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Theorem gcdaddmlem 16462

Description: Lemma for gcdaddm 16463. (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 16441	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
4	1, 2, 3	mp2an 689	. . . . . 6 ⊢ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)
5	4	simpli 483	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ 𝑀
6		gcdcl 16444	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
7	1, 2, 6	mp2an 689	. . . . . . . . 9 ⊢ (𝑀 gcd 𝑁) ∈ ℕ₀
8	7	nn0zi 12584	. . . . . . . 8 ⊢ (𝑀 gcd 𝑁) ∈ ℤ
9		gcdaddmlem.1	. . . . . . . . 9 ⊢ 𝐾 ∈ ℤ
10		1z 12589	. . . . . . . . 9 ⊢ 1 ∈ ℤ
11		dvds2ln 16229	. . . . . . . . 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 699	. . . . . . . 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 1457	. . . . . . 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 12560	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
16	2, 15	ax-mp 5	. . . . . . . 8 ⊢ 𝑁 ∈ ℂ
17	16	mullidi 11216	. . . . . . 7 ⊢ (1 · 𝑁) = 𝑁
18	17	oveq2i 7412	. . . . . 6 ⊢ ((𝐾 · 𝑀) + (1 · 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
19	14, 18	breqtri 5163	. . . . 5 ⊢ (𝑀 gcd 𝑁) ∥ ((𝐾 · 𝑀) + 𝑁)
20		zmulcl 12608	. . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ)
21	9, 1, 20	mp2an 689	. . . . . . 7 ⊢ (𝐾 · 𝑀) ∈ ℤ
22		zaddcl 12599	. . . . . . 7 ⊢ (((𝐾 · 𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) + 𝑁) ∈ ℤ)
23	21, 2, 22	mp2an 689	. . . . . 6 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ
24		dvdslegcd 16442	. . . . . . 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 1457	. . . . 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 695	. . . 4 ⊢ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
28		gcddvds 16441	. . . . . . 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 689	. . . . . 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 16444	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ ((𝐾 · 𝑀) + 𝑁) ∈ ℤ) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℕ₀)
32	1, 23, 31	mp2an 689	. . . . . . . . 9 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℕ₀
33	32	nn0zi 12584	. . . . . . . 8 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℤ
34		znegcl 12594	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → -𝐾 ∈ ℤ)
35	9, 34	ax-mp 5	. . . . . . . . 9 ⊢ -𝐾 ∈ ℤ
36		dvds2ln 16229	. . . . . . . . 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 699	. . . . . . . 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 1457	. . . . . . 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 12560	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
41	9, 40	ax-mp 5	. . . . . . . . 9 ⊢ 𝐾 ∈ ℂ
42		zcn 12560	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
43	1, 42	ax-mp 5	. . . . . . . . 9 ⊢ 𝑀 ∈ ℂ
44	41, 43	mulneg1i 11657	. . . . . . . 8 ⊢ (-𝐾 · 𝑀) = -(𝐾 · 𝑀)
45		zcn 12560	. . . . . . . . . 10 ⊢ (((𝐾 · 𝑀) + 𝑁) ∈ ℤ → ((𝐾 · 𝑀) + 𝑁) ∈ ℂ)
46	23, 45	ax-mp 5	. . . . . . . . 9 ⊢ ((𝐾 · 𝑀) + 𝑁) ∈ ℂ
47	46	mullidi 11216	. . . . . . . 8 ⊢ (1 · ((𝐾 · 𝑀) + 𝑁)) = ((𝐾 · 𝑀) + 𝑁)
48	44, 47	oveq12i 7413	. . . . . . 7 ⊢ ((-𝐾 · 𝑀) + (1 · ((𝐾 · 𝑀) + 𝑁))) = (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
49	41, 43	mulcli 11218	. . . . . . . . . 10 ⊢ (𝐾 · 𝑀) ∈ ℂ
50	49	negcli 11525	. . . . . . . . . 10 ⊢ -(𝐾 · 𝑀) ∈ ℂ
51	49	negidi 11526	. . . . . . . . . 10 ⊢ ((𝐾 · 𝑀) + -(𝐾 · 𝑀)) = 0
52	49, 50, 51	addcomli 11403	. . . . . . . . 9 ⊢ (-(𝐾 · 𝑀) + (𝐾 · 𝑀)) = 0
53	52	oveq1i 7411	. . . . . . . 8 ⊢ ((-(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (0 + 𝑁)
54	50, 49, 16	addassi 11221	. . . . . . . 8 ⊢ ((-(𝐾 · 𝑀) + (𝐾 · 𝑀)) + 𝑁) = (-(𝐾 · 𝑀) + ((𝐾 · 𝑀) + 𝑁))
55	16	addlidi 11399	. . . . . . . 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 5163	. . . . 5 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∥ 𝑁
59		dvdslegcd 16442	. . . . . . 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 1457	. . . . 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 695	. . . 4 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ≤ (𝑀 gcd 𝑁))
63	27, 62	anim12i 612	. . 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 12561	. . . 4 ⊢ (𝑀 gcd 𝑁) ∈ ℝ
65	33	zrei 12561	. . . 4 ⊢ (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) ∈ ℝ
66	64, 65	letri3i 11327	. . 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 233	. 2 ⊢ ((¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
68		pm4.57 987	. . 3 ⊢ (¬ (¬ (𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) ↔ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∨ (𝑀 = 0 ∧ 𝑁 = 0)))
69		oveq2 7409	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = (𝐾 · 0))
70	41	mul01i 11401	. . . . . . . . . 10 ⊢ (𝐾 · 0) = 0
71	69, 70	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐾 · 𝑀) = 0)
72	71	oveq1d 7416	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = (0 + 𝑁))
73	72, 55	eqtrdi 2780	. . . . . . 7 ⊢ (𝑀 = 0 → ((𝐾 · 𝑀) + 𝑁) = 𝑁)
74	73	eqeq1d 2726	. . . . . 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 7410	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
77		oveq12 7410	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)) = (0 gcd 0))
78	75, 77	sylbir 234	. . . . . 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 216	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
81	80, 79	jaoi 854	. . 3 ⊢ (((𝑀 = 0 ∧ ((𝐾 · 𝑀) + 𝑁) = 0) ∨ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁)))
82	68, 81	sylbi 216	. 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 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5138 (class class class)co 7401 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 ≤ cle 11246 -cneg 11442 ℕ₀cn0 12469 ℤcz 12555 ∥ 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-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: gcdaddm 16463

W3C validator