Theorem lcmgcd 16571

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 16889 or of Bézout's identity bezout 16512; see e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 16512 and https://math.stackexchange.com/a/470827 16512. 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 16558 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 16474	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ0)
2	1	nn0cnd 12558	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℂ)
3	2	mul02d 11436	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 · (𝑀 gcd 𝑁)) = 0)
4		0z 12593	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
5		lcmcom 16557	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 lcm 0) = (0 lcm 𝑁))
6	4, 5	mpan2 690	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = (0 lcm 𝑁))
7		lcm0val 16558	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0)
8	6, 7	eqtr3d 2770	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (0 lcm 𝑁) = 0)
9	8	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 lcm 𝑁) = 0)
10	9	oveq1d 7429	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((0 lcm 𝑁) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁)))
11		zcn 12587	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
12	11	adantl 481	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
13	12	mul02d 11436	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 · 𝑁) = 0)
14	13	abs00bd 15264	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(0 · 𝑁)) = 0)
15	3, 10, 14	3eqtr4d 2778	. . . . 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 7429	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (0 lcm 𝑁))
19	18	oveq1d 7429	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((0 lcm 𝑁) · (𝑀 gcd 𝑁)))
20	17	oveq1d 7429	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 · 𝑁) = (0 · 𝑁))
21	20	fveq2d 6895	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(0 · 𝑁)))
22	16, 19, 21	3eqtr4d 2778	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
23		lcm0val 16558	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0)
24	23	adantr 480	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 0) = 0)
25	24	oveq1d 7429	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 0) · (𝑀 gcd 𝑁)) = (0 · (𝑀 gcd 𝑁)))
26		zcn 12587	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℂ)
28	27	mul01d 11437	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 0) = 0)
29	28	abs00bd 15264	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 0)) = 0)
30	3, 25, 29	3eqtr4d 2778	. . . . 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 7430	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm 0))
34	33	oveq1d 7429	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 0) · (𝑀 gcd 𝑁)))
35	32	oveq2d 7430	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 · 𝑁) = (𝑀 · 0))
36	35	fveq2d 6895	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (abs‘(𝑀 · 𝑁)) = (abs‘(𝑀 · 0)))
37	31, 34, 36	3eqtr4d 2778	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
38	22, 37	jaodan 956	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁)))
39		neanior 3031	. . . . 5 ⊢ ((𝑀 ≠ 0 ∧ 𝑁 ≠ 0) ↔ ¬ (𝑀 = 0 ∨ 𝑁 = 0))
40		nnabscl 15298	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
41		nnabscl 15298	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
42	40, 41	anim12i 612	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
43	42	an4s 659	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
44	39, 43	sylan2br 594	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘𝑀) ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ))
45		lcmgcdlem 16570	. . . . 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 16569	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))
49		gcdabs 16499	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))
50	48, 49	oveq12d 7432	. . . 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 15296	. . . . . . 7 ⊢ (𝑀 ∈ ℂ → (abs‘(abs‘𝑀)) = (abs‘𝑀))
53		absidm 15296	. . . . . . 7 ⊢ (𝑁 ∈ ℂ → (abs‘(abs‘𝑁)) = (abs‘𝑁))
54	52, 53	oveqan12d 7433	. . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))) = ((abs‘𝑀) · (abs‘𝑁)))
55	26, 11, 54	syl2an 595	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))) = ((abs‘𝑀) · (abs‘𝑁)))
56		nn0abscl 15285	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ0)
57	56	nn0cnd 12558	. . . . . . 7 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ)
58	57	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑀) ∈ ℂ)
59		nn0abscl 15285	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℕ0)
60	59	nn0cnd 12558	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℂ)
61	60	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑁) ∈ ℂ)
62	58, 61	absmuld 15427	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘((abs‘𝑀) · (abs‘𝑁))) = ((abs‘(abs‘𝑀)) · (abs‘(abs‘𝑁))))
63	27, 12	absmuld 15427	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
64	55, 62, 63	3eqtr4d 2778	. . . 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 2776	. 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 846   = wceq 1534   ∈ wcel 2099   ≠ wne 2936   class class class wbr 5142  ‘cfv 6542  (class class class)co 7414  ℂcc 11130  0cc0 11132   · cmul 11137  ℕcn 12236  ℤcz 12582  abscabs 15207   ∥ cdvds 16224   gcd cgcd 16462   lcm clcm 16552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9459  df-inf 9460  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-z 12583  df-uz 12847  df-rp 13001  df-fl 13783  df-mod 13861  df-seq 13993  df-exp 14053  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-dvds 16225  df-gcd 16463  df-lcm 16554

This theorem is referenced by:  lcmid  16573  lcm1  16574  lcmgcdnn  16575  nzprmdif  43750

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