Theorem pythagtriplem19 15538

Description: Lemma for pythagtrip 15539. 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		nnz 11399	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
2		nnz 11399	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
3	1, 2	anim12i 590	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
4		nnne0 11053	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
5	4	neneqd 2799	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ¬ 𝐴 = 0)
6	5	intnanrd 963	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
7	6	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
8		gcdn0cl 15224	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
9	3, 7, 8	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ)
10	9	3adant3 1081	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ)
11	10	3ad2ant1 1082	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐴 gcd 𝐵) ∈ ℕ)
12		gcddvds 15225	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
13	1, 2, 12	syl2an 494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
14	13	3adant3 1081	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
15	14	simpld 475	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∥ 𝐴)
16	10	nnzd 11481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℤ)
17	10	nnne0d 11065	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ≠ 0)
18	1	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
19		dvdsval2 14986	. . . . . . . . 9 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ (𝐴 gcd 𝐵) ≠ 0 ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ))
20	16, 17, 18, 19	syl3anc 1326	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ))
21	15, 20	mpbid 222	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ)
22		nnre 11027	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
23	22	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
24	10	nnred 11035	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℝ)
25		nngt0 11049	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 0 < 𝐴)
26	25	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐴)
27	10	nngt0d 11064	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐴 gcd 𝐵))
28	23, 24, 26, 27	divgt0d 10959	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐴 / (𝐴 gcd 𝐵)))
29		elnnz 11387	. . . . . . 7 ⊢ ((𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐴 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐴 / (𝐴 gcd 𝐵))))
30	21, 28, 29	sylanbrc 698	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ)
31	30	3ad2ant1 1082	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ)
32	14	simprd 479	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∥ 𝐵)
33	2	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℤ)
34		dvdsval2 14986	. . . . . . . . 9 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ (𝐴 gcd 𝐵) ≠ 0 ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ (𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ))
35	16, 17, 33, 34	syl3anc 1326	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ (𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ))
36	32, 35	mpbid 222	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ)
37		nnre 11027	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
38	37	3ad2ant2 1083	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
39		nngt0 11049	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
40	39	3ad2ant2 1083	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
41	38, 24, 40, 27	divgt0d 10959	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐵 / (𝐴 gcd 𝐵)))
42		elnnz 11387	. . . . . . 7 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐵 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐵 / (𝐴 gcd 𝐵))))
43	36, 41, 42	sylanbrc 698	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ)
44	43	3ad2ant1 1082	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ)
45		dvdssq 15280	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
46	16, 18, 45	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
47		dvdssq 15280	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
48	16, 33, 47	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐵 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
49	46, 48	anbi12d 747	. . . . . . . . . . . . 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))))
50	14, 49	mpbid 222	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2) ∧ ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
51	10	nnsqcld 13029	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℕ)
52	51	nnzd 11481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
53		nnsqcl 12933	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℕ)
54	53	3ad2ant1 1082	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℕ)
55	54	nnzd 11481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℤ)
56		nnsqcl 12933	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → (𝐵↑2) ∈ ℕ)
