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Theorem dvdssq 12168

Description: Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
dvdssq	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2)))

**Proof of Theorem dvdssq**
Step	Hyp	Ref	Expression
1		0z 9328	. . . 4 ⊢ 0 ∈ ℤ
2		zdceq 9392	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0)
3	1, 2	mpan2 425	. . 3 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0)
4		exmiddc 837	. . 3 ⊢ (DECID 𝑀 = 0 → (𝑀 = 0 ∨ ¬ 𝑀 = 0))
5		0dvds 11954	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
6		zcn 9322	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
7		sqeq0 10673	. . . . . . . . 9 ⊢ (𝑁 ∈ ℂ → ((𝑁↑2) = 0 ↔ 𝑁 = 0))
8	6, 7	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → ((𝑁↑2) = 0 ↔ 𝑁 = 0))
9	5, 8	bitr4d 191	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ (𝑁↑2) = 0))
10		zsqcl 10681	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁↑2) ∈ ℤ)
11		0dvds 11954	. . . . . . . 8 ⊢ ((𝑁↑2) ∈ ℤ → (0 ∥ (𝑁↑2) ↔ (𝑁↑2) = 0))
12	10, 11	syl 14	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0 ∥ (𝑁↑2) ↔ (𝑁↑2) = 0))
13	9, 12	bitr4d 191	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 0 ∥ (𝑁↑2)))
14	13	adantl 277	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 0 ∥ (𝑁↑2)))
15		breq1 4032	. . . . . 6 ⊢ (𝑀 = 0 → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
16		sq0i 10702	. . . . . . 7 ⊢ (𝑀 = 0 → (𝑀↑2) = 0)
17	16	breq1d 4039	. . . . . 6 ⊢ (𝑀 = 0 → ((𝑀↑2) ∥ (𝑁↑2) ↔ 0 ∥ (𝑁↑2)))
18	15, 17	bibi12d 235	. . . . 5 ⊢ (𝑀 = 0 → ((𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2)) ↔ (0 ∥ 𝑁 ↔ 0 ∥ (𝑁↑2))))
19	14, 18	imbitrrid 156	. . . 4 ⊢ (𝑀 = 0 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2))))
20		df-ne 2365	. . . . 5 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
21		nnabscl 11244	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
22		zdceq 9392	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
23	1, 22	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 0)
24		exmiddc 837	. . . . . . . . . . 11 ⊢ (DECID 𝑁 = 0 → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
25		nnz 9336	. . . . . . . . . . . . . . 15 ⊢ ((abs‘𝑀) ∈ ℕ → (abs‘𝑀) ∈ ℤ)
26		dvds0 11949	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑀) ∈ ℤ → (abs‘𝑀) ∥ 0)
27		zsqcl 10681	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘𝑀) ∈ ℤ → ((abs‘𝑀)↑2) ∈ ℤ)
28		dvds0 11949	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘𝑀)↑2) ∈ ℤ → ((abs‘𝑀)↑2) ∥ 0)
29	27, 28	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑀) ∈ ℤ → ((abs‘𝑀)↑2) ∥ 0)
30	26, 29	2thd 175	. . . . . . . . . . . . . . 15 ⊢ ((abs‘𝑀) ∈ ℤ → ((abs‘𝑀) ∥ 0 ↔ ((abs‘𝑀)↑2) ∥ 0))
31	25, 30	syl 14	. . . . . . . . . . . . . 14 ⊢ ((abs‘𝑀) ∈ ℕ → ((abs‘𝑀) ∥ 0 ↔ ((abs‘𝑀)↑2) ∥ 0))
32	31	adantr 276	. . . . . . . . . . . . 13 ⊢ (((abs‘𝑀) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 0 ↔ ((abs‘𝑀)↑2) ∥ 0))
33		breq2 4033	. . . . . . . . . . . . . 14 ⊢ (𝑁 = 0 → ((abs‘𝑀) ∥ 𝑁 ↔ (abs‘𝑀) ∥ 0))
34		sq0i 10702	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = 0 → (𝑁↑2) = 0)
35	34	breq2d 4041	. . . . . . . . . . . . . 14 ⊢ (𝑁 = 0 → (((abs‘𝑀)↑2) ∥ (𝑁↑2) ↔ ((abs‘𝑀)↑2) ∥ 0))
36	33, 35	bibi12d 235	. . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2)) ↔ ((abs‘𝑀) ∥ 0 ↔ ((abs‘𝑀)↑2) ∥ 0)))
37	32, 36	imbitrrid 156	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (((abs‘𝑀) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2))))
38		df-ne 2365	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
39		nnabscl 11244	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
40		dvdssqlem 12167	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ) → ((abs‘𝑀) ∥ (abs‘𝑁) ↔ ((abs‘𝑀)↑2) ∥ ((abs‘𝑁)↑2)))
41	39, 40	sylan2 286	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∥ (abs‘𝑁) ↔ ((abs‘𝑀)↑2) ∥ ((abs‘𝑁)↑2)))
42		simpl 109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ)
43		dvdsabsb 11953	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
44	25, 42, 43	syl2an 289	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
45		nnsqcl 10680	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs‘𝑀) ∈ ℕ → ((abs‘𝑀)↑2) ∈ ℕ)
46	45	nnzd 9438	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘𝑀) ∈ ℕ → ((abs‘𝑀)↑2) ∈ ℤ)
47	10	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑁↑2) ∈ ℤ)
