Theorem lcmgcd 12595

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 12527; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12527 and https://math.stackexchange.com/a/470827 12527. 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 12582 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 12482	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2	1	nn0cnd 9420	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
3	2	mul02d 8534	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
4		0z 9453	. . . . . . . . . 10 |- 0 e. ZZ
5		lcmcom 12581	. . . . . . . . . 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 12582	. . . . . . . . 9 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
8	6, 7	eqtr3d 2264	. . . . . . . 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 6015	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( 0 lcm N ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
11		zcn 9447	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
12	11	adantl 277	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
13	12	mul02d 8534	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
14	13	abs00bd 11572	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( 0 x. N ) ) = 0 )
15	3, 10, 14	3eqtr4d 2272	. . . . 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 6015	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = ( 0 lcm N ) )
19	18	oveq1d 6015	. . . 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 6015	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = ( 0 x. N ) )
21	20	fveq2d 5630	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( abs ` ( M x. N ) ) = ( abs ` ( 0 x. N ) ) )
22	16, 19, 21	3eqtr4d 2272	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
23		lcm0val 12582	. . . . . . . 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 6015	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm 0 ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
26		zcn 9447	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
27	26	adantr 276	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. CC )
28	27	mul01d 8535	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
29	28	abs00bd 11572	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. 0 ) ) = 0 )
30	3, 25, 29	3eqtr4d 2272	. . . . 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 6016	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = ( M lcm 0 ) )
34	33	oveq1d 6015	. . . 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 6016	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = ( M x. 0 ) )
36	35	fveq2d 5630	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( abs ` ( M x. N ) ) = ( abs ` ( M x. 0 ) ) )
37	31, 34, 36	3eqtr4d 2272	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
38	22, 37	jaodan 802	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
39		neanior 2487	. . . . 5 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
40		nnabscl 11606	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
41		nnabscl 11606	. . . . . . 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 590	. . . . 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 12594	. . . . 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 12593	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
49		gcdabs 12504	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
50	48, 49	oveq12d 6018	. . . 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 11604	. . . . . . 7 |- ( M e. CC -> ( abs ` ( abs ` M ) ) = ( abs ` M ) )
53		absidm 11604	. . . . . . 7 |- ( N e. CC -> ( abs ` ( abs ` N ) ) = ( abs ` N ) )
54	52, 53	oveqan12d 6019	. . . . . 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 11591	. . . . . . . 8 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
57	56	nn0cnd 9420	. . . . . . 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 11591	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. NN0 )
60	59	nn0cnd 9420	. . . . . . 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 11700	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) = ( ( abs ` ( abs ` M ) ) x. ( abs ` ( abs ` N ) ) ) )
63	27, 12	absmuld 11700	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
64	55, 62, 63	3eqtr4d 2272	. . . 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 2270	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
67		lcmmndc 12579	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> 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 e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
70	38, 66, 69	mpjaodan 803	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 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   0cc0 7995   x. cmul 8000   NNcn 9106   ZZcz 9442   abscabs 11503   || cdvds 12293   gcd cgcd 12469   lcm clcm 12577

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-dvds 12294  df-gcd 12470  df-lcm 12578

This theorem is referenced by:  lcmid  12597  lcm1  12598  lcmgcdnn  12599