Theorem pythagtriplem19 12980

Description: Lemma for pythagtrip 12981. Introduce 𝑘 and remove the relative primality requirement. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
pythagtriplem19	⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))

Distinct variable groups:   𝐴,𝑚,𝑛,𝑘   𝐵,𝑚,𝑛,𝑘   𝐶,𝑚,𝑛,𝑘

Proof of Theorem pythagtriplem19

Step	Hyp	Ref	Expression
1		gcdnncl 12663	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ)
2	1	3adant3 1044	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ)
3	2	3ad2ant1 1045	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐴 gcd 𝐵) ∈ ℕ)
4		nnz 9596	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
5		nnz 9596	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
6		gcddvds 12659	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
7	4, 5, 6	syl2an 289	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
8	7	3adant3 1044	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
9	8	simpld 112	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∥ 𝐴)
10	2	nnzd 9699	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℤ)
11	2	nnne0d 9282	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ≠ 0)
12	4	3ad2ant1 1045	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
13		dvdsval2 12476	. . . . . . . . 9 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ (𝐴 gcd 𝐵) ≠ 0 ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ))
14	10, 11, 12, 13	syl3anc 1274	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ))
15	9, 14	mpbid 147	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ)
16		nnre 9244	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
17	16	3ad2ant1 1045	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
18	2	nnred 9250	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℝ)
19		nngt0 9262	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴)
20	19	3ad2ant1 1045	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐴)
21	2	nngt0d 9281	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐴 gcd 𝐵))
22	17, 18, 20, 21	divgt0d 9209	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐴 / (𝐴 gcd 𝐵)))
23		elnnz 9587	. . . . . . 7 ⊢ ((𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐴 / (𝐴 gcd 𝐵))))
24	15, 22, 23	sylanbrc 417	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ)
25	24	3ad2ant1 1045	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ)
26	8	simprd 114	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∥ 𝐵)
27	5	3ad2ant2 1046	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℤ)
28		dvdsval2 12476	. . . . . . . . 9 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ (𝐴 gcd 𝐵) ≠ 0 ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ (𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ))
29	10, 11, 27, 28	syl3anc 1274	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ (𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ))
30	26, 29	mpbid 147	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ)
31		nnre 9244	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
32	31	3ad2ant2 1046	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
33		nngt0 9262	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
34	33	3ad2ant2 1046	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
35	32, 18, 34, 21	divgt0d 9209	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐵 / (𝐴 gcd 𝐵)))
36		elnnz 9587	. . . . . . 7 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐵 / (𝐴 gcd 𝐵))))
37	30, 35, 36	sylanbrc 417	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ)
38	37	3ad2ant1 1045	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ)
39		dvdssq 12727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
40	10, 12, 39	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
41		dvdssq 12727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
42	10, 27, 41	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
43	40, 42	anbi12d 473	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵) ↔ (((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2) ∧ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2))))
44	8, 43	mpbid 147	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2) ∧ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
45	2	nnsqcld 11056	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℕ)
46	45	nnzd 9699	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
47		nnsqcl 10971	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℕ)
48	47	3ad2ant1 1045	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℕ)
49	48	nnzd 9699	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℤ)
50		nnsqcl 10971	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → (𝐵↑2) ∈ ℕ)
51	50	3ad2ant2 1046	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℕ)
52	51	nnzd 9699	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℤ)
53		dvds2add 12511	. . . . . . . . . . . . 13 ⊢ ((((𝐴 gcd 𝐵)↑2) ∈ ℤ ∧ (𝐴↑2) ∈ ℤ ∧ (𝐵↑2) ∈ ℤ) → ((((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2) ∧ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)) → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2))))
54	46, 49, 52, 53	syl3anc 1274	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2) ∧ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)) → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2))))
55	44, 54	mpd 13	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
56	55	adantr 276	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
57		simpr 110	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))
58	56, 57	breqtrd 4135	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2))
59		nnz 9596	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
60	59	3ad2ant3 1047	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℤ)
