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Theorem gcdaddmlem 16558

Description: Lemma for gcdaddm 16559. (Contributed by Paul Chapman, 31-Mar-2011.)

**Hypotheses**
Ref	Expression
gcdaddmlem.1	⊢ 𝐾 ∈ ℤ
gcdaddmlem.2	⊢ 𝑀 ∈ ℤ
gcdaddmlem.3	⊢ 𝑁 ∈ ℤ

**Assertion**
Ref	Expression
gcdaddmlem	⊢ (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁))

**Proof of Theorem gcdaddmlem**
Step	Hyp	Ref	Expression
1		gcdaddmlem.2	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
2		gcdaddmlem.3	. . . . . . 7 ⊢ 𝑁 ∈ ℤ
3		gcddvds 16537	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
4	1, 2, 3	mp2an 692	. . . . . 6 ⊢ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)
5	4	simpli 483	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ 𝑀
6		gcdcl 16540	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
7	1, 2, 6	mp2an 692	. . . . . . . . 9 ⊢ (𝑀 gcd 𝑁) ∈ ℕ₀
8	7	nn0zi 12640	. . . . . . . 8 ⊢ (𝑀 gcd 𝑁) ∈ ℤ
9		gcdaddmlem.1	. . . . . . . . 9 ⊢ 𝐾 ∈ ℤ
10		1z 12645	. . . . . . . . 9 ⊢ 1 ∈ ℤ
11		dvds2ln 16323	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℤ ∧ 1 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) → (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + (1 · 𝑁))))
12	9, 10, 11	mpanl12 702	. . . . . . . 8 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) → (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + (1 · 𝑁))))
13	8, 1, 2, 12	mp3an 1460	. . . . . . 7 ⊢ (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) → (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + (1 · 𝑁)))
14	4, 13	ax-mp 5	. . . . . 6 ⊢ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + (1 · 𝑁))
15		zcn 12616	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
16	2, 15	ax-mp 5	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
17	16	mullidi 11264	. . . . . . 7 ⊢ (1 · 𝑁) = 𝑁
18	17	oveq2i 7442	. . . . . 6 ⊢ ((𝐾 · 𝑀) + (1 · 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
19	14, 18	breqtri 5173	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + 𝑁)
20		zmulcl 12664	. . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ)
21	9, 1, 20	mp2an 692	. . . . . . 7 ⊢ (𝐾 · 𝑀) ∈ ℤ
22		zaddcl 12655	. . . . . . 7 ⊢ (((𝐾 · 𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) + 𝑁) ∈ ℤ)
23	21, 2, 22	mp2an 692	. . . . . 6 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ
24		dvdslegcd 16538	. . . . . . 7 ⊢ ((((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0)) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + 𝑁)) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁))))
25	24	ex 412	. . . . . 6 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ) → (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + 𝑁)) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))))
26	8, 1, 23, 25	mp3an 1460	. . . . 5 ⊢ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + 𝑁)) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁))))
27	5, 19, 26	mp2ani 698	. . . 4 ⊢ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
28		gcddvds 16537	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ) → ((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((𝐾 · 𝑀) + 𝑁)))
29	1, 23, 28	mp2an 692	. . . . . 6 ⊢ ((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((𝐾 · 𝑀) + 𝑁))
30	29	simpli 483	. . . . 5 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀
31		gcdcl 16540	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℕ₀)
32	1, 23, 31	mp2an 692	. . . . . . . . 9 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℕ₀
33	32	nn0zi 12640	. . . . . . . 8 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ
34		znegcl 12650	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → -𝐾 ∈ ℤ)
35	9, 34	ax-mp 5	. . . . . . . . 9 ⊢ -𝐾 ∈ ℤ
36		dvds2ln 16323	. . . . . . . . 9 ⊢ (((-𝐾 ∈ ℤ ∧ 1 ∈ ℤ) ∧ ((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ)) → (((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((𝐾 · 𝑀) + 𝑁)) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁)))))
37	35, 10, 36	mpanl12 702	. . . . . . . 8 ⊢ (((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ) → (((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((𝐾 · 𝑀) + 𝑁)) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁)))))
38	33, 1, 23, 37	mp3an 1460	. . . . . . 7 ⊢ (((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((𝐾 · 𝑀) + 𝑁)) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))))
