Theorem dvdsabseq 12616

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 12561	. . 3 ⊢ (𝑀 ∥ 𝑁 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2		simpr 110	. . . . . . 7 ⊢ ((𝑀 ∥ 𝑁 ∧ 𝑁 ∥ 𝑀) → 𝑁 ∥ 𝑀)
3		breq1 4133	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑁 ∥ 𝑀 ↔ 0 ∥ 𝑀))
4		0dvds 12580	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (0 ∥ 𝑀 ↔ 𝑀 = 0))
5	4	adantr 276	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑀 ↔ 𝑀 = 0))
6		zcn 9651	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
7	6	abs00ad 11833	. . . . . . . . . . . 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 466	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑀) = 0))
12		fveq2 5695	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (abs‘𝑁) = (abs‘0))
13		abs0 11826	. . . . . . . . . . 11 ⊢ (abs‘0) = 0
14	12, 13	eqtrdi 2287	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (abs‘𝑁) = 0)
15	14	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑁) = 0)
16	15	eqeq2d 2250	. . . . . . . 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 535	. . . . . . 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 2428	. . . . . . . 8 ⊢ (¬ 𝑁 = 0 → 𝑁 ≠ 0)
25	24	adantr 276	. . . . . . 7 ⊢ ((¬ 𝑁 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ≠ 0)
26		dvdsleabs2 12615	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑀 ∥ 𝑁 → (abs‘𝑀) ≤ (abs‘𝑁)))
27	21, 23, 25, 26	syl3anc 1278	. . . . . 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 4133	. . . . . . . . . . . . . . 15 ⊢ (𝑀 = 0 → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
30		0dvds 12580	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
31		zcn 9651	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
32	31	abs00ad 11833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → ((abs‘𝑁) = 0 ↔ 𝑁 = 0))
33		eqcom 2240	. . . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 ∥ 𝑁 ↔ 0 = (abs‘𝑁)))
38		fveq2 5695	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 = 0 → (abs‘𝑀) = (abs‘0))
39	38, 13	eqtrdi 2287	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (abs‘𝑀) = 0)
40	39	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (abs‘𝑀) = 0)
41	40	eqeq1d 2247	. . . . . . . . . . . . . 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 1491	. . . . . . . . . 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 535	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ∈ ℤ)
49		neqne 2428	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑀 = 0 → 𝑀 ≠ 0)
50	49	adantr 276	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑀 ≠ 0)
51		dvdsleabs2 12615	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑁 ∥ 𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
52	47, 48, 50, 51	syl3anc 1278	. . . . . . . . . . . 12 ⊢ ((¬ 𝑀 = 0 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∥ 𝑀 → (abs‘𝑁) ≤ (abs‘𝑀)))
53		eqcom 2240	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑀) = (abs‘𝑁) ↔ (abs‘𝑁) = (abs‘𝑀))
54	31	abscld 11949	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℝ)
55	6	abscld 11949	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℝ)
56		letri3 8406	. . . . . . . . . . . . . . . . 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 9657	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
66		zdceq 9722	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0)
67	65, 66	mpan2 429	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0)
68		exmiddc 848	. . . . . . . . . . 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 730	. . . . . . . 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 9722	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
77	65, 76	mpan2 429	. . . . . 6 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 0)
78		exmiddc 848	. . . . . 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 730	. . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420   class class class wbr 4130  ‘cfv 5377  ℝcr 8178  0cc0 8179   ≤ cle 8361  ℤcz 9646  abscabs 11765   ∥ cdvds 12556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-dvds 12557

This theorem is used by:  dvdseq  12617