Theorem dvdsabseq 12014

Description: If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)

Assertion

Ref	Expression
dvdsabseq	⊢ ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → (abs‘𝑀) = (abs‘𝑁))

Proof of Theorem dvdsabseq

Step	Hyp	Ref	Expression
1		dvdszrcl 11959	. . 3 ⊢ (𝑀 ∥ 𝑁 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2		simpr 110	. . . . . . 7 ⊢ ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → 𝑁 ∥ 𝑀)
3		breq1 4037	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑁 ∥ 𝑀 ↔ 0 ∥ 𝑀))
4		0dvds 11978	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (0 ∥ 𝑀 ↔ 𝑀 = 0))
5	4	adantr 276	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀 ↔ 𝑀 = 0))
6		zcn 9333	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
7	6	abs00ad 11232	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → ((abs‘𝑀) = 0 ↔ 𝑀 = 0))
8	7	bicomd 141	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑀 = 0 ↔ (abs‘𝑀) = 0))
9	8	adantr 276	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 0 ↔ (abs‘𝑀) = 0))
10	5, 9	bitrd 188	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀 ↔ (abs‘𝑀) = 0))
11	3, 10	sylan9bb 462	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑀) = 0))
12		fveq2 5559	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (abs‘𝑁) = (abs‘0))
13		abs0 11225	. . . . . . . . . . 11 ⊢ (abs‘0) = 0
14	12, 13	eqtrdi 2245	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (abs‘𝑁) = 0)
15	14	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑁) = 0)
16	15	eqeq2d 2208	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑀) = 0))
17	11, 16	bitr4d 191	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑀) = (abs‘𝑁)))
18	2, 17	imbitrid 154	. . . . . 6 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → (abs‘𝑀) = (abs‘𝑁)))
19	18	expd 258	. . . . 5 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → (abs‘𝑀) = (abs‘𝑁))))
20	19	expcom 116	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → (abs‘𝑀) = (abs‘𝑁)))))
21		simprl 529	. . . . . . 7 ⊢ ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
22		simpr 110	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
23	22	adantl 277	. . . . . . 7 ⊢ ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
24		neqne 2375	. . . . . . . 8 ⊢ (¬ 𝑁 = 0 → 𝑁 ≠ 0)
25	24	adantr 276	. . . . . . 7 ⊢ ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ≠ 0)
26		dvdsleabs2 12013	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀 ∥ 𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
27	21, 23, 25, 26	syl3anc 1249	. . . . . 6 ⊢ ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
28		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁) → 𝑀 ∥ 𝑁)
29		breq1 4037	. . . . . . . . . . . . . . 15 ⊢ (𝑀 = 0 → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
30		0dvds 11978	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
31		zcn 9333	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
32	31	abs00ad 11232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → ((abs‘𝑁) = 0 ↔ 𝑁 = 0))
33		eqcom 2198	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘𝑁) = 0 ↔ 0 = (abs‘𝑁))
34	32, 33	bitr3di 195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ↔ 0 = (abs‘𝑁)))
35	30, 34	bitrd 188	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
36	35	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
37	29, 36	sylan9bb 462	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
38		fveq2 5559	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 = 0 → (abs‘𝑀) = (abs‘0))
39	38, 13	eqtrdi 2245	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (abs‘𝑀) = 0)
40	39	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑀) = 0)
41	40	eqeq1d 2205	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑀) = (abs‘𝑁) ↔ 0 = (abs‘𝑁)))
42	37, 41	bitr4d 191	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) = (abs‘𝑁)))
43	28, 42	imbitrid 154	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁) → (abs‘𝑀) = (abs‘𝑁)))
44	43	a1dd 48	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁) → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
45	44	expcomd 1452	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁)))))
46	45	expcom 116	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 0 → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))))
47	22	adantl 277	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
48		simprl 529	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
49		neqne 2375	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑀 = 0 → 𝑀 ≠ 0)
50	49	adantr 276	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ≠ 0)
51		dvdsleabs2 12013	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑁 ∥ 𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
52	47, 48, 50, 51	syl3anc 1249	. . . . . . . . . . . 12 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∥ 𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
53		eqcom 2198	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑁) = (abs‘𝑀))
54	31	abscld 11348	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
55	6	abscld 11348	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℝ)
56		letri3 8109	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘𝑁) ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ) → ((abs‘𝑁) = (abs‘𝑀) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
57	54, 55, 56	syl2anr 290	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑁) = (abs‘𝑀) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
58	53, 57	bitrid 192	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) = (abs‘𝑁) ↔ ((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁))))
59	58	biimprd 158	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑁) ≤ (abs‘𝑀) ∧ (abs‘𝑀) ≤ (abs‘𝑁)) → (abs‘𝑀) = (abs‘𝑁)))
60	59	expd 258	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑁) ≤ (abs‘𝑀) → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
61	60	adantl 277	. . . . . . . . . . . 12 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((abs‘𝑁) ≤ (abs‘𝑀) → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
62	52, 61	syld 45	. . . . . . . . . . 11 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∥ 𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))
63	62	a1d 22	. . . . . . . . . 10 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁)))))
64	63	expcom 116	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 = 0 → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁))))))
65		0z 9339	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
66		zdceq 9403	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0)
67	65, 66	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0)
68		exmiddc 837	. . . . . . . . . . 11 ⊢ (DECID 𝑀 = 0 → (𝑀 = 0 ∨ ¬ 𝑀 = 0))
69	67, 68	syl 14	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (𝑀 = 0 ∨ ¬ 𝑀 = 0))
70	69	adantr 276	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 0 ∨ ¬ 𝑀 = 0))
71	46, 64, 70	mpjaod 719	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → ((abs‘𝑀) ≤ (abs‘𝑁) → (abs‘𝑀) = (abs‘𝑁)))))
72	71	com34 83	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 → ((abs‘𝑀) ≤ (abs‘𝑁) → (𝑁 ∥ 𝑀 → (abs‘𝑀) = (abs‘𝑁)))))
73	72	adantl 277	. . . . . 6 ⊢ ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 → ((abs‘𝑀) ≤ (abs‘𝑁) → (𝑁 ∥ 𝑀 → (abs‘𝑀) = (abs‘𝑁)))))
74	27, 73	mpdd 41	. . . . 5 ⊢ ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → (abs‘𝑀) = (abs‘𝑁))))
75	74	expcom 116	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑁 = 0 → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → (abs‘𝑀) = (abs‘𝑁)))))
76		zdceq 9403	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
77	65, 76	mpan2 425	. . . . . 6 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 0)
78		exmiddc 837	. . . . . 6 ⊢ (DECID 𝑁 = 0 → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
79	77, 78	syl 14	. . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
80	79	adantl 277	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
81	20, 75, 80	mpjaod 719	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → (abs‘𝑀) = (abs‘𝑁))))
82	1, 81	mpcom 36	. 2 ⊢ (𝑀 ∥ 𝑁 → (𝑁 ∥ 𝑀 → (abs‘𝑀) = (abs‘𝑁)))
83	82	imp 124	1 ⊢ ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → (abs‘𝑀) = (abs‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167   ≠ wne 2367   class class class wbr 4034  ‘cfv 5259  ℝcr 7880  0cc0 7881   ≤ cle 8064  ℤcz 9328  abscabs 11164   ∥ cdvds 11954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000  ax-caucvg 8001

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-frec 6450  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-n0 9252  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-seqfrec 10542  df-exp 10633  df-cj 11009  df-re 11010  df-im 11011  df-rsqrt 11165  df-abs 11166  df-dvds 11955

This theorem is referenced by:  dvdseq  12015