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Theorem dvdssq 12198

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 9337	. . . 4 ⊢ 0 ∈ ℤ
2		zdceq 9401	. . . 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 11976	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
6		zcn 9331	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
7		sqeq0 10694	. . . . . . . . 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 10702	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁↑2) ∈ ℤ)
11		0dvds 11976	. . . . . . . 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 4036	. . . . . 6 ⊢ (𝑀 = 0 → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
16		sq0i 10723	. . . . . . 7 ⊢ (𝑀 = 0 → (𝑀↑2) = 0)
17	16	breq1d 4043	. . . . . 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 2368	. . . . 5 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
21		nnabscl 11265	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
22		zdceq 9401	. . . . . . . . . . . 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 9345	. . . . . . . . . . . . . . 15 ⊢ ((abs‘𝑀) ∈ ℕ → (abs‘𝑀) ∈ ℤ)
26		dvds0 11971	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑀) ∈ ℤ → (abs‘𝑀) ∥ 0)
27		zsqcl 10702	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘𝑀) ∈ ℤ → ((abs‘𝑀)↑2) ∈ ℤ)
28		dvds0 11971	. . . . . . . . . . . . . . . . 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 4037	. . . . . . . . . . . . . 14 ⊢ (𝑁 = 0 → ((abs‘𝑀) ∥ 𝑁 ↔ (abs‘𝑀) ∥ 0))
34		sq0i 10723	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = 0 → (𝑁↑2) = 0)
35	34	breq2d 4045	. . . . . . . . . . . . . 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 2368	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
39		nnabscl 11265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
40		dvdssqlem 12197	. . . . . . . . . . . . . . . . 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 11975	. . . . . . . . . . . . . . . . 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 10701	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs‘𝑀) ∈ ℕ → ((abs‘𝑀)↑2) ∈ ℕ)
46	45	nnzd 9447	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘𝑀) ∈ ℕ → ((abs‘𝑀)↑2) ∈ ℤ)
47	10	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑁↑2) ∈ ℤ)
48		dvdsabsb 11975	. . . . . . . . . . . . . . . . . 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 11246	. . . . . . . . . . . . . . . . . . . 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 4045	. . . . . . . . . . . . . . . . . 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 11974	. . . . . . . . 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 10702	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
67	66	adantr 276	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀↑2) ∈ ℤ)
68		absdvdsb 11974	. . . . . . . . . 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 9331	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
71		abssq 11246	. . . . . . . . . . . . . 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 2202	. . . . . . . . . . . 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 4043	. . . . . . . . . 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 2167 ≠ wne 2367 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 ℂcc 7877 0cc0 7879 ℕcn 8990 2c2 9041 ℤcz 9326 ↑cexp 10630 abscabs 11162 ∥ cdvds 11952

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-sup 7050 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-gcd 12121

This theorem is referenced by: pythagtriplem19 12451 4sqlem9 12555 4sqlem10 12556 lgsdir 15276 2sqlem8a 15363

W3C validator