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Theorem sqgcd 16499

Description: Square distributes over gcd. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
sqgcd	⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁)↑2) = ((𝑀↑2) gcd (𝑁↑2)))

**Proof of Theorem sqgcd**
Step	Hyp	Ref	Expression
1		gcdnncl 16445	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℕ)
2	1	nnsqcld 14204	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁)↑2) ∈ ℕ)
3	2	nncnd 12225	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁)↑2) ∈ ℂ)
4	3	mulridd 11228	. 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 gcd 𝑁)↑2) · 1) = ((𝑀 gcd 𝑁)↑2))
5		nnsqcl 14090	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑀↑2) ∈ ℕ)
6	5	nnzd 12582	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀↑2) ∈ ℤ)
7	6	adantr 480	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀↑2) ∈ ℤ)
8		nnsqcl 14090	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℕ)
9	8	nnzd 12582	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) ∈ ℤ)
10	9	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁↑2) ∈ ℤ)
11		nnz 12576	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
12		nnz 12576	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
13		gcddvds 16441	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
14	11, 12, 13	syl2an 595	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
15	14	simpld 494	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∥ 𝑀)
16	1	nnzd 12582	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℤ)
17	11	adantr 480	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℤ)
18		dvdssqim 16493	. . . . . . 7 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 → ((𝑀 gcd 𝑁)↑2) ∥ (𝑀↑2)))
19	16, 17, 18	syl2anc 583	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ 𝑀 → ((𝑀 gcd 𝑁)↑2) ∥ (𝑀↑2)))
20	15, 19	mpd 15	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁)↑2) ∥ (𝑀↑2))
21	14	simprd 495	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∥ 𝑁)
22	12	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
23		dvdssqim 16493	. . . . . . 7 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 → ((𝑀 gcd 𝑁)↑2) ∥ (𝑁↑2)))
24	16, 22, 23	syl2anc 583	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ 𝑁 → ((𝑀 gcd 𝑁)↑2) ∥ (𝑁↑2)))
25	21, 24	mpd 15	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁)↑2) ∥ (𝑁↑2))
26		gcddiv 16490	. . . . 5 ⊢ ((((𝑀↑2) ∈ ℤ ∧ (𝑁↑2) ∈ ℤ ∧ ((𝑀 gcd 𝑁)↑2) ∈ ℕ) ∧ (((𝑀 gcd 𝑁)↑2) ∥ (𝑀↑2) ∧ ((𝑀 gcd 𝑁)↑2) ∥ (𝑁↑2))) → (((𝑀↑2) gcd (𝑁↑2)) / ((𝑀 gcd 𝑁)↑2)) = (((𝑀↑2) / ((𝑀 gcd 𝑁)↑2)) gcd ((𝑁↑2) / ((𝑀 gcd 𝑁)↑2))))
27	7, 10, 2, 20, 25, 26	syl32anc 1375	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀↑2) gcd (𝑁↑2)) / ((𝑀 gcd 𝑁)↑2)) = (((𝑀↑2) / ((𝑀 gcd 𝑁)↑2)) gcd ((𝑁↑2) / ((𝑀 gcd 𝑁)↑2))))
28		nncn 12217	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
29	28	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℂ)
30	1	nncnd 12225	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℂ)
31	1	nnne0d 12259	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ≠ 0)
