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Theorem gcdaddmlem 15960

Description: Lemma for gcdaddm 15961. (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 15939	. . . . . . 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 487	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ 𝑀
6		gcdcl 15942	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
7	1, 2, 6	mp2an 692	. . . . . . . . 9 ⊢ (𝑀 gcd 𝑁) ∈ ℕ₀
8	7	nn0zi 12081	. . . . . . . 8 ⊢ (𝑀 gcd 𝑁) ∈ ℤ
9		gcdaddmlem.1	. . . . . . . . 9 ⊢ 𝐾 ∈ ℤ
10		1z 12086	. . . . . . . . 9 ⊢ 1 ∈ ℤ
11		dvds2ln 15727	. . . . . . . . 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 1462	. . . . . . 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 12060	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
16	2, 15	ax-mp 5	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
17	16	mulid2i 10717	. . . . . . 7 ⊢ (1 · 𝑁) = 𝑁
18	17	oveq2i 7175	. . . . . 6 ⊢ ((𝐾 · 𝑀) + (1 · 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
19	14, 18	breqtri 5052	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + 𝑁)
20		zmulcl 12105	. . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ)
21	9, 1, 20	mp2an 692	. . . . . . 7 ⊢ (𝐾 · 𝑀) ∈ ℤ
22		zaddcl 12096	. . . . . . 7 ⊢ (((𝐾 · 𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) + 𝑁) ∈ ℤ)
23	21, 2, 22	mp2an 692	. . . . . 6 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ
24		dvdslegcd 15940	. . . . . . 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 416	. . . . . 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 1462	. . . . 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 15939	. . . . . . 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 487	. . . . 5 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑀
31		gcdcl 15942	. . . . . . . . . 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 12081	. . . . . . . 8 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ
34		znegcl 12091	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → -𝐾 ∈ ℤ)
35	9, 34	ax-mp 5	. . . . . . . . 9 ⊢ -𝐾 ∈ ℤ
36		dvds2ln 15727	. . . . . . . . 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 1462	. . . . . . 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 12060	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
41	9, 40	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ ℂ
42		zcn 12060	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
43	1, 42	ax-mp 5	. . . . . . . . 9 ⊢ 𝑀 ∈ ℂ
44	41, 43	mulneg1i 11157	. . . . . . . 8 ⊢ (-𝐾 · 𝑀) = -(𝐾 · 𝑀)
45		zcn 12060	. . . . . . . . . 10 ⊢ (((𝐾 · 𝑀) + 𝑁) ∈ ℤ → ((𝐾 · 𝑀) + 𝑁) ∈ ℂ)
46	23, 45	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℂ
47	46	mulid2i 10717	. . . . . . . 8 ⊢ (1 · ((𝐾 · 𝑀) + 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
48	44, 47	oveq12i 7176	. . . . . . 7 ⊢ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
49	41, 43	mulcli 10719	. . . . . . . . . 10 ⊢ (𝐾 · 𝑀) ∈ ℂ
50	49	negcli 11025	. . . . . . . . . 10 ⊢ -(𝐾 · 𝑀) ∈ ℂ
51	49	negidi 11026	. . . . . . . . . 10 ⊢ ((𝐾 · 𝑀) + -(𝐾 · 𝑀)) = 0
52	49, 50, 51	addcomli 10903	. . . . . . . . 9 ⊢ (-(𝐾 · 𝑀) + (𝐾 · 𝑀)) = 0
53	52	oveq1i 7174	. . . . . . . 8 ⊢ ((-(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (0 + 𝑁)
54	50, 49, 16	addassi 10722	. . . . . . . 8 ⊢ ((-(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
55	16	addid2i 10899	. . . . . . . 8 ⊢ (0 + 𝑁) = 𝑁
56	53, 54, 55	3eqtr3i 2769	. . . . . . 7 ⊢ (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁)) = 𝑁
57	48, 56	eqtri 2761	. . . . . 6 ⊢ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = 𝑁
58	39, 57	breqtri 5052	. . . . 5 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑁
59		dvdslegcd 15940	. . . . . . 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 416	. . . . . 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 1462	. . . . 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 616	. . 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 12061	. . . 4 ⊢ (𝑀 gcd 𝑁) ∈ ℝ
65	33	zrei 12061	. . . 4 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℝ
66	64, 65	letri3i 10827	. . 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		pm4.57 990	. . 3 ⊢ (¬ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ↔ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∨ (𝑀 = 0 ∧ 𝑁 = 0)))
69		oveq2 7172	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = (𝐾 · 0))
70	41	mul01i 10901	. . . . . . . . . 10 ⊢ (𝐾 · 0) = 0
71	69, 70	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = 0)
72	71	oveq1d 7179	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = (0 + 𝑁))
73	72, 55	eqtrdi 2789	. . . . . . 7 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = 𝑁)
74	73	eqeq1d 2740	. . . . . 6 ⊢ (𝑀 = 0 → (((𝐾 · 𝑀) + 𝑁) = 0 ↔ 𝑁 = 0))
75	74	pm5.32i 578	. . . . 5 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ↔ (𝑀 = 0 ∧ 𝑁 = 0))
76		oveq12 7173	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
77		oveq12 7173	. . . . . . 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 2776	. . . . 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 856	. . 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 185	1 ⊢ (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 class class class wbr 5027 (class class class)co 7164 ℂcc 10606 0cc0 10608 1c1 10609 + caddc 10611 · cmul 10613 ≤ cle 10747 -cneg 10942 ℕ₀cn0 11969 ℤcz 12055 ∥ cdvds 15692 gcd cgcd 15930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-sup 8972 df-inf 8973 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-rp 12466 df-seq 13454 df-exp 13515 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-dvds 15693 df-gcd 15931

This theorem is referenced by: gcdaddm 15961

W3C validator