48		dvdsabsb 11953	. . . . . . . . . . . . . . . . . 18 ⊢ ((((abs‘𝑀)↑2) ∈ ℤ ∧ (𝑁↑2) ∈ ℤ) → (((abs‘𝑀)↑2) ∥ (𝑁↑2) ↔ ((abs‘𝑀)↑2) ∥ (abs‘(𝑁↑2))))
49	46, 47, 48	syl2an 289	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (((abs‘𝑀)↑2) ∥ (𝑁↑2) ↔ ((abs‘𝑀)↑2) ∥ (abs‘(𝑁↑2))))
50	6	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℂ)
51		abssq 11225	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℂ → ((abs‘𝑁)↑2) = (abs‘(𝑁↑2)))
52	50, 51	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((abs‘𝑁)↑2) = (abs‘(𝑁↑2)))
53	52	breq2d 4041	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (((abs‘𝑀)↑2) ∥ ((abs‘𝑁)↑2) ↔ ((abs‘𝑀)↑2) ∥ (abs‘(𝑁↑2))))
54	53	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (((abs‘𝑀)↑2) ∥ ((abs‘𝑁)↑2) ↔ ((abs‘𝑀)↑2) ∥ (abs‘(𝑁↑2))))
55	49, 54	bitr4d 191	. . . . . . . . . . . . . . . 16 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (((abs‘𝑀)↑2) ∥ (𝑁↑2) ↔ ((abs‘𝑀)↑2) ∥ ((abs‘𝑁)↑2)))
56	41, 44, 55	3bitr4d 220	. . . . . . . . . . . . . . 15 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2)))
57	56	anassrs 400	. . . . . . . . . . . . . 14 ⊢ ((((abs‘𝑀) ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2)))
58	57	expcom 116	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 → (((abs‘𝑀) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2))))
59	38, 58	sylbir 135	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 = 0 → (((abs‘𝑀) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2))))
60	37, 59	jaoi 717	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∨ ¬ 𝑁 = 0) → (((abs‘𝑀) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2))))
61	23, 24, 60	3syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (((abs‘𝑀) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2))))
62	61	anabsi7 581	. . . . . . . . 9 ⊢ (((abs‘𝑀) ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2)))
63	21, 62	sylan 283	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2)))
64		absdvdsb 11952	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁))
65	64	adantlr 477	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁))
66		zsqcl 10681	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
67	66	adantr 276	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀↑2) ∈ ℤ)
68		absdvdsb 11952	. . . . . . . . . 10 ⊢ (((𝑀↑2) ∈ ℤ ∧ (𝑁↑2) ∈ ℤ) → ((𝑀↑2) ∥ (𝑁↑2) ↔ (abs‘(𝑀↑2)) ∥ (𝑁↑2)))
69	67, 10, 68	syl2an 289	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ 𝑁 ∈ ℤ) → ((𝑀↑2) ∥ (𝑁↑2) ↔ (abs‘(𝑀↑2)) ∥ (𝑁↑2)))
70		zcn 9322	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
71		abssq 11225	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℂ → ((abs‘𝑀)↑2) = (abs‘(𝑀↑2)))
72	70, 71	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → ((abs‘𝑀)↑2) = (abs‘(𝑀↑2)))
73	72	eqcomd 2199	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (abs‘(𝑀↑2)) = ((abs‘𝑀)↑2))
74	73	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘(𝑀↑2)) = ((abs‘𝑀)↑2))
75	74	breq1d 4039	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ((abs‘(𝑀↑2)) ∥ (𝑁↑2) ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2)))
76	75	adantr 276	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ 𝑁 ∈ ℤ) → ((abs‘(𝑀↑2)) ∥ (𝑁↑2) ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2)))
77	69, 76	bitrd 188	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ 𝑁 ∈ ℤ) → ((𝑀↑2) ∥ (𝑁↑2) ↔ ((abs‘𝑀)↑2) ∥ (𝑁↑2)))
78	63, 65, 77	3bitr4d 220	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2)))
79	78	an32s 568	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2)))
80	79	expcom 116	. . . . 5 ⊢ (𝑀 ≠ 0 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2))))
81	20, 80	sylbir 135	. . . 4 ⊢ (¬ 𝑀 = 0 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2))))
82	19, 81	jaoi 717	. . 3 ⊢ ((𝑀 = 0 ∨ ¬ 𝑀 = 0) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2))))
83	3, 4, 82	3syl 17	. 2 ⊢ (𝑀 ∈ ℤ → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2))))
84	83	anabsi5 579	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 0cc0 7872 ℕcn 8982 2c2 9033 ℤcz 9317 ↑cexp 10609 abscabs 11141 ∥ cdvds 11930

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-sup 7043 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-dvds 11931 df-gcd 12080

This theorem is referenced by: pythagtriplem19 12420 4sqlem9 12524 4sqlem10 12525 lgsdir 15151 2sqlem8a 15209

W3C validator