32	29, 30, 31	sqdivd 14121	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 / (𝑀 gcd 𝑁))↑2) = ((𝑀↑2) / ((𝑀 gcd 𝑁)↑2)))
33		nncn 12217	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
34	33	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
35	34, 30, 31	sqdivd 14121	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁 / (𝑀 gcd 𝑁))↑2) = ((𝑁↑2) / ((𝑀 gcd 𝑁)↑2)))
36	32, 35	oveq12d 7419	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 / (𝑀 gcd 𝑁))↑2) gcd ((𝑁 / (𝑀 gcd 𝑁))↑2)) = (((𝑀↑2) / ((𝑀 gcd 𝑁)↑2)) gcd ((𝑁↑2) / ((𝑀 gcd 𝑁)↑2))))
37		gcddiv 16490	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 gcd 𝑁) ∈ ℕ) ∧ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) → ((𝑀 gcd 𝑁) / (𝑀 gcd 𝑁)) = ((𝑀 / (𝑀 gcd 𝑁)) gcd (𝑁 / (𝑀 gcd 𝑁))))
38	17, 22, 1, 14, 37	syl31anc 1370	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) / (𝑀 gcd 𝑁)) = ((𝑀 / (𝑀 gcd 𝑁)) gcd (𝑁 / (𝑀 gcd 𝑁))))
39	30, 31	dividd 11985	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) / (𝑀 gcd 𝑁)) = 1)
40	38, 39	eqtr3d 2766	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 / (𝑀 gcd 𝑁)) gcd (𝑁 / (𝑀 gcd 𝑁))) = 1)
41		dvdsval2 16197	. . . . . . . . 9 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ (𝑀 gcd 𝑁) ≠ 0 ∧ 𝑀 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ (𝑀 / (𝑀 gcd 𝑁)) ∈ ℤ))
42	16, 31, 17, 41	syl3anc 1368	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ (𝑀 / (𝑀 gcd 𝑁)) ∈ ℤ))
43	15, 42	mpbid 231	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 / (𝑀 gcd 𝑁)) ∈ ℤ)
44		nnre 12216	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
45	44	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
46	1	nnred 12224	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 gcd 𝑁) ∈ ℝ)
47		nngt0 12240	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
48	47	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < 𝑀)
49	1	nngt0d 12258	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝑀 gcd 𝑁))
50	45, 46, 48, 49	divgt0d 12146	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝑀 / (𝑀 gcd 𝑁)))
51		elnnz 12565	. . . . . . 7 ⊢ ((𝑀 / (𝑀 gcd 𝑁)) ∈ ℕ ↔ ((𝑀 / (𝑀 gcd 𝑁)) ∈ ℤ ∧ 0 < (𝑀 / (𝑀 gcd 𝑁))))
52	43, 50, 51	sylanbrc 582	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 / (𝑀 gcd 𝑁)) ∈ ℕ)
53		dvdsval2 16197	. . . . . . . . 9 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ (𝑀 gcd 𝑁) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑁 / (𝑀 gcd 𝑁)) ∈ ℤ))
54	16, 31, 22, 53	syl3anc 1368	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑁 / (𝑀 gcd 𝑁)) ∈ ℤ))
55	21, 54	mpbid 231	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 / (𝑀 gcd 𝑁)) ∈ ℤ)
56		nnre 12216	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
57	56	adantl 481	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
58		nngt0 12240	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
59	58	adantl 481	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < 𝑁)
60	57, 46, 59, 49	divgt0d 12146	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝑁 / (𝑀 gcd 𝑁)))
