Theorem lcmgcd 16515

Description: The product of two numbers' least common multiple and greatest common divisor is the absolute value of the product of the two numbers. In particular, that absolute value is the least common multiple of two coprime numbers, for which (𝑀 gcd 𝑁) = 1.

Multiple methods exist for proving this, and it is often proven either as a consequence of the fundamental theorem of arithmetic 1arith 16836 or of Bézout's identity bezout 16451; see e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 16451 and https://math.stackexchange.com/a/470827 16451. This proof uses the latter to first confirm it for positive integers 𝑀 and 𝑁 (the "Second Proof" in the above Stack Exchange page), then shows that implies it for all nonzero integer inputs, then finally uses lcm0val 16502 to show it applies when either or both inputs are zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

Ref	Expression
lcmgcd	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))

Proof of Theorem lcmgcd

Step	Hyp	Ref	Expression
1		gcdcl 16414	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ0)
2	1	nn0cnd 12441	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℂ)
3	2	mul02d 11308	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 · (𝑀 gcd 𝑁)) = 0)
4		0z 12476	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
5		lcmcom 16501	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 lcm 0) = (0 lcm 𝑁))
6	4, 5	mpan2 691	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = (0 lcm 𝑁))
7		lcm0val 16502	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0)
8	6, 7	eqtr3d 2768	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (0 lcm 𝑁) = 0)
9	8	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 lcm 𝑁) = 0)
10	9	oveq1d 7361	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁)))
11		zcn 12470	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
12	11	adantl 481	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
13	12	mul02d 11308	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 · 𝑁) = 0)
14	13	abs00bd 15195	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(0 · 𝑁)) = 0)
15	3, 10, 14	3eqtr4d 2776	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(0 · 𝑁)))
16	15	adantr 480	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(0 · 𝑁)))
17		simpr 484	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → 𝑀 = 0)
18	17	oveq1d 7361	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (0 lcm 𝑁))
19	18	oveq1d 7361	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((0 lcm 𝑁) · (𝑀 gcd 𝑁)))
20	17	oveq1d 7361	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 · 𝑁) = (0 · 𝑁))
21	20	fveq2d 6826	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(0 · 𝑁)))
22	16, 19, 21	3eqtr4d 2776	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
23		lcm0val 16502	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0)
24	23	adantr 480	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 0) = 0)
25	24	oveq1d 7361	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁)))
26		zcn 12470	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℂ)
28	27	mul01d 11309	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 0) = 0)
29	28	abs00bd 15195	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 0)) = 0)
30	3, 25, 29	3eqtr4d 2776	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 0)))
31	30	adantr 480	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 0)))
32		simpr 484	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → 𝑁 = 0)
33	32	oveq2d 7362	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm 0))
34	33	oveq1d 7361	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 0) · (𝑀 gcd 𝑁)))
35	32	oveq2d 7362	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0))
36	35	fveq2d 6826	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(𝑀 · 0)))
37	31, 34, 36	3eqtr4d 2776	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
38	22, 37	jaodan 959	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
39		neanior 3021	. . . . 5 ⊢ ((𝑀 ≠ 0 ∧ 𝑁 ≠ 0) ↔ ¬ (𝑀 = 0 ∨ 𝑁 = 0))
40		nnabscl 15230	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
41		nnabscl 15230	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
42	40, 41	anim12i 613	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
43	42	an4s 660	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
44	39, 43	sylan2br 595	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
45		lcmgcdlem 16514	. . . . 5 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ) → ((((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) = (abs‘((abs‘𝑀) · (abs‘𝑁))) ∧ ((0 ∈ ℕ ∧ ((abs‘𝑀) ∥ 0 ∧ (abs‘𝑁) ∥ 0)) → ((abs‘𝑀) lcm (abs‘𝑁)) ∥ 0)))
46	45	simpld 494	. . . 4 ⊢ (((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ) → (((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) = (abs‘((abs‘𝑀) · (abs‘𝑁))))
47	44, 46	syl 17	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) = (abs‘((abs‘𝑀) · (abs‘𝑁))))
48		lcmabs 16513	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))
49		gcdabs 16439	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))
50	48, 49	oveq12d 7364	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) = ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)))
51	50	adantr 480	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) = ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)))
52		absidm 15228	. . . . . . 7 ⊢ (𝑀 ∈ ℂ → (abs‘(abs‘𝑀)) = (abs‘𝑀))
53		absidm 15228	. . . . . . 7 ⊢ (𝑁 ∈ ℂ → (abs‘(abs‘𝑁)) = (abs‘𝑁))
54	52, 53	oveqan12d 7365	. . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))) = ((abs‘𝑀) · (abs‘𝑁)))
55	26, 11, 54	syl2an 596	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))) = ((abs‘𝑀) · (abs‘𝑁)))
56		nn0abscl 15216	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ0)
57	56	nn0cnd 12441	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ)
58	57	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑀) ∈ ℂ)
59		nn0abscl 15216	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ0)
60	59	nn0cnd 12441	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℂ)
61	60	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑁) ∈ ℂ)
62	58, 61	absmuld 15361	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘((abs‘𝑀) · (abs‘𝑁))) = ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))))
63	27, 12	absmuld 15361	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
64	55, 62, 63	3eqtr4d 2776	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘((abs‘𝑀) · (abs‘𝑁))) = (abs‘(𝑀 · 𝑁)))
65	64	adantr 480	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘((abs‘𝑀) · (abs‘𝑁))) = (abs‘(𝑀 · 𝑁)))
66	47, 51, 65	3eqtr3d 2774	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
67	38, 66	pm2.61dan 812	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  ℂcc 11001  0cc0 11003   · cmul 11008  ℕcn 12122  ℤcz 12465  abscabs 15138   ∥ cdvds 16160   gcd cgcd 16402   lcm clcm 16496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-inf 9327  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-n0 12379  df-z 12466  df-uz 12730  df-rp 12888  df-fl 13693  df-mod 13771  df-seq 13906  df-exp 13966  df-cj 15003  df-re 15004  df-im 15005  df-sqrt 15139  df-abs 15140  df-dvds 16161  df-gcd 16403  df-lcm 16498

This theorem is referenced by:  lcmid  16517  lcm1  16518  lcmgcdnn  16519  nzprmdif  44351