61		dvdssq 12727	. . . . . . . . . . 11 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2)))
62	10, 60, 61	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2)))
63	62	adantr 276	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2)))
64	58, 63	mpbird 167	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐴 gcd 𝐵) ∥ 𝐶)
65		dvdsval2 12476	. . . . . . . . . 10 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ (𝐴 gcd 𝐵) ≠ 0 ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ))
66	10, 11, 60, 65	syl3anc 1274	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ))
67	66	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ))
68	64, 67	mpbid 147	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ)
69		nnre 9244	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
70	69	3ad2ant3 1047	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
71		nngt0 9262	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 0 < 𝐶)
72	71	3ad2ant3 1047	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
73	70, 18, 72, 21	divgt0d 9209	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 / (𝐴 gcd 𝐵)))
74	73	adantr 276	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶 / (𝐴 gcd 𝐵)))
75		elnnz 9587	. . . . . . 7 ⊢ ((𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐶 / (𝐴 gcd 𝐵))))
76	68, 74, 75	sylanbrc 417	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ)
77	76	3adant3 1044	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ)
78	48	nncnd 9251	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℂ)
79	51	nncnd 9251	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℂ)
80	45	nncnd 9251	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℂ)
81	45	nnap0d 9283	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) # 0)
82	78, 79, 80, 81	divdirapd 9103	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴↑2) + (𝐵↑2)) / ((𝐴 gcd 𝐵)↑2)) = (((𝐴↑2) / ((𝐴 gcd 𝐵)↑2)) + ((𝐵↑2) / ((𝐴 gcd 𝐵)↑2))))
83	82	3ad2ant1 1045	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (((𝐴↑2) + (𝐵↑2)) / ((𝐴 gcd 𝐵)↑2)) = (((𝐴↑2) / ((𝐴 gcd 𝐵)↑2)) + ((𝐵↑2) / ((𝐴 gcd 𝐵)↑2))))
84		nncn 9245	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
85	84	3ad2ant3 1047	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℂ)
86	2	nncnd 9251	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℂ)
87	2	nnap0d 9283	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) # 0)
88	85, 86, 87	sqdivapd 11048	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐶 / (𝐴 gcd 𝐵))↑2) = ((𝐶↑2) / ((𝐴 gcd 𝐵)↑2)))
89	88	3ad2ant1 1045	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ((𝐶 / (𝐴 gcd 𝐵))↑2) = ((𝐶↑2) / ((𝐴 gcd 𝐵)↑2)))
90		oveq1 6057	. . . . . . . 8 ⊢ (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) / ((𝐴 gcd 𝐵)↑2)) = ((𝐶↑2) / ((𝐴 gcd 𝐵)↑2)))
91	90	3ad2ant2 1046	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (((𝐴↑2) + (𝐵↑2)) / ((𝐴 gcd 𝐵)↑2)) = ((𝐶↑2) / ((𝐴 gcd 𝐵)↑2)))
92	89, 91	eqtr4d 2268	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ((𝐶 / (𝐴 gcd 𝐵))↑2) = (((𝐴↑2) + (𝐵↑2)) / ((𝐴 gcd 𝐵)↑2)))
93		nncn 9245	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
94	93	3ad2ant1 1045	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℂ)
95	94, 86, 87	sqdivapd 11048	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵))↑2) = ((𝐴↑2) / ((𝐴 gcd 𝐵)↑2)))
96		nncn 9245	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
97	96	3ad2ant2 1046	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℂ)
98	97, 86, 87	sqdivapd 11048	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐵 / (𝐴 gcd 𝐵))↑2) = ((𝐵↑2) / ((𝐴 gcd 𝐵)↑2)))
99	95, 98	oveq12d 6068	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴 / (𝐴 gcd 𝐵))↑2) + ((𝐵 / (𝐴 gcd 𝐵))↑2)) = (((𝐴↑2) / ((𝐴 gcd 𝐵)↑2)) + ((𝐵↑2) / ((𝐴 gcd 𝐵)↑2))))
100	99	3ad2ant1 1045	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (((𝐴 / (𝐴 gcd 𝐵))↑2) + ((𝐵 / (𝐴 gcd 𝐵))↑2)) = (((𝐴↑2) / ((𝐴 gcd 𝐵)↑2)) + ((𝐵↑2) / ((𝐴 gcd 𝐵)↑2))))
101	83, 92, 100	3eqtr4rd 2276	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (((𝐴 / (𝐴 gcd 𝐵))↑2) + ((𝐵 / (𝐴 gcd 𝐵))↑2)) = ((𝐶 / (𝐴 gcd 𝐵))↑2))
102		gcddiv 12715	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝐴 gcd 𝐵) ∈ ℕ) ∧ ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵)) → ((𝐴 gcd 𝐵) / (𝐴 gcd 𝐵)) = ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))))
103	12, 27, 2, 8, 102	syl31anc 1277	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) / (𝐴 gcd 𝐵)) = ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))))
104	86, 87	dividapd 9060	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) / (𝐴 gcd 𝐵)) = 1)
105	103, 104	eqtr3d 2267	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
106	105	3ad2ant1 1045	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
107		simp3 1026	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)))
108		pythagtriplem18 12979	. . . . 5 ⊢ ((((𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ ∧ (𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ ∧ (𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ) ∧ (((𝐴 / (𝐴 gcd 𝐵))↑2) + ((𝐵 / (𝐴 gcd 𝐵))↑2)) = ((𝐶 / (𝐴 gcd 𝐵))↑2) ∧ (((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1 ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))))
109	25, 38, 77, 101, 106, 107, 108	syl312anc 1295	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))))
110	94, 86, 87	divcanap2d 9066	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) = 𝐴)
111	110	eqcomd 2238	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 = ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))))
112	97, 86, 87	divcanap2d 9066	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) = 𝐵)
113	112	eqcomd 2238	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))))
