Theorem lcmgcd 12219

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 12151; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12151 and https://math.stackexchange.com/a/470827 12151. 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 12206 to show it applies when either or both inputs are zero. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

Ref	Expression
lcmgcd	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )

Proof of Theorem lcmgcd

Step	Hyp	Ref	Expression
1		gcdcl 12106	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2	1	nn0cnd 9298	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
3	2	mul02d 8413	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
4		0z 9331	. . . . . . . . . 10 |- 0 e. ZZ
5		lcmcom 12205	. . . . . . . . . 10 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N lcm 0 ) = ( 0 lcm N ) )
6	4, 5	mpan2 425	. . . . . . . . 9 |- ( N e. ZZ -> ( N lcm 0 ) = ( 0 lcm N ) )
7		lcm0val 12206	. . . . . . . . 9 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
8	6, 7	eqtr3d 2228	. . . . . . . 8 |- ( N e. ZZ -> ( 0 lcm N ) = 0 )
9	8	adantl 277	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 lcm N ) = 0 )
10	9	oveq1d 5934	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( 0 lcm N ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
11		zcn 9325	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
12	11	adantl 277	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
13	12	mul02d 8413	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
14	13	abs00bd 11213	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( 0 x. N ) ) = 0 )
15	3, 10, 14	3eqtr4d 2236	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( 0 lcm N ) x. ( M gcd N ) ) = ( abs ` ( 0 x. N ) ) )
16	15	adantr 276	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( 0 lcm N ) x. ( M gcd N ) ) = ( abs ` ( 0 x. N ) ) )
17		simpr 110	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 )
18	17	oveq1d 5934	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = ( 0 lcm N ) )
19	18	oveq1d 5934	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( ( 0 lcm N ) x. ( M gcd N ) ) )
20	17	oveq1d 5934	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = ( 0 x. N ) )
21	20	fveq2d 5559	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( abs ` ( M x. N ) ) = ( abs ` ( 0 x. N ) ) )
22	16, 19, 21	3eqtr4d 2236	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
23		lcm0val 12206	. . . . . . . 8 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
24	23	adantr 276	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm 0 ) = 0 )
25	24	oveq1d 5934	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm 0 ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
26		zcn 9325	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
27	26	adantr 276	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. CC )
28	27	mul01d 8414	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
29	28	abs00bd 11213	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. 0 ) ) = 0 )
30	3, 25, 29	3eqtr4d 2236	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm 0 ) x. ( M gcd N ) ) = ( abs ` ( M x. 0 ) ) )
31	30	adantr 276	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( M lcm 0 ) x. ( M gcd N ) ) = ( abs ` ( M x. 0 ) ) )
32		simpr 110	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> N = 0 )
33	32	oveq2d 5935	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = ( M lcm 0 ) )
34	33	oveq1d 5934	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( ( M lcm 0 ) x. ( M gcd N ) ) )
35	32	oveq2d 5935	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = ( M x. 0 ) )
36	35	fveq2d 5559	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( abs ` ( M x. N ) ) = ( abs ` ( M x. 0 ) ) )
37	31, 34, 36	3eqtr4d 2236	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
38	22, 37	jaodan 798	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
39		neanior 2451	. . . . 5 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
40		nnabscl 11247	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
41		nnabscl 11247	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
42	40, 41	anim12i 338	. . . . . 6 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( abs ` M ) e. NN /\ ( abs ` N ) e. NN ) )
43	42	an4s 588	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( ( abs ` M ) e. NN /\ ( abs ` N ) e. NN ) )
44	39, 43	sylan2br 288	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( abs ` M ) e. NN /\ ( abs ` N ) e. NN ) )
45		lcmgcdlem 12218	. . . . 5 |- ( ( ( abs ` M ) e. NN /\ ( abs ` N ) e. NN ) -> ( ( ( ( abs ` M ) lcm ( abs ` N ) ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) = ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) /\ ( ( 0 e. NN /\ ( ( abs ` M ) || 0 /\ ( abs ` N ) || 0 ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || 0 ) ) )
46	45	simpld 112	. . . 4 |- ( ( ( abs ` M ) e. NN /\ ( abs ` N ) e. NN ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) = ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) )
47	44, 46	syl 14	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) = ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) )
48		lcmabs 12217	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
49		gcdabs 12128	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
50	48, 49	oveq12d 5937	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) = ( ( M lcm N ) x. ( M gcd N ) ) )
51	50	adantr 276	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) x. ( ( abs ` M ) gcd ( abs ` N ) ) ) = ( ( M lcm N ) x. ( M gcd N ) ) )
52		absidm 11245	. . . . . . 7 |- ( M e. CC -> ( abs ` ( abs ` M ) ) = ( abs ` M ) )
53		absidm 11245	. . . . . . 7 |- ( N e. CC -> ( abs ` ( abs ` N ) ) = ( abs ` N ) )
54	52, 53	oveqan12d 5938	. . . . . 6 |- ( ( M e. CC /\ N e. CC ) -> ( ( abs ` ( abs ` M ) ) x. ( abs ` ( abs ` N ) ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
55	26, 11, 54	syl2an 289	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( abs ` M ) ) x. ( abs ` ( abs ` N ) ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
56		nn0abscl 11232	. . . . . . . 8 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
57	56	nn0cnd 9298	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. CC )
58	57	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` M ) e. CC )
59		nn0abscl 11232	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. NN0 )
60	59	nn0cnd 9298	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) e. CC )
61	60	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` N ) e. CC )
62	58, 61	absmuld 11341	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) = ( ( abs ` ( abs ` M ) ) x. ( abs ` ( abs ` N ) ) ) )
63	27, 12	absmuld 11341	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
64	55, 62, 63	3eqtr4d 2236	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) = ( abs ` ( M x. N ) ) )
65	64	adantr 276	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) = ( abs ` ( M x. N ) ) )
66	47, 51, 65	3eqtr3d 2234	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
67		lcmmndc 12203	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
68		exmiddc 837	. . 3 |- (DECID ( M = 0 \/ N = 0 ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
69	67, 68	syl 14	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
70	38, 66, 69	mpjaodan 799	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   x. cmul 7879   NNcn 8984   ZZcz 9320   abscabs 11144   || cdvds 11933   gcd cgcd 12082   lcm clcm 12201

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-dvds 11934  df-gcd 12083  df-lcm 12202

This theorem is referenced by:  lcmid  12221  lcm1  12222  lcmgcdnn  12223