61		elnnz 12565	. . . . . . 7 ⊢ ((𝑁 / (𝑀 gcd 𝑁)) ∈ ℕ ↔ ((𝑁 / (𝑀 gcd 𝑁)) ∈ ℤ ∧ 0 < (𝑁 / (𝑀 gcd 𝑁))))
62	55, 60, 61	sylanbrc 582	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 / (𝑀 gcd 𝑁)) ∈ ℕ)
63		2nn 12282	. . . . . . 7 ⊢ 2 ∈ ℕ
64		rppwr 16498	. . . . . . 7 ⊢ (((𝑀 / (𝑀 gcd 𝑁)) ∈ ℕ ∧ (𝑁 / (𝑀 gcd 𝑁)) ∈ ℕ ∧ 2 ∈ ℕ) → (((𝑀 / (𝑀 gcd 𝑁)) gcd (𝑁 / (𝑀 gcd 𝑁))) = 1 → (((𝑀 / (𝑀 gcd 𝑁))↑2) gcd ((𝑁 / (𝑀 gcd 𝑁))↑2)) = 1))
65	63, 64	mp3an3 1446	. . . . . 6 ⊢ (((𝑀 / (𝑀 gcd 𝑁)) ∈ ℕ ∧ (𝑁 / (𝑀 gcd 𝑁)) ∈ ℕ) → (((𝑀 / (𝑀 gcd 𝑁)) gcd (𝑁 / (𝑀 gcd 𝑁))) = 1 → (((𝑀 / (𝑀 gcd 𝑁))↑2) gcd ((𝑁 / (𝑀 gcd 𝑁))↑2)) = 1))
66	52, 62, 65	syl2anc 583	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 / (𝑀 gcd 𝑁)) gcd (𝑁 / (𝑀 gcd 𝑁))) = 1 → (((𝑀 / (𝑀 gcd 𝑁))↑2) gcd ((𝑁 / (𝑀 gcd 𝑁))↑2)) = 1))
67	40, 66	mpd 15	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 / (𝑀 gcd 𝑁))↑2) gcd ((𝑁 / (𝑀 gcd 𝑁))↑2)) = 1)
68	27, 36, 67	3eqtr2d 2770	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀↑2) gcd (𝑁↑2)) / ((𝑀 gcd 𝑁)↑2)) = 1)
69	6, 9	anim12i 612	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀↑2) ∈ ℤ ∧ (𝑁↑2) ∈ ℤ))
70	5	nnne0d 12259	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑀↑2) ≠ 0)
71	70	neneqd 2937	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → ¬ (𝑀↑2) = 0)
72	71	intnanrd 489	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → ¬ ((𝑀↑2) = 0 ∧ (𝑁↑2) = 0))
73	72	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ¬ ((𝑀↑2) = 0 ∧ (𝑁↑2) = 0))
74		gcdn0cl 16440	. . . . . 6 ⊢ ((((𝑀↑2) ∈ ℤ ∧ (𝑁↑2) ∈ ℤ) ∧ ¬ ((𝑀↑2) = 0 ∧ (𝑁↑2) = 0)) → ((𝑀↑2) gcd (𝑁↑2)) ∈ ℕ)
75	69, 73, 74	syl2anc 583	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀↑2) gcd (𝑁↑2)) ∈ ℕ)
76	75	nncnd 12225	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀↑2) gcd (𝑁↑2)) ∈ ℂ)
77	2	nnne0d 12259	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁)↑2) ≠ 0)
78		ax-1cn 11164	. . . . 5 ⊢ 1 ∈ ℂ
79		divmul 11872	. . . . 5 ⊢ ((((𝑀↑2) gcd (𝑁↑2)) ∈ ℂ ∧ 1 ∈ ℂ ∧ (((𝑀 gcd 𝑁)↑2) ∈ ℂ ∧ ((𝑀 gcd 𝑁)↑2) ≠ 0)) → ((((𝑀↑2) gcd (𝑁↑2)) / ((𝑀 gcd 𝑁)↑2)) = 1 ↔ (((𝑀 gcd 𝑁)↑2) · 1) = ((𝑀↑2) gcd (𝑁↑2))))
80	78, 79	mp3an2 1445	. . . 4 ⊢ ((((𝑀↑2) gcd (𝑁↑2)) ∈ ℂ ∧ (((𝑀 gcd 𝑁)↑2) ∈ ℂ ∧ ((𝑀 gcd 𝑁)↑2) ≠ 0)) → ((((𝑀↑2) gcd (𝑁↑2)) / ((𝑀 gcd 𝑁)↑2)) = 1 ↔ (((𝑀 gcd 𝑁)↑2) · 1) = ((𝑀↑2) gcd (𝑁↑2))))
81	76, 3, 77, 80	syl12anc 834	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((((𝑀↑2) gcd (𝑁↑2)) / ((𝑀 gcd 𝑁)↑2)) = 1 ↔ (((𝑀 gcd 𝑁)↑2) · 1) = ((𝑀↑2) gcd (𝑁↑2))))
82	68, 81	mpbid 231	. 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝑀 gcd 𝑁)↑2) · 1) = ((𝑀↑2) gcd (𝑁↑2)))
83	4, 82	eqtr3d 2766	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 gcd 𝑁)↑2) = ((𝑀↑2) gcd (𝑁↑2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 class class class wbr 5138 (class class class)co 7401 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11245 / cdiv 11868 ℕcn 12209 2c2 12264 ℤcz 12555 ↑cexp 14024 ∥ cdvds 16194 gcd cgcd 16432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-dvds 16195 df-gcd 16433

This theorem is referenced by: dvdssqlem 16500 nn0gcdsq 16687 pythagtriplem3 16750

W3C validator