57	56	3ad2ant2 1083	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℕ)
58	57	nnzd 11481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℤ)
59		dvds2add 15015	. . . . . . . . . . . . 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))))
60	52, 55, 58, 59	syl3anc 1326	. . . . . . . . . . . 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))))
61	50, 60	mpd 15	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
62	61	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
63		simpr 477	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))
64	62, 63	breqtrd 4679	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2))
65		nnz 11399	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
66	65	3ad2ant3 1084	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℤ)
67		dvdssq 15280	. . . . . . . . . . 11 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2)))
68	16, 66, 67	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2)))
69	68	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑2) ∥ (𝐶↑2)))
70	64, 69	mpbird 247	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐴 gcd 𝐵) ∥ 𝐶)
71		dvdsval2 14986	. . . . . . . . . 10 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ (𝐴 gcd 𝐵) ≠ 0 ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ))
72	16, 17, 66, 71	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ))
73	72	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ))
74	70, 73	mpbid 222	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ)
75		nnre 11027	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
76	75	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
77		nngt0 11049	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 0 < 𝐶)
78	77	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
79	76, 24, 78, 27	divgt0d 10959	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 / (𝐴 gcd 𝐵)))
80	79	adantr 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶 / (𝐴 gcd 𝐵)))
81		elnnz 11387	. . . . . . 7 ⊢ ((𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ ↔ ((𝐶 / (𝐴 gcd 𝐵)) ∈ ℤ ∧ 0 < (𝐶 / (𝐴 gcd 𝐵))))
82	74, 80, 81	sylanbrc 698	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → (𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ)
83	82	3adant3 1081	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → (𝐶 / (𝐴 gcd 𝐵)) ∈ ℕ)
84		nncn 11028	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
85	84	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℂ)
86	10	nncnd 11036	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℂ)
87	85, 86, 17	sqdivd 13021	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵))↑2) = ((𝐴↑2) / ((𝐴 gcd 𝐵)↑2)))
88		nncn 11028	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
89	88	3ad2ant2 1083	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℂ)
90	89, 86, 17	sqdivd 13021	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐵 / (𝐴 gcd 𝐵))↑2) = ((𝐵↑2) / ((𝐴 gcd 𝐵)↑2)))
91	87, 90	oveq12d 6668	. . . . . . . 8 ⊢ ((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))))
92	91	3ad2ant1 1082	. . . . . . 7 ⊢ (((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))))
93	54	nncnd 11036	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℂ)
94	57	nncnd 11036	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℂ)
95	51	nncnd 11036	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ∈ ℂ)
96	51	nnne0d 11065	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) ≠ 0)
97	93, 94, 95, 96	divdird 10839	. . . . . . . 8 ⊢ ((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))))
98	97	3ad2ant1 1082	. . . . . . 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)) = (((𝐴↑2) / ((𝐴 gcd 𝐵)↑2)) + ((𝐵↑2) / ((𝐴 gcd 𝐵)↑2))))
99	92, 98	eqtr4d 2659	. . . . . 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) + (𝐵↑2)) / ((𝐴 gcd 𝐵)↑2)))
100		nncn 11028	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
101	100	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℂ)
102	101, 86, 17	sqdivd 13021	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐶 / (𝐴 gcd 𝐵))↑2) = ((𝐶↑2) / ((𝐴 gcd 𝐵)↑2)))
103	102	3ad2ant1 1082	. . . . . . 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)))
104		oveq1 6657	. . . . . . . 8 ⊢ (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) / ((𝐴 gcd 𝐵)↑2)) = ((𝐶↑2) / ((𝐴 gcd 𝐵)↑2)))
105	104	3ad2ant2 1083	. . . . . . 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)))
106	103, 105	eqtr4d 2659	. . . . . 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)))
107	99, 106	eqtr4d 2659	. . . . 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))
108		gcddiv 15268	. . . . . . . 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))))
109	18, 33, 10, 14, 108	syl31anc 1329	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) / (𝐴 gcd 𝐵)) = ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))))
