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Theorem gcdaddmlem 16435

Description: Lemma for gcdaddm 16436. (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 16414	. . . . . . 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 16417	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
7	1, 2, 6	mp2an 692	. . . . . . . . 9 ⊢ (𝑀 gcd 𝑁) ∈ ℕ₀
8	7	nn0zi 12500	. . . . . . . 8 ⊢ (𝑀 gcd 𝑁) ∈ ℤ
9		gcdaddmlem.1	. . . . . . . . 9 ⊢ 𝐾 ∈ ℤ
10		1z 12505	. . . . . . . . 9 ⊢ 1 ∈ ℤ
11		dvds2ln 16200	. . . . . . . . 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 1463	. . . . . . 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 12476	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
16	2, 15	ax-mp 5	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
17	16	mullidi 11120	. . . . . . 7 ⊢ (1 · 𝑁) = 𝑁
18	17	oveq2i 7360	. . . . . 6 ⊢ ((𝐾 · 𝑀) + (1 · 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
19	14, 18	breqtri 5117	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + 𝑁)
20		zmulcl 12524	. . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ)
21	9, 1, 20	mp2an 692	. . . . . . 7 ⊢ (𝐾 · 𝑀) ∈ ℤ
22		zaddcl 12515	. . . . . . 7 ⊢ (((𝐾 · 𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) + 𝑁) ∈ ℤ)
23	21, 2, 22	mp2an 692	. . . . . 6 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ
24		dvdslegcd 16415	. . . . . . 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 1463	. . . . 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 16414	. . . . . . 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 16417	. . . . . . . . . 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 12500	. . . . . . . 8 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ
34		znegcl 12510	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → -𝐾 ∈ ℤ)
35	9, 34	ax-mp 5	. . . . . . . . 9 ⊢ -𝐾 ∈ ℤ
36		dvds2ln 16200	. . . . . . . . 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 1463	. . . . . . 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 12476	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
41	9, 40	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ ℂ
42		zcn 12476	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
43	1, 42	ax-mp 5	. . . . . . . . 9 ⊢ 𝑀 ∈ ℂ
44	41, 43	mulneg1i 11566	. . . . . . . 8 ⊢ (-𝐾 · 𝑀) = -(𝐾 · 𝑀)
45		zcn 12476	. . . . . . . . . 10 ⊢ (((𝐾 · 𝑀) + 𝑁) ∈ ℤ → ((𝐾 · 𝑀) + 𝑁) ∈ ℂ)
46	23, 45	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℂ
47	46	mullidi 11120	. . . . . . . 8 ⊢ (1 · ((𝐾 · 𝑀) + 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
48	44, 47	oveq12i 7361	. . . . . . 7 ⊢ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
49	41, 43	mulcli 11122	. . . . . . . . . 10 ⊢ (𝐾 · 𝑀) ∈ ℂ
50	49	negcli 11432	. . . . . . . . . 10 ⊢ -(𝐾 · 𝑀) ∈ ℂ
51	49	negidi 11433	. . . . . . . . . 10 ⊢ ((𝐾 · 𝑀) + -(𝐾 · 𝑀)) = 0
52	49, 50, 51	addcomli 11308	. . . . . . . . 9 ⊢ (-(𝐾 · 𝑀) + (𝐾 · 𝑀)) = 0
53	52	oveq1i 7359	. . . . . . . 8 ⊢ ((-(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (0 + 𝑁)
54	50, 49, 16	addassi 11125	. . . . . . . 8 ⊢ ((-(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
55	16	addlidi 11304	. . . . . . . 8 ⊢ (0 + 𝑁) = 𝑁
56	53, 54, 55	3eqtr3i 2760	. . . . . . 7 ⊢ (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁)) = 𝑁
57	48, 56	eqtri 2752	. . . . . 6 ⊢ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = 𝑁
58	39, 57	breqtri 5117	. . . . 5 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑁
59		dvdslegcd 16415	. . . . . . 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 1463	. . . . 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 12477	. . . 4 ⊢ (𝑀 gcd 𝑁) ∈ ℝ
65	33	zrei 12477	. . . 4 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℝ
66	64, 65	letri3i 11232	. . 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 7357	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = (𝐾 · 0))
70	41	mul01i 11306	. . . . . . . . . 10 ⊢ (𝐾 · 0) = 0
71	69, 70	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = 0)
72	71	oveq1d 7364	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = (0 + 𝑁))
73	72, 55	eqtrdi 2780	. . . . . . 7 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = 𝑁)
74	73	eqeq1d 2731	. . . . . 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 7358	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
77		oveq12 7358	. . . . . . 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 2767	. . . . 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 1540 ∈ wcel 2109 class class class wbr 5092 (class class class)co 7349 ℂcc 11007 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 ≤ cle 11150 -cneg 11348 ℕ₀cn0 12384 ℤcz 12471 ∥ cdvds 16163 gcd cgcd 16405

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-z 12472 df-uz 12736 df-rp 12894 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406

This theorem is referenced by: gcdaddm 16436

W3C validator