Theorem pythagtriplem19 16172

Description: Lemma for pythagtrip 16173. 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 15858	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ)
2	1	3adant3 1128	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ)
3	2	3ad2ant1 1129	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐴 gcd 𝐵) ∈ ℕ)
4		nnz 12007	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
5		nnz 12007	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
6		gcddvds 15854	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
7	4, 5, 6	syl2an 597	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
8	7	3adant3 1128	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
9	8	simpld 497	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∥ 𝐴)
10	2	nnzd 12089	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℤ)
11	2	nnne0d 11690	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ≠ 0)
12	4	3ad2ant1 1129	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
13		dvdsval2 15612	. . . . . . . . 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 1367	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ))
15	9, 14	mpbid 234	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ)
16		nnre 11647	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
17	16	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
18	2	nnred 11655	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℝ)
19		nngt0 11671	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴)
20	19	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐴)
21	2	nngt0d 11689	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐴 gcd 𝐵))
22	17, 18, 20, 21	divgt0d 11577	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐴 / (𝐴 gcd 𝐵)))
23		elnnz 11994	. . . . . . 7 ⊢ ((𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐴 / (𝐴 gcd 𝐵))))
24	15, 22, 23	sylanbrc 585	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ)
25	24	3ad2ant1 1129	. . . . 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 498	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∥ 𝐵)
27	5	3ad2ant2 1130	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℤ)
28		dvdsval2 15612	. . . . . . . . 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 1367	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ (𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ))
30	26, 29	mpbid 234	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ)
31		nnre 11647	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
32	31	3ad2ant2 1130	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
33		nngt0 11671	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
34	33	3ad2ant2 1130	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
35	32, 18, 34, 21	divgt0d 11577	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐵 / (𝐴 gcd 𝐵)))
36		elnnz 11994	. . . . . . 7 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐵 / (𝐴 gcd 𝐵))))
37	30, 35, 36	sylanbrc 585	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ)
38	37	3ad2ant1 1129	. . . . 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 15913	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
40	10, 12, 39	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
41		dvdssq 15913	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
42	10, 27, 41	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
43	40, 42	anbi12d 632	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2) ∧ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
45	2	nnsqcld 13608	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℕ)
46	45	nnzd 12089	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
47		nnsqcl 13496	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℕ)
48	47	3ad2ant1 1129	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℕ)
49	48	nnzd 12089	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℤ)
50		nnsqcl 13496	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → (𝐵↑2) ∈ ℕ)
51	50	3ad2ant2 1130	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℕ)
52	51	nnzd 12089	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℤ)
53		dvds2add 15645	. . . . . . . . . . . . 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 1367	. . . . . . . . . . . 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 15	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
56	55	adantr 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
57		simpr 487	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))
58	56, 57	breqtrd 5094	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2))
59		nnz 12007	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
60	59	3ad2ant3 1131	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℤ)
61		dvdssq 15913	. . . . . . . . . . 11 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2)))
62	10, 60, 61	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2)))
63	62	adantr 483	. . . . . . . . 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 259	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐴 gcd 𝐵) ∥ 𝐶)
65		dvdsval2 15612	. . . . . . . . . 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 1367	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ))
67	66	adantr 483	. . . . . . . 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 234	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ)
69		nnre 11647	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
70	69	3ad2ant3 1131	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
71		nngt0 11671	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 0 < 𝐶)
72	71	3ad2ant3 1131	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
73	70, 18, 72, 21	divgt0d 11577	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 / (𝐴 gcd 𝐵)))
74	73	adantr 483	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶 / (𝐴 gcd 𝐵)))
75		elnnz 11994	. . . . . . 7 ⊢ ((𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐶 / (𝐴 gcd 𝐵))))
76	68, 74, 75	sylanbrc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ)
77	76	3adant3 1128	. . . . 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 11656	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℂ)
79	51	nncnd 11656	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℂ)
80	45	nncnd 11656	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℂ)
81	45	nnne0d 11690	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ≠ 0)
82	78, 79, 80, 81	divdird 11456	. . . . . . 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 1129	. . . . . 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 11648	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
85	84	3ad2ant3 1131	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℂ)
86	2	nncnd 11656	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℂ)