39	29, 38	ax-mp 5	. . . . . 6 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁)))
40		zcn 12616	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
41	9, 40	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ ℂ
42		zcn 12616	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
43	1, 42	ax-mp 5	. . . . . . . . 9 ⊢ 𝑀 ∈ ℂ
44	41, 43	mulneg1i 11707	. . . . . . . 8 ⊢ (-𝐾 · 𝑀) = -(𝐾 · 𝑀)
45		zcn 12616	. . . . . . . . . 10 ⊢ (((𝐾 · 𝑀) + 𝑁) ∈ ℤ → ((𝐾 · 𝑀) + 𝑁) ∈ ℂ)
46	23, 45	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℂ
47	46	mullidi 11264	. . . . . . . 8 ⊢ (1 · ((𝐾 · 𝑀) + 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
48	44, 47	oveq12i 7443	. . . . . . 7 ⊢ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
49	41, 43	mulcli 11266	. . . . . . . . . 10 ⊢ (𝐾 · 𝑀) ∈ ℂ
50	49	negcli 11575	. . . . . . . . . 10 ⊢ -(𝐾 · 𝑀) ∈ ℂ
51	49	negidi 11576	. . . . . . . . . 10 ⊢ ((𝐾 · 𝑀) + -(𝐾 · 𝑀)) = 0
52	49, 50, 51	addcomli 11451	. . . . . . . . 9 ⊢ (-(𝐾 · 𝑀) + (𝐾 · 𝑀)) = 0
53	52	oveq1i 7441	. . . . . . . 8 ⊢ ((-(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (0 + 𝑁)
54	50, 49, 16	addassi 11269	. . . . . . . 8 ⊢ ((-(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
55	16	addlidi 11447	. . . . . . . 8 ⊢ (0 + 𝑁) = 𝑁
56	53, 54, 55	3eqtr3i 2771	. . . . . . 7 ⊢ (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁)) = 𝑁
57	48, 56	eqtri 2763	. . . . . 6 ⊢ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = 𝑁
58	39, 57	breqtri 5173	. . . . 5 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑁
59		dvdslegcd 16538	. . . . . . 7 ⊢ ((((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑁) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ≤ (𝑀 gcd 𝑁)))
60	59	ex 412	. . . . . 6 ⊢ (((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑁) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ≤ (𝑀 gcd 𝑁))))
61	33, 1, 2, 60	mp3an 1460	. . . . 5 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑁) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ≤ (𝑀 gcd 𝑁)))
62	30, 58, 61	mp2ani 698	. . . 4 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ≤ (𝑀 gcd 𝑁))
63	27, 62	anim12i 613	. . 3 ⊢ ((¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ≤ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ≤ (𝑀 gcd 𝑁)))
64	8	zrei 12617	. . . 4 ⊢ (𝑀 gcd 𝑁) ∈ ℝ
65	33	zrei 12617	. . . 4 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℝ
66	64, 65	letri3i 11375	. . 3 ⊢ ((𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ↔ ((𝑀 gcd 𝑁) ≤ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ≤ (𝑀 gcd 𝑁)))
67	63, 66	sylibr 234	. 2 ⊢ ((¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
68		pm4.57 992	. . 3 ⊢ (¬ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ↔ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∨ (𝑀 = 0 ∧ 𝑁 = 0)))
69		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = (𝐾 · 0))
70	41	mul01i 11449	. . . . . . . . . 10 ⊢ (𝐾 · 0) = 0
71	69, 70	eqtrdi 2791	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = 0)
72	71	oveq1d 7446	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = (0 + 𝑁))
73	72, 55	eqtrdi 2791	. . . . . . 7 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = 𝑁)
74	73	eqeq1d 2737	. . . . . 6 ⊢ (𝑀 = 0 → (((𝐾 · 𝑀) + 𝑁) = 0 ↔ 𝑁 = 0))
75	74	pm5.32i 574	. . . . 5 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ↔ (𝑀 = 0 ∧ 𝑁 = 0))
76		oveq12 7440	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
77		oveq12 7440	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) = (0 gcd 0))
78	75, 77	sylbir 235	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) = (0 gcd 0))
79	76, 78	eqtr4d 2778	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
80	75, 79	sylbi 217	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
81	80, 79	jaoi 857	. . 3 ⊢ (((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∨ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
82	68, 81	sylbi 217	. 2 ⊢ (¬ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
83	67, 82	pm2.61i 182	1 ⊢ (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ≤ cle 11294 -cneg 11491 ℕ₀cn0 12524 ℤcz 12611 ∥ cdvds 16287 gcd cgcd 16528

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529

This theorem is referenced by: gcdaddm 16559

W3C validator