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Theorem lcmgcdeq 16669

Description: Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.)

**Assertion**
Ref	Expression
lcmgcdeq	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 gcd 𝑁) ↔ (abs‘𝑀) = (abs‘𝑁)))

**Proof of Theorem lcmgcdeq**
Step	Hyp	Ref	Expression
1		dvdslcm 16655	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
2	1	simpld 499	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 lcm 𝑁))
3	2	adantr 485	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → 𝑀 ∥ (𝑀 lcm 𝑁))
4		gcddvds 16560	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
5	4	simprd 500	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∥ 𝑁)
6		breq1 5116	. . . . . . 7 ⊢ ((𝑀 lcm 𝑁) = (𝑀 gcd 𝑁) → ((𝑀 lcm 𝑁) ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ 𝑁))
7	5, 6	syl5ibrcom 250	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 gcd 𝑁) → (𝑀 lcm 𝑁) ∥ 𝑁))
8	7	imp 411	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (𝑀 lcm 𝑁) ∥ 𝑁)
9		lcmcl 16658	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ₀)
10	9	nn0zd 12615	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℤ)
11		dvdstr 16351	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
12	10, 11	syl3an2 1180	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
13	12	3com12 1139	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
14	13	3expb 1136	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
15	14	anidms 576	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
16	15	adantr 485	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
17	3, 8, 16	mp2and 711	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → 𝑀 ∥ 𝑁)
18		absdvdsb 16331	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁))
19		zabscl 15363	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℤ)
20		dvdsabsb 16332	. . . . . . 7 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
21	19, 20	sylan 591	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
22	18, 21	bitrd 282	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
23	22	adantr 485	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
24	17, 23	mpbid 235	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (abs‘𝑀) ∥ (abs‘𝑁))
25	1	simprd 500	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 lcm 𝑁))
26	25	adantr 485	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → 𝑁 ∥ (𝑀 lcm 𝑁))
27	4	simpld 499	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∥ 𝑀)
28		breq1 5116	. . . . . . 7 ⊢ ((𝑀 lcm 𝑁) = (𝑀 gcd 𝑁) → ((𝑀 lcm 𝑁) ∥ 𝑀 ↔ (𝑀 gcd 𝑁) ∥ 𝑀))
29	27, 28	syl5ibrcom 250	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 gcd 𝑁) → (𝑀 lcm 𝑁) ∥ 𝑀))
30	29	imp 411	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (𝑀 lcm 𝑁) ∥ 𝑀)
31		dvdstr 16351	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
32	10, 31	syl3an2 1180	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
33	32	3coml 1143	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
34	33	3expb 1136	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
35	34	anidms 576	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
36	35	adantr 485	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
37	26, 30, 36	mp2and 711	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → 𝑁 ∥ 𝑀)
38		absdvdsb 16331	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑁) ∥ 𝑀))
39		zabscl 15363	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℤ)
40		dvdsabsb 16332	. . . . . . . 8 ⊢ (((abs‘𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((abs‘𝑁) ∥ 𝑀 ↔ (abs‘𝑁) ∥ (abs‘𝑀)))
41	39, 40	sylan 591	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((abs‘𝑁) ∥ 𝑀 ↔ (abs‘𝑁) ∥ (abs‘𝑀)))