87	85, 86, 11	sqdivd 13526	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐶 / (𝐴 gcd 𝐵))↑2) = ((𝐶↑2) / ((𝐴 gcd 𝐵)↑2)))
88	87	3ad2ant1 1129	. . . . . . 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)))
89		oveq1 7165	. . . . . . . 8 ⊢ (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) / ((𝐴 gcd 𝐵)↑2)) = ((𝐶↑2) / ((𝐴 gcd 𝐵)↑2)))
90	89	3ad2ant2 1130	. . . . . . 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)))
91	88, 90	eqtr4d 2861	. . . . . 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)))
92		nncn 11648	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
93	92	3ad2ant1 1129	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℂ)
94	93, 86, 11	sqdivd 13526	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵))↑2) = ((𝐴↑2) / ((𝐴 gcd 𝐵)↑2)))
95		nncn 11648	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
96	95	3ad2ant2 1130	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℂ)
97	96, 86, 11	sqdivd 13526	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐵 / (𝐴 gcd 𝐵))↑2) = ((𝐵↑2) / ((𝐴 gcd 𝐵)↑2)))
98	94, 97	oveq12d 7176	. . . . . . 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))))
99	98	3ad2ant1 1129	. . . . . 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))))
100	83, 91, 99	3eqtr4rd 2869	. . . . 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))
101		gcddiv 15901	. . . . . . . 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))))
102	12, 27, 2, 8, 101	syl31anc 1369	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) / (𝐴 gcd 𝐵)) = ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))))
103	86, 11	dividd 11416	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) / (𝐴 gcd 𝐵)) = 1)
104	102, 103	eqtr3d 2860	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
105	104	3ad2ant1 1129	. . . . 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)
106		simp3 1134	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)))
107		pythagtriplem18 16171	. . . . 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))))
108	25, 38, 77, 100, 105, 106, 107	syl312anc 1387	. . . 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))))
109	93, 86, 11	divcan2d 11420	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) = 𝐴)
110	109	eqcomd 2829	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 = ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))))
111	96, 86, 11	divcan2d 11420	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) = 𝐵)
112	111	eqcomd 2829	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))))
113	85, 86, 11	divcan2d 11420	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))) = 𝐶)
114	113	eqcomd 2829	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 = ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))))
115	110, 112, 114	3jca 1124	. . . . . . . 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)))))
116	115	3ad2ant1 1129	. . . . . . 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)))))
117		oveq2 7166	. . . . . . . . . 10 ⊢ ((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) → ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))))
118	117	eqeq2d 2834	. . . . . . . . 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)))))
119	118	3ad2ant1 1129	. . . . . . . 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)))))
120		oveq2 7166	. . . . . . . . . 10 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) → ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))))
121	120	eqeq2d 2834	. . . . . . . . 9 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) → (𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) ↔ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛)))))
122	121	3ad2ant2 1130	. . . . . . . 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)))))
123		oveq2 7166	. . . . . . . . . 10 ⊢ ((𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2)) → ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))
124	123	eqeq2d 2834	. . . . . . . . 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)))))
125	124	3ad2ant3 1131	. . . . . . . 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)))))
126	119, 122, 125	3anbi123d 1432	. . . . . . 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))))))
127	116, 126	syl5ibcom 247	. . . . . 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))))))
128	127	reximdv 3275	. . . . 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))))))
129	128	reximdv 3275	. . . 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))))))
130	108, 129	mpd 15	. . 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)))))
131		oveq1 7165	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · ((𝑚↑2) − (𝑛↑2))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))))
132	131	eqeq2d 2834	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ↔ 𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2)))))
133		oveq1 7165	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · (2 · (𝑚 · 𝑛))) = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))))
134	133	eqeq2d 2834	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ↔ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛)))))
135		oveq1 7165	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · ((𝑚↑2) + (𝑛↑2))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))
136	135	eqeq2d 2834	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))) ↔ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2)))))
137	132, 134, 136	3anbi123d 1432	. . . . 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))))))
138	137	2rexbidv 3302	. . . 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))))))
139	138	rspcev 3625	. . 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)))))
140	3, 130, 139	syl2anc 586	. 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)))))
141		rexcom 3357	. . 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)))))
142		rexcom 3357	. . . 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)))))
143	142	rexbii 3249	. . 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)))))
144	141, 143	bitri 277	. 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)))))
145	140, 144	sylib 220	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   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141   class class class wbr 5068  (class class class)co 7158  ℂcc 10537  ℝcr 10538  0cc0 10539  1c1 10540   + caddc 10542   · cmul 10544   < clt 10677   − cmin 10872   / cdiv 11299  ℕcn 11640  2c2 11695  ℤcz 11984  ↑cexp 13432   ∥ cdvds 15609   gcd cgcd 15845

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-fz 12896  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-dvds 15610  df-gcd 15846  df-prm 16018

This theorem is referenced by:  pythagtrip  16173