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Theorem gcdaddmlem 16728

Description: Lemma for gcdaddm 16729. (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 16705	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
4	1, 2, 3	mp2an 705	. . . . . 6 ⊢ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)
5	4	simpli 489	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ 𝑀
6		gcdcl 16708	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
7	1, 2, 6	mp2an 705	. . . . . . . . 9 ⊢ (𝑀 gcd 𝑁) ∈ ℕ₀
8	7	nn0zi 12751	. . . . . . . 8 ⊢ (𝑀 gcd 𝑁) ∈ ℤ
9		gcdaddmlem.1	. . . . . . . . 9 ⊢ 𝐾 ∈ ℤ
10		1z 12756	. . . . . . . . 9 ⊢ 1 ∈ ℤ
11		dvds2ln 16491	. . . . . . . . 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 715	. . . . . . . 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 1491	. . . . . . 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 12728	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
16	2, 15	ax-mp 5	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
17	16	mullidi 11344	. . . . . . 7 ⊢ (1 · 𝑁) = 𝑁
18	17	oveq2i 7457	. . . . . 6 ⊢ ((𝐾 · 𝑀) + (1 · 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
19	14, 18	breqtri 5134	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + 𝑁)
20		zmulcl 12775	. . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ)
21	9, 1, 20	mp2an 705	. . . . . . 7 ⊢ (𝐾 · 𝑀) ∈ ℤ
22		zaddcl 12766	. . . . . . 7 ⊢ (((𝐾 · 𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) + 𝑁) ∈ ℤ)
23	21, 2, 22	mp2an 705	. . . . . 6 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ
24		dvdslegcd 16706	. . . . . . 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 418	. . . . . 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 1491	. . . . 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 711	. . . 4 ⊢ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
28		gcddvds 16705	. . . . . . 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 705	. . . . . 6 ⊢ ((𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀 ∧ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ ((𝐾 · 𝑀) + 𝑁))
30	29	simpli 489	. . . . 5 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀
31		gcdcl 16708	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℕ₀)
32	1, 23, 31	mp2an 705	. . . . . . . . 9 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℕ₀
33	32	nn0zi 12751	. . . . . . . 8 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ
34		znegcl 12761	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → -𝐾 ∈ ℤ)
35	9, 34	ax-mp 5	. . . . . . . . 9 ⊢ -𝐾 ∈ ℤ
36		dvds2ln 16491	. . . . . . . . 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 715	. . . . . . . 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 1491	. . . . . . 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 12728	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
41	9, 40	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ ℂ
42		zcn 12728	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
43	1, 42	ax-mp 5	. . . . . . . . 9 ⊢ 𝑀 ∈ ℂ
44	41, 43	mulneg1i 11792	. . . . . . . 8 ⊢ ( -𝐾 · 𝑀) = -(𝐾 · 𝑀)
45		zcn 12728	. . . . . . . . . 10 ⊢ (((𝐾 · 𝑀) + 𝑁) ∈ ℤ → ((𝐾 · 𝑀) + 𝑁) ∈ ℂ)
46	23, 45	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℂ
47	46	mullidi 11344	. . . . . . . 8 ⊢ (1 · ((𝐾 · 𝑀) + 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
48	44, 47	oveq12i 7458	. . . . . . 7 ⊢ (( -𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = ( -(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
49	41, 43	mulcli 11346	. . . . . . . . . 10 ⊢ (𝐾 · 𝑀) ∈ ℂ
50	49	negcli 11656	. . . . . . . . . 10 ⊢ -(𝐾 · 𝑀) ∈ ℂ
51	49	negidi 11657	. . . . . . . . . 10 ⊢ ((𝐾 · 𝑀) + -(𝐾 · 𝑀)) = 0
52	49, 50, 51	addcomli 11532	. . . . . . . . 9 ⊢ ( -(𝐾 · 𝑀) + (𝐾 · 𝑀)) = 0
53	52	oveq1i 7456	. . . . . . . 8 ⊢ (( -(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (0 + 𝑁)
54	50, 49, 16	addassi 11349	. . . . . . . 8 ⊢ (( -(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = ( -(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
55	16	addlidi 11528	. . . . . . . 8 ⊢ (0 + 𝑁) = 𝑁
56	53, 54, 55	3eqtr3i 2794	. . . . . . 7 ⊢ ( -(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁)) = 𝑁
57	48, 56	eqtri 2786	. . . . . 6 ⊢ (( -𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = 𝑁
58	39, 57	breqtri 5134	. . . . 5 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑁
59		dvdslegcd 16706	. . . . . . 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 418	. . . . . 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 1491	. . . . 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 711	. . . 4 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ≤ (𝑀 gcd 𝑁))
63	27, 62	anim12i 625	. . 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 12729	. . . 4 ⊢ (𝑀 gcd 𝑁) ∈ ℝ
65	33	zrei 12729	. . . 4 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℝ
66	64, 65	letri3i 11456	. . 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 237	. 2 ⊢ ((¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
68		pm4.57 1006	. . 3 ⊢ (¬ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ↔ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∨ (𝑀 = 0 ∧ 𝑁 = 0)))
69		oveq2 7454	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = (𝐾 · 0))
70	41	mul01i 11530	. . . . . . . . . 10 ⊢ (𝐾 · 0) = 0
71	69, 70	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = 0)
72	71	oveq1d 7461	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = (0 + 𝑁))
73	72, 55	eqtrdi 2814	. . . . . . 7 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = 𝑁)
74	73	eqeq1d 2765	. . . . . 6 ⊢ (𝑀 = 0 → (((𝐾 · 𝑀) + 𝑁) = 0 ↔ 𝑁 = 0))
75	74	pm5.32i 585	. . . . 5 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ↔ (𝑀 = 0 ∧ 𝑁 = 0))
76		oveq12 7455	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
77		oveq12 7455	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) = (0 gcd 0))
78	75, 77	sylbir 238	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) = (0 gcd 0))
79	76, 78	eqtr4d 2801	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
80	75, 79	sylbi 220	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
81	80, 79	jaoi 871	. . 3 ⊢ (((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∨ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
82	68, 81	sylbi 220	. 2 ⊢ (¬ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
83	67, 82	pm2.61i 184	1 ⊢ (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 class class class wbr 5106 (class class class)co 7446 ℂcc 11228 0cc0 11230 1c1 11231 + caddc 11233 · cmul 11235 ≤ cle 11374 -cneg 11572 ℕ₀cn0 12636 ℤcz 12723 ∥ cdvds 16454 gcd cgcd 16696

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-er 8748 df-en 9005 df-dom 9006 df-sdom 9007 df-sup 9464 df-inf 9465 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-n0 12637 df-z 12724 df-uz 12996 df-rp 13151 df-seq 14175 df-exp 14235 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-dvds 16455 df-gcd 16697

This theorem is used by: gcdaddm 16729

W3C validator