42	38, 41	bitrd 282	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑁) ∥ (abs‘𝑀)))
43	42	ancoms 463	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑁) ∥ (abs‘𝑀)))
44	43	adantr 485	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑁) ∥ (abs‘𝑀)))
45	37, 44	mpbid 235	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (abs‘𝑁) ∥ (abs‘𝑀))
46		nn0abscl 15362	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ₀)
47		nn0abscl 15362	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ₀)
48	46, 47	anim12i 624	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∈ ℕ₀ ∧ (abs‘𝑁) ∈ ℕ₀))
49		dvdseq 16371	. . . . . 6 ⊢ ((((abs‘𝑀) ∈ ℕ₀ ∧ (abs‘𝑁) ∈ ℕ₀) ∧ ((abs‘𝑀) ∥ (abs‘𝑁) ∧ (abs‘𝑁) ∥ (abs‘𝑀))) → (abs‘𝑀) = (abs‘𝑁))
50	48, 49	sylan 591	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((abs‘𝑀) ∥ (abs‘𝑁) ∧ (abs‘𝑁) ∥ (abs‘𝑀))) → (abs‘𝑀) = (abs‘𝑁))
51	50	ex 417	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) ∥ (abs‘𝑁) ∧ (abs‘𝑁) ∥ (abs‘𝑀)) → (abs‘𝑀) = (abs‘𝑁)))
52	51	adantr 485	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (((abs‘𝑀) ∥ (abs‘𝑁) ∧ (abs‘𝑁) ∥ (abs‘𝑀)) → (abs‘𝑀) = (abs‘𝑁)))
53	24, 45, 52	mp2and 711	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (abs‘𝑀) = (abs‘𝑁))
54		lcmid 16666	. . . . . . . 8 ⊢ ((abs‘𝑀) ∈ ℤ → ((abs‘𝑀) lcm (abs‘𝑀)) = (abs‘(abs‘𝑀)))
55	19, 54	syl 18	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((abs‘𝑀) lcm (abs‘𝑀)) = (abs‘(abs‘𝑀)))
56		gcdid 16584	. . . . . . . 8 ⊢ ((abs‘𝑀) ∈ ℤ → ((abs‘𝑀) gcd (abs‘𝑀)) = (abs‘(abs‘𝑀)))
57	19, 56	syl 18	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → ((abs‘𝑀) gcd (abs‘𝑀)) = (abs‘(abs‘𝑀)))
58	55, 57	eqtr4d 2807	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((abs‘𝑀) lcm (abs‘𝑀)) = ((abs‘𝑀) gcd (abs‘𝑀)))
59		oveq2 7419	. . . . . . 7 ⊢ ((abs‘𝑀) = (abs‘𝑁) → ((abs‘𝑀) lcm (abs‘𝑀)) = ((abs‘𝑀) lcm (abs‘𝑁)))
60		oveq2 7419	. . . . . . 7 ⊢ ((abs‘𝑀) = (abs‘𝑁) → ((abs‘𝑀) gcd (abs‘𝑀)) = ((abs‘𝑀) gcd (abs‘𝑁)))
61	59, 60	eqeq12d 2785	. . . . . 6 ⊢ ((abs‘𝑀) = (abs‘𝑁) → (((abs‘𝑀) lcm (abs‘𝑀)) = ((abs‘𝑀) gcd (abs‘𝑀)) ↔ ((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁))))
62	58, 61	syl5ibcom 248	. . . . 5 ⊢ (𝑀 ∈ ℤ → ((abs‘𝑀) = (abs‘𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁))))
63	62	imp 411	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ (abs‘𝑀) = (abs‘𝑁)) → ((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁)))
64	63	adantlr 727	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (abs‘𝑀) = (abs‘𝑁)) → ((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁)))
65		lcmabs 16662	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))
66		gcdabs 16588	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))
67	65, 66	eqeq12d 2785	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁)) ↔ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)))
68	67	adantr 485	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (abs‘𝑀) = (abs‘𝑁)) → (((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁)) ↔ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)))
69	64, 68	mpbid 235	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (abs‘𝑀) = (abs‘𝑁)) → (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁))
70	53, 69	impbida 812	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 gcd 𝑁) ↔ (abs‘𝑀) = (abs‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ℕ₀cn0 12503 ℤcz 12590 abscabs 15284 ∥ cdvds 16309 gcd cgcd 16551 lcm clcm 16645

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9401 df-inf 9402 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-z 12591 df-uz 12862 df-rp 13016 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-dvds 16310 df-gcd 16552 df-lcm 16647

This theorem is referenced by: (None)

W3C validator