110	86, 17	dividd 10799	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) / (𝐴 gcd 𝐵)) = 1)
111	109, 110	eqtr3d 2658	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵)) gcd (𝐵 / (𝐴 gcd 𝐵))) = 1)
112	111	3ad2ant1 1082	. . . . 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)
113		simp3 1063	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵))) → ¬ 2 ∥ (𝐴 / (𝐴 gcd 𝐵)))
114		pythagtriplem18 15537	. . . . 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))))
115	31, 44, 83, 107, 112, 113, 114	syl312anc 1347	. . . 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))))
116	85, 86, 17	divcan2d 10803	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) = 𝐴)
117	116	eqcomd 2628	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 = ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))))
118	89, 86, 17	divcan2d 10803	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) = 𝐵)
119	118	eqcomd 2628	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))))
120	101, 86, 17	divcan2d 10803	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))) = 𝐶)
121	120	eqcomd 2628	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 = ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))))
122	117, 119, 121	3jca 1242	. . . . . . . 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)))))
123	122	3ad2ant1 1082	. . . . . . 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)))))
124		oveq2 6658	. . . . . . . . . 10 ⊢ ((𝐴 / (𝐴 gcd 𝐵)) = ((𝑚↑2) − (𝑛↑2)) → ((𝐴 gcd 𝐵) · (𝐴 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))))
125	124	eqeq2d 2632	. . . . . . . . 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)))))
126	125	3ad2ant1 1082	. . . . . . . 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)))))
127		oveq2 6658	. . . . . . . . . 10 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) → ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))))
128	127	eqeq2d 2632	. . . . . . . . 9 ⊢ ((𝐵 / (𝐴 gcd 𝐵)) = (2 · (𝑚 · 𝑛)) → (𝐵 = ((𝐴 gcd 𝐵) · (𝐵 / (𝐴 gcd 𝐵))) ↔ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛)))))
129	128	3ad2ant2 1083	. . . . . . . 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)))))
130		oveq2 6658	. . . . . . . . . 10 ⊢ ((𝐶 / (𝐴 gcd 𝐵)) = ((𝑚↑2) + (𝑛↑2)) → ((𝐴 gcd 𝐵) · (𝐶 / (𝐴 gcd 𝐵))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))
131	130	eqeq2d 2632	. . . . . . . . 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)))))
132	131	3ad2ant3 1084	. . . . . . . 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)))))
133	126, 129, 132	3anbi123d 1399	. . . . . . 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))))))
134	123, 133	syl5ibcom 235	. . . . . 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))))))
135	134	reximdv 3016	. . . . 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))))))
136	135	reximdv 3016	. . . 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))))))
137	115, 136	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)))))
138		oveq1 6657	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · ((𝑚↑2) − (𝑛↑2))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2))))
139	138	eqeq2d 2632	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐴 = (𝑘 · ((𝑚↑2) − (𝑛↑2))) ↔ 𝐴 = ((𝐴 gcd 𝐵) · ((𝑚↑2) − (𝑛↑2)))))
140		oveq1 6657	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · (2 · (𝑚 · 𝑛))) = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛))))
141	140	eqeq2d 2632	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐵 = (𝑘 · (2 · (𝑚 · 𝑛))) ↔ 𝐵 = ((𝐴 gcd 𝐵) · (2 · (𝑚 · 𝑛)))))
142		oveq1 6657	. . . . . . 7 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝑘 · ((𝑚↑2) + (𝑛↑2))) = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2))))
143	142	eqeq2d 2632	. . . . . 6 ⊢ (𝑘 = (𝐴 gcd 𝐵) → (𝐶 = (𝑘 · ((𝑚↑2) + (𝑛↑2))) ↔ 𝐶 = ((𝐴 gcd 𝐵) · ((𝑚↑2) + (𝑛↑2)))))
144	139, 141, 143	3anbi123d 1399	. . . . 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))))))
145	144	2rexbidv 3057	. . . 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))))))
146	145	rspcev 3309	. . 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)))))
147	11, 137, 146	syl2anc 693	. 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)))))
148		rexcom 3099	. . 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)))))
149		rexcom 3099	. . . 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)))))
150	149	rexbii 3041	. . 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)))))
151	148, 150	bitri 264	. 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)))))
152	147, 151	sylib 208	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   class class class wbr 4653  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   < clt 10074   − cmin 10266   / cdiv 10684  ℕcn 11020  2c2 11070  ℤcz 11377  ↑cexp 12860   ∥ cdvds 14983   gcd cgcd 15216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386

This theorem is referenced by:  pythagtrip  15539