Theorem dvdsabseq 11836

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 11783	. . 3 ⊢ (𝑀 ∥ 𝑁 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2		simpr 110	. . . . . . 7 ⊢ ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → 𝑁 ∥ 𝑀)
3		breq1 4003	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑁 ∥ 𝑀 ↔ 0 ∥ 𝑀))
4		0dvds 11802	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (0 ∥ 𝑀 ↔ 𝑀 = 0))
5	4	adantr 276	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀 ↔ 𝑀 = 0))
6		zcn 9247	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
7	6	abs00ad 11058	. . . . . . . . . . . 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 5511	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (abs‘𝑁) = (abs‘0))
13		abs0 11051	. . . . . . . . . . 11 ⊢ (abs‘0) = 0
14	12, 13	eqtrdi 2226	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (abs‘𝑁) = 0)
15	14	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑁) = 0)
16	15	eqeq2d 2189	. . . . . . . 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 2355	. . . . . . . 8 ⊢ (¬ 𝑁 = 0 → 𝑁 ≠ 0)
25	24	adantr 276	. . . . . . 7 ⊢ ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ≠ 0)
26		dvdsleabs2 11835	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀 ∥ 𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
27	21, 23, 25, 26	syl3anc 1238	. . . . . 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 4003	. . . . . . . . . . . . . . 15 ⊢ (𝑀 = 0 → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
30		0dvds 11802	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
31		zcn 9247	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
32	31	abs00ad 11058	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → ((abs‘𝑁) = 0 ↔ 𝑁 = 0))
33		eqcom 2179	. . . . . . . . . . . . . . . . . 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 5511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 = 0 → (abs‘𝑀) = (abs‘0))
39	38, 13	eqtrdi 2226	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (abs‘𝑀) = 0)
40	39	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑀) = 0)
41	40	eqeq1d 2186	. . . . . . . . . . . . . 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 1441	. . . . . . . . . 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 2355	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑀 = 0 → 𝑀 ≠ 0)
50	49	adantr 276	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ≠ 0)
51		dvdsleabs2 11835	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑁 ∥ 𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
52	47, 48, 50, 51	syl3anc 1238	. . . . . . . . . . . 12 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∥ 𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
53		eqcom 2179	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑁) = (abs‘𝑀))
54	31	abscld 11174	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
55	6	abscld 11174	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℝ)
56		letri3 8028	. . . . . . . . . . . . . . . . 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 9253	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
66		zdceq 9317	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0)
67	65, 66	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0)
68		exmiddc 836	. . . . . . . . . . 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 718	. . . . . . . 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 9317	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
77	65, 76	mpan2 425	. . . . . 6 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 0)
78		exmiddc 836	. . . . . 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 718	. . 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 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148   ≠ wne 2347   class class class wbr 4000  ‘cfv 5212  ℝcr 7801  0cc0 7802   ≤ cle 7983  ℤcz 9242  abscabs 10990   ∥ cdvds 11778

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779

This theorem is referenced by:  dvdseq  11837