114	85, 86, 87	divcanap2d 9066	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))) = 𝐶)
115	114	eqcomd 2238	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 = ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))))
116	111, 113, 115	3jca 1204	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 = ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵)))))
117	116	3ad2ant1 1045	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐴 = ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵)))))
118		oveq2 6058	. . . . . . . . . 10 ⊢ ((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) → ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))))
119	118	eqeq2d 2244	. . . . . . . . 9 ⊢ ((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) → (𝐴 = ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) ↔ 𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2)))))
120	119	3ad2ant1 1045	. . . . . . . 8 ⊢ (((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))) → (𝐴 = ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) ↔ 𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2)))))
121		oveq2 6058	. . . . . . . . . 10 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) → ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))))
122	121	eqeq2d 2244	. . . . . . . . 9 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) → (𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) ↔ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛)))))
123	122	3ad2ant2 1046	. . . . . . . 8 ⊢ (((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))) → (𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) ↔ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛)))))
124		oveq2 6058	. . . . . . . . . 10 ⊢ ((𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2)) → ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))
125	124	eqeq2d 2244	. . . . . . . . 9 ⊢ ((𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2)) → (𝐶 = ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))) ↔ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2)))))
126	125	3ad2ant3 1047	. . . . . . . 8 ⊢ (((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))) → (𝐶 = ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))) ↔ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2)))))
127	120, 123, 126	3anbi123d 1349	. . . . . . 7 ⊢ (((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))) → ((𝐴 = ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵)))) ↔ (𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))))
128	117, 127	syl5ibcom 155	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))) → (𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))))
129	128	reximdv 2643	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (∃𝑚 ∈ ℕ ((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))) → ∃𝑚 ∈ ℕ (𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))))
130	129	reximdv 2643	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) ∧ (𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) ∧ (𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))))
131	109, 130	mpd 13	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2)))))
132		oveq1 6057	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · ((𝑚↑2) − (𝑛↑2))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))))
133	132	eqeq2d 2244	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ↔ 𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2)))))
134		oveq1 6057	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · (2 · (𝑚 · 𝑛))) = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))))
135	134	eqeq2d 2244	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ↔ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛)))))
136		oveq1 6057	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · ((𝑚↑2) + (𝑛↑2))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))
137	136	eqeq2d 2244	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))) ↔ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2)))))
138	133, 135, 137	3anbi123d 1349	. . . . 5 ⊢ (𝑘 = (𝐴 gcd 𝐵) → ((𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ (𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))))
139	138	2rexbidv 2567	. . . 4 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))))
140	139	rspcev 2921	. . 3 ⊢ (((𝐴 gcd 𝐵) ∈ ℕ ∧ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))) → ∃𝑘 ∈ ℕ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
141	3, 131, 140	syl2anc 411	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ∃𝑘 ∈ ℕ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
142		rexcom 2707	. . 3 ⊢ (∃𝑘 ∈ ℕ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑛 ∈ ℕ ∃𝑘 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
143		rexcom 2707	. . . 4 ⊢ (∃𝑘 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
144	143	rexbii 2549	. . 3 ⊢ (∃𝑛 ∈ ℕ ∃𝑘 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
145	142, 144	bitri 184	. 2 ⊢ (∃𝑘 ∈ ℕ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))) ↔ ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))
146	141, 145	sylib 122	1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ ∃𝑘 ∈ ℕ (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ∧ 𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ∧ 𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∃wrex 2521   class class class wbr 4109  (class class class)co 6050  ℂcc 8125  ℝcr 8126  0cc0 8127  1c1 8128   + caddc 8130   · cmul 8132   < clt 8308   − cmin 8444   / cdiv 8946  ℕcn 9237  2c2 9288  ℤcz 9577  ↑cexp 10900   ∥ cdvds 12473   gcd cgcd 12649

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650  df-prm 12805

This theorem is referenced by:  pythagtrip  12981