3an2i 1468	. . . . 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 16740	. . . . . . 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 7371	. . . . . 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 12293	. . . . . 6 ⊢ (2 · 1) = 2
119	117, 118	eqtrdi 2784	. . . . 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 2768	. . . 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 5121	. . 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 16608	. . . 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 587	. . 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 232	. 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 888	. 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)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 class class class wbr 5095 (class class class)co 7355 ℂcc 11014 0cc0 11016 1c1 11017 + caddc 11019 · cmul 11021 − cmin 11354 -cneg 11355 ℕcn 12135 2c2 12190 ℕ₀cn0 12391 ℤcz 12478 ↑cexp 13978 ∥ cdvds 16173 gcd cgcd 16415 ℙcprime 16592

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 ax-pre-sup 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-sup 9336 df-inf 9337 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-div 11785 df-nn 12136 df-2 12198 df-3 12199 df-n0 12392 df-z 12479 df-uz 12743 df-rp 12901 df-fz 13418 df-fl 13706 df-mod 13784 df-seq 13919 df-exp 13979 df-cj 15016 df-re 15017 df-im 15018 df-sqrt 15152 df-abs 15153 df-dvds 16174 df-gcd 16416 df-prm 16593

This theorem is referenced by: pythagtriplem6 16743 pythagtriplem7 16744 flt4lem3 42756

W3C validator