Theorem lcmgcd 12643

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 12575; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12575 and https://math.stackexchange.com/a/470827 12575. 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 12630 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 12530	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ0)
2	1	nn0cnd 9450	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℂ)
3	2	mul02d 8564	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 · (𝑀 gcd 𝑁)) = 0)
4		0z 9483	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
5		lcmcom 12629	. . . . . . . . . 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 12630	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0)
8	6, 7	eqtr3d 2264	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (0 lcm 𝑁) = 0)
9	8	adantl 277	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 lcm 𝑁) = 0)
10	9	oveq1d 6028	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁)))
11		zcn 9477	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
12	11	adantl 277	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
13	12	mul02d 8564	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 · 𝑁) = 0)
14	13	abs00bd 11620	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(0 · 𝑁)) = 0)
15	3, 10, 14	3eqtr4d 2272	. . . . 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 6028	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (0 lcm 𝑁))
19	18	oveq1d 6028	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((0 lcm 𝑁) · (𝑀 gcd 𝑁)))
20	17	oveq1d 6028	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 · 𝑁) = (0 · 𝑁))
21	20	fveq2d 5639	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(0 · 𝑁)))
22	16, 19, 21	3eqtr4d 2272	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
23		lcm0val 12630	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0)
24	23	adantr 276	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 0) = 0)
25	24	oveq1d 6028	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁)))
26		zcn 9477	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
27	26	adantr 276	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℂ)
28	27	mul01d 8565	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 0) = 0)
29	28	abs00bd 11620	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 0)) = 0)
30	3, 25, 29	3eqtr4d 2272	. . . . 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 6029	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm 0))
34	33	oveq1d 6028	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 0) · (𝑀 gcd 𝑁)))
35	32	oveq2d 6029	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0))
36	35	fveq2d 5639	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(𝑀 · 0)))
37	31, 34, 36	3eqtr4d 2272	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
38	22, 37	jaodan 802	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
39		neanior 2487	. . . . 5 ⊢ ((𝑀 ≠ 0 ∧ 𝑁 ≠ 0) ↔ ¬ (𝑀 = 0 ∨ 𝑁 = 0))
40		nnabscl 11654	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
41		nnabscl 11654	. . . . . . 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 590	. . . . 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 12642	. . . . 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 12641	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))
49		gcdabs 12552	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))
50	48, 49	oveq12d 6031	. . . 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 11652	. . . . . . 7 ⊢ (𝑀 ∈ ℂ → (abs‘(abs‘𝑀)) = (abs‘𝑀))
53		absidm 11652	. . . . . . 7 ⊢ (𝑁 ∈ ℂ → (abs‘(abs‘𝑁)) = (abs‘𝑁))
54	52, 53	oveqan12d 6032	. . . . . 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 11639	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ0)
57	56	nn0cnd 9450	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ)
58	57	adantr 276	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑀) ∈ ℂ)
59		nn0abscl 11639	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ0)
60	59	nn0cnd 9450	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℂ)
61	60	adantl 277	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑁) ∈ ℂ)
62	58, 61	absmuld 11748	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘((abs‘𝑀) · (abs‘𝑁))) = ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))))
63	27, 12	absmuld 11748	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
64	55, 62, 63	3eqtr4d 2272	. . . 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 2270	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
67		lcmmndc 12627	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))
68		exmiddc 841	. . 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 803	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713  DECID wdc 839   = wceq 1395   ∈ wcel 2200   ≠ wne 2400   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  ℂcc 8023  0cc0 8025   · cmul 8030  ℕcn 9136  ℤcz 9472  abscabs 11551   ∥ cdvds 12341   gcd cgcd 12517   lcm clcm 12625

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7177  df-inf 7178  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-fl 10523  df-mod 10578  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-dvds 12342  df-gcd 12518  df-lcm 12626

This theorem is referenced by:  lcmid  12645  lcm1  12646  lcmgcdnn  12647