Theorem lcmgcd 12783

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 or of Bézout's identity bezout 12715; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12715 and https://math.stackexchange.com/a/470827 12715. 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 12770 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 12670	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ0)
2	1	nn0cnd 9560	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℂ)
3	2	mul02d 8670	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 · (𝑀 gcd 𝑁)) = 0)
4		0z 9593	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
5		lcmcom 12769	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 lcm 0) = (0 lcm 𝑁))
6	4, 5	mpan2 425	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = (0 lcm 𝑁))
7		lcm0val 12770	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0)
8	6, 7	eqtr3d 2269	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (0 lcm 𝑁) = 0)
9	8	adantl 277	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 lcm 𝑁) = 0)
10	9	oveq1d 6067	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁)))
11		zcn 9587	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
12	11	adantl 277	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
13	12	mul02d 8670	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 · 𝑁) = 0)
14	13	abs00bd 11759	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(0 · 𝑁)) = 0)
15	3, 10, 14	3eqtr4d 2277	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(0 · 𝑁)))
16	15	adantr 276	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(0 · 𝑁)))
17		simpr 110	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → 𝑀 = 0)
18	17	oveq1d 6067	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (0 lcm 𝑁))
19	18	oveq1d 6067	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((0 lcm 𝑁) · (𝑀 gcd 𝑁)))
20	17	oveq1d 6067	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 · 𝑁) = (0 · 𝑁))
21	20	fveq2d 5676	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(0 · 𝑁)))
22	16, 19, 21	3eqtr4d 2277	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
23		lcm0val 12770	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0)
24	23	adantr 276	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 0) = 0)
25	24	oveq1d 6067	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁)))
26		zcn 9587	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
27	26	adantr 276	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℂ)
28	27	mul01d 8671	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 0) = 0)
29	28	abs00bd 11759	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 0)) = 0)
30	3, 25, 29	3eqtr4d 2277	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 0)))
31	30	adantr 276	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 0)))
32		simpr 110	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → 𝑁 = 0)
33	32	oveq2d 6068	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm 0))
34	33	oveq1d 6067	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 0) · (𝑀 gcd 𝑁)))
35	32	oveq2d 6068	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0))
36	35	fveq2d 5676	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(𝑀 · 0)))
37	31, 34, 36	3eqtr4d 2277	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
38	22, 37	jaodan 805	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
39		neanior 2501	. . . . 5 ⊢ ((𝑀 ≠ 0 ∧ 𝑁 ≠ 0) ↔ ¬ (𝑀 = 0 ∨ 𝑁 = 0))
40		nnabscl 11793	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
41		nnabscl 11793	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
42	40, 41	anim12i 338	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
43	42	an4s 592	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
44	39, 43	sylan2br 288	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
45		lcmgcdlem 12782	. . . . 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 112	. . . 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 14	. . 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 12781	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))
49		gcdabs 12692	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))
50	48, 49	oveq12d 6070	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) lcm (abs‘𝑁)) · ((abs‘𝑀) gcd (abs‘𝑁))) = ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)))
51	50	adantr 276	. . 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 11791	. . . . . . 7 ⊢ (𝑀 ∈ ℂ → (abs‘(abs‘𝑀)) = (abs‘𝑀))
53		absidm 11791	. . . . . . 7 ⊢ (𝑁 ∈ ℂ → (abs‘(abs‘𝑁)) = (abs‘𝑁))
54	52, 53	oveqan12d 6071	. . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))) = ((abs‘𝑀) · (abs‘𝑁)))
55	26, 11, 54	syl2an 289	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))) = ((abs‘𝑀) · (abs‘𝑁)))
56		nn0abscl 11778	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ0)
57	56	nn0cnd 9560	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ)
58	57	adantr 276	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑀) ∈ ℂ)
59		nn0abscl 11778	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ0)
60	59	nn0cnd 9560	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℂ)
61	60	adantl 277	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑁) ∈ ℂ)
62	58, 61	absmuld 11887	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘((abs‘𝑀) · (abs‘𝑁))) = ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))))
63	27, 12	absmuld 11887	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
64	55, 62, 63	3eqtr4d 2277	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘((abs‘𝑀) · (abs‘𝑁))) = (abs‘(𝑀 · 𝑁)))
65	64	adantr 276	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘((abs‘𝑀) · (abs‘𝑁))) = (abs‘(𝑀 · 𝑁)))
66	47, 51, 65	3eqtr3d 2275	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
67		lcmmndc 12767	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))
68		exmiddc 844	. . 3 ⊢ (DECID (𝑀 = 0 ∨ 𝑁 = 0) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0)))
69	67, 68	syl 14	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0)))
70	38, 66, 69	mpjaodan 806	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2205   ≠ wne 2414   class class class wbr 4111  ‘cfv 5354  (class class class)co 6052  ℂcc 8130  0cc0 8132   · cmul 8137  ℕcn 9242  ℤcz 9582  abscabs 11690   ∥ cdvds 12481   gcd cgcd 12657   lcm clcm 12765

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-fz 10349  df-fzo 10484  df-fl 10637  df-mod 10692  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-dvds 12482  df-gcd 12658  df-lcm 12766

This theorem is referenced by:  lcmid  12785  lcm1  12786  lcmgcdnn  12787