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Theorem dvdssq 12385

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 9385	. . . 4 ⊢ 0 ∈ ℤ
2		zdceq 9450	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0)
3	1, 2	mpan2 425	. . 3 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0)
4		exmiddc 838	. . 3 ⊢ (DECID 𝑀 = 0 → (𝑀 = 0 ∨ ¬ 𝑀 = 0))
5		0dvds 12155	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
6		zcn 9379	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
7		sqeq0 10749	. . . . . . . . 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 10757	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁↑2) ∈ ℤ)
11		0dvds 12155	. . . . . . . 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 4048	. . . . . 6 ⊢ (𝑀 = 0 → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
16		sq0i 10778	. . . . . . 7 ⊢ (𝑀 = 0 → (𝑀↑2) = 0)
17	16	breq1d 4055	. . . . . 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 2377	. . . . 5 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
21		nnabscl 11444	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
22		zdceq 9450	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
23	1, 22	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 0)
24		exmiddc 838	. . . . . . . . . . 11 ⊢ (DECID 𝑁 = 0 → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
25		nnz 9393	. . . . . . . . . . . . . . 15 ⊢ ((abs‘𝑀) ∈ ℕ → (abs‘𝑀) ∈ ℤ)
26		dvds0 12150	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑀) ∈ ℤ → (abs‘𝑀) ∥ 0)
27		zsqcl 10757	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘𝑀) ∈ ℤ → ((abs‘𝑀)↑2) ∈ ℤ)
28		dvds0 12150	. . . . . . . . . . . . . . . . 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 4049	. . . . . . . . . . . . . 14 ⊢ (𝑁 = 0 → ((abs‘𝑀) ∥ 𝑁 ↔ (abs‘𝑀) ∥ 0))
34		sq0i 10778	. . . . . . . . . . . . . . 15 ⊢ (𝑁 = 0 → (𝑁↑2) = 0)
35	34	breq2d 4057	. . . . . . . . . . . . . 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 2377	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
39		nnabscl 11444	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
40		dvdssqlem 12384	. . . . . . . . . . . . . . . . 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 12154	. . . . . . . . . . . . . . . . 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 10756	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs‘𝑀) ∈ ℕ → ((abs‘𝑀)↑2) ∈ ℕ)
46	45	nnzd 9496	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘𝑀) ∈ ℕ → ((abs‘𝑀)↑2) ∈ ℤ)
47	10	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑁↑2) ∈ ℤ)
48		dvdsabsb 12154	. . . . . . . . . . . . . . . . . 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 11425	. . . . . . . . . . . . . . . . . . . 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 4057	. . . . . . . . . . . . . . . . . 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 718	. . . . . . . . . . 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 12153	. . . . . . . . 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 10757	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
67	66	adantr 276	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀↑2) ∈ ℤ)
68		absdvdsb 12153	. . . . . . . . . 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 9379	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
71		abssq 11425	. . . . . . . . . . . . . 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 2211	. . . . . . . . . . . 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 4055	. . . . . . . . . 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 718	. . 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 710 DECID wdc 836 = wceq 1373 ∈ wcel 2176 ≠ wne 2376 class class class wbr 4045 ‘cfv 5272 (class class class)co 5946 ℂcc 7925 0cc0 7927 ℕcn 9038 2c2 9089 ℤcz 9374 ↑cexp 10685 abscabs 11341 ∥ cdvds 12131

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4160 ax-sep 4163 ax-nul 4171 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-iinf 4637 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045 ax-arch 8046 ax-caucvg 8047

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4046 df-opab 4107 df-mpt 4108 df-tr 4144 df-id 4341 df-po 4344 df-iso 4345 df-iord 4414 df-on 4416 df-ilim 4417 df-suc 4419 df-iom 4640 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-1st 6228 df-2nd 6229 df-recs 6393 df-frec 6479 df-sup 7088 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-inn 9039 df-2 9097 df-3 9098 df-4 9099 df-n0 9298 df-z 9375 df-uz 9651 df-q 9743 df-rp 9778 df-fz 10133 df-fzo 10267 df-fl 10415 df-mod 10470 df-seqfrec 10595 df-exp 10686 df-cj 11186 df-re 11187 df-im 11188 df-rsqrt 11342 df-abs 11343 df-dvds 12132 df-gcd 12308

This theorem is referenced by: pythagtriplem19 12638 4sqlem9 12742 4sqlem10 12743 lgsdir 15545 2sqlem8a 15632

W3C validator