Theorem lcmgcd 12775

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 12707; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12707 and https://math.stackexchange.com/a/470827 12707. 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 12762 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 12662	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2	1	nn0cnd 9555	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
3	2	mul02d 8665	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
4		0z 9588	. . . . . . . . . 10 |- 0 e. ZZ
5		lcmcom 12761	. . . . . . . . . 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 12762	. . . . . . . . 9 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
8	6, 7	eqtr3d 2267	. . . . . . . 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 6065	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( 0 lcm N ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
11		zcn 9582	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
12	11	adantl 277	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
13	12	mul02d 8665	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
14	13	abs00bd 11751	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( 0 x. N ) ) = 0 )
15	3, 10, 14	3eqtr4d 2275	. . . . 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 6065	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = ( 0 lcm N ) )
19	18	oveq1d 6065	. . . 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 6065	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = ( 0 x. N ) )
21	20	fveq2d 5674	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( abs ` ( M x. N ) ) = ( abs ` ( 0 x. N ) ) )
22	16, 19, 21	3eqtr4d 2275	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
23		lcm0val 12762	. . . . . . . 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 6065	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm 0 ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
26		zcn 9582	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
27	26	adantr 276	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. CC )
28	27	mul01d 8666	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
29	28	abs00bd 11751	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. 0 ) ) = 0 )
30	3, 25, 29	3eqtr4d 2275	. . . . 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 6066	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = ( M lcm 0 ) )
34	33	oveq1d 6065	. . . 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 6066	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = ( M x. 0 ) )
36	35	fveq2d 5674	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( abs ` ( M x. N ) ) = ( abs ` ( M x. 0 ) ) )
37	31, 34, 36	3eqtr4d 2275	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
38	22, 37	jaodan 805	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
39		neanior 2499	. . . . 5 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
40		nnabscl 11785	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
41		nnabscl 11785	. . . . . . 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 592	. . . . 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 12774	. . . . 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 12773	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
49		gcdabs 12684	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
50	48, 49	oveq12d 6068	. . . 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 11783	. . . . . . 7 |- ( M e. CC -> ( abs ` ( abs ` M ) ) = ( abs ` M ) )
53		absidm 11783	. . . . . . 7 |- ( N e. CC -> ( abs ` ( abs ` N ) ) = ( abs ` N ) )
54	52, 53	oveqan12d 6069	. . . . . 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 11770	. . . . . . . 8 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
57	56	nn0cnd 9555	. . . . . . 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 11770	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. NN0 )
60	59	nn0cnd 9555	. . . . . . 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 11879	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) = ( ( abs ` ( abs ` M ) ) x. ( abs ` ( abs ` N ) ) ) )
63	27, 12	absmuld 11879	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
64	55, 62, 63	3eqtr4d 2275	. . . 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 2273	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
67		lcmmndc 12759	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> 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 e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
70	38, 66, 69	mpjaodan 806	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 716  DECID wdc 842   = wceq 1398   e. wcel 2203   =/= wne 2412   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   x. cmul 8132   NNcn 9237   ZZcz 9577   abscabs 11682   || cdvds 12473   gcd cgcd 12649   lcm clcm 12757

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650  df-lcm 12758

This theorem is referenced by:  lcmid  12777  lcm1  12778  lcmgcdnn  12779