Theorem lcmgcd 12081

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 12015; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12015 and https://math.stackexchange.com/a/470827 12015. 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 12068 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 11970	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2	1	nn0cnd 9234	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
3	2	mul02d 8352	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
4		0z 9267	. . . . . . . . . 10 |- 0 e. ZZ
5		lcmcom 12067	. . . . . . . . . 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 12068	. . . . . . . . 9 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
8	6, 7	eqtr3d 2212	. . . . . . . 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 5893	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( 0 lcm N ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
11		zcn 9261	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
12	11	adantl 277	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
13	12	mul02d 8352	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
14	13	abs00bd 11078	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( 0 x. N ) ) = 0 )
15	3, 10, 14	3eqtr4d 2220	. . . . 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 5893	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = ( 0 lcm N ) )
19	18	oveq1d 5893	. . . 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 5893	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = ( 0 x. N ) )
21	20	fveq2d 5521	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( abs ` ( M x. N ) ) = ( abs ` ( 0 x. N ) ) )
22	16, 19, 21	3eqtr4d 2220	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
23		lcm0val 12068	. . . . . . . 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 5893	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm 0 ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
26		zcn 9261	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
27	26	adantr 276	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. CC )
28	27	mul01d 8353	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
29	28	abs00bd 11078	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. 0 ) ) = 0 )
30	3, 25, 29	3eqtr4d 2220	. . . . 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 5894	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = ( M lcm 0 ) )
34	33	oveq1d 5893	. . . 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 5894	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = ( M x. 0 ) )
36	35	fveq2d 5521	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( abs ` ( M x. N ) ) = ( abs ` ( M x. 0 ) ) )
37	31, 34, 36	3eqtr4d 2220	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
38	22, 37	jaodan 797	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
39		neanior 2434	. . . . 5 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
40		nnabscl 11112	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
41		nnabscl 11112	. . . . . . 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 12080	. . . . 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 12079	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
49		gcdabs 11992	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
50	48, 49	oveq12d 5896	. . . 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 11110	. . . . . . 7 |- ( M e. CC -> ( abs ` ( abs ` M ) ) = ( abs ` M ) )
53		absidm 11110	. . . . . . 7 |- ( N e. CC -> ( abs ` ( abs ` N ) ) = ( abs ` N ) )
54	52, 53	oveqan12d 5897	. . . . . 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 11097	. . . . . . . 8 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
57	56	nn0cnd 9234	. . . . . . 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 11097	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. NN0 )
60	59	nn0cnd 9234	. . . . . . 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 11206	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) = ( ( abs ` ( abs ` M ) ) x. ( abs ` ( abs ` N ) ) ) )
63	27, 12	absmuld 11206	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
64	55, 62, 63	3eqtr4d 2220	. . . 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 2218	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) x. ( M gcd N ) ) = ( abs ` ( M x. N ) ) )
67		lcmmndc 12065	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
68		exmiddc 836	. . 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 798	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 708  DECID wdc 834   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   0cc0 7814   x. cmul 7819   NNcn 8922   ZZcz 9256   abscabs 11009   || cdvds 11797   gcd cgcd 11946   lcm clcm 12063

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-sup 6986  df-inf 6987  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-fz 10012  df-fzo 10146  df-fl 10273  df-mod 10326  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-dvds 11798  df-gcd 11947  df-lcm 12064

This theorem is referenced by:  lcmid  12083  lcm1  12084  lcmgcdnn  12085