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Theorem lcmgcdeq 16831

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 16817	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
2	1	simpld 500	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 lcm 𝑁))
3	2	adantr 486	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → 𝑀 ∥ (𝑀 lcm 𝑁))
4		gcddvds 16717	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
5	4	simprd 501	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∥ 𝑁)
6		breq1 5111	. . . . . . 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 412	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (𝑀 lcm 𝑁) ∥ 𝑁)
9		lcmcl 16820	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ₀)
10	9	nn0zd 12758	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℤ)
11		dvdstr 16508	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
12	10, 11	syl3an2 1182	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
13	12	3com12 1141	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
14	13	3expb 1138	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
15	14	anidms 577	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑁) → 𝑀 ∥ 𝑁))
16	15	adantr 486	. . . . 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 712	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → 𝑀 ∥ 𝑁)
18		absdvdsb 16488	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ 𝑁))
19		zabscl 15523	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℤ)
20		dvdsabsb 16489	. . . . . . 7 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
21	19, 20	sylan 592	. . . . . 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 486	. . . 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 501	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 lcm 𝑁))
26	25	adantr 486	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → 𝑁 ∥ (𝑀 lcm 𝑁))
27	4	simpld 500	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∥ 𝑀)
28		breq1 5111	. . . . . . 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 412	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (𝑀 lcm 𝑁) ∥ 𝑀)
31		dvdstr 16508	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
32	10, 31	syl3an2 1182	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
33	32	3coml 1145	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
34	33	3expb 1138	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
35	34	anidms 577	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝑀) → 𝑁 ∥ 𝑀))
36	35	adantr 486	. . . . 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 712	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → 𝑁 ∥ 𝑀)
38		absdvdsb 16488	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑁) ∥ 𝑀))
39		zabscl 15523	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℤ)
40		dvdsabsb 16489	. . . . . . . 8 ⊢ (((abs‘𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((abs‘𝑁) ∥ 𝑀 ↔ (abs‘𝑁) ∥ (abs‘𝑀)))
41	39, 40	sylan 592	. . . . . . 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 464	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∥ 𝑀 ↔ (abs‘𝑁) ∥ (abs‘𝑀)))
44	43	adantr 486	. . . 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 15522	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ₀)
47		nn0abscl 15522	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ₀)
48	46, 47	anim12i 625	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) ∈ ℕ₀ ∧ (abs‘𝑁) ∈ ℕ₀))
49		dvdseq 16528	. . . . . 6 ⊢ ((((abs‘𝑀) ∈ ℕ₀ ∧ (abs‘𝑁) ∈ ℕ₀) ∧ ((abs‘𝑀) ∥ (abs‘𝑁) ∧ (abs‘𝑁) ∥ (abs‘𝑀))) → (abs‘𝑀) = (abs‘𝑁))
50	48, 49	sylan 592	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((abs‘𝑀) ∥ (abs‘𝑁) ∧ (abs‘𝑁) ∥ (abs‘𝑀))) → (abs‘𝑀) = (abs‘𝑁))
51	50	ex 418	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) ∥ (abs‘𝑁) ∧ (abs‘𝑁) ∥ (abs‘𝑀)) → (abs‘𝑀) = (abs‘𝑁)))
52	51	adantr 486	. . 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 712	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)) → (abs‘𝑀) = (abs‘𝑁))
54		lcmid 16828	. . . . . . . 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 16743	. . . . . . . 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 2801	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((abs‘𝑀) lcm (abs‘𝑀)) = ((abs‘𝑀) gcd (abs‘𝑀)))
59		oveq2 7459	. . . . . . 7 ⊢ ((abs‘𝑀) = (abs‘𝑁) → ((abs‘𝑀) lcm (abs‘𝑀)) = ((abs‘𝑀) lcm (abs‘𝑁)))
60		oveq2 7459	. . . . . . 7 ⊢ ((abs‘𝑀) = (abs‘𝑁) → ((abs‘𝑀) gcd (abs‘𝑀)) = ((abs‘𝑀) gcd (abs‘𝑁)))
61	59, 60	eqeq12d 2779	. . . . . 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 412	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ (abs‘𝑀) = (abs‘𝑁)) → ((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁)))
64	63	adantlr 728	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (abs‘𝑀) = (abs‘𝑁)) → ((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁)))
65		lcmabs 16824	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))
66		gcdabs 16748	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))
67	65, 66	eqeq12d 2779	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) lcm (abs‘𝑁)) = ((abs‘𝑀) gcd (abs‘𝑁)) ↔ (𝑀 lcm 𝑁) = (𝑀 gcd 𝑁)))
68	67	adantr 486	. . 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 813	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 gcd 𝑁) ↔ (abs‘𝑀) = (abs‘𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℕ₀cn0 12646 ℤcz 12733 abscabs 15444 ∥ cdvds 16466 gcd cgcd 16708 lcm clcm 16807

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-sup 9471 df-inf 9472 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-rp 13161 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467 df-gcd 16709 df-lcm 16809

This theorem is used by: (None)

W3C validator