Theorem lcmgcd 12615

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 12547; see, e.g., https://proofwiki.org/wiki/Product_of_GCD_and_LCM 12547 and https://math.stackexchange.com/a/470827 12547. 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 12602 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 12502	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
2	1	nn0cnd 9435	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. CC )
3	2	mul02d 8549	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. ( M gcd N ) ) = 0 )
4		0z 9468	. . . . . . . . . 10 |- 0 e. ZZ
5		lcmcom 12601	. . . . . . . . . 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 12602	. . . . . . . . 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 6022	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( 0 lcm N ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
11		zcn 9462	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
12	11	adantl 277	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
13	12	mul02d 8549	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
14	13	abs00bd 11592	. . . . . 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 6022	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = ( 0 lcm N ) )
19	18	oveq1d 6022	. . . 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 6022	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = ( 0 x. N ) )
21	20	fveq2d 5633	. . . 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 12602	. . . . . . . 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 6022	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm 0 ) x. ( M gcd N ) ) = ( 0 x. ( M gcd N ) ) )
26		zcn 9462	. . . . . . . . 9 |- ( M e. ZZ -> M e. CC )
27	26	adantr 276	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. CC )
28	27	mul01d 8550	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
29	28	abs00bd 11592	. . . . . 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 6023	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = ( M lcm 0 ) )
34	33	oveq1d 6022	. . . 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 6023	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = ( M x. 0 ) )
36	35	fveq2d 5633	. . . 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 11626	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
41		nnabscl 11626	. . . . . . 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 12614	. . . . 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 12613	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
49		gcdabs 12524	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
50	48, 49	oveq12d 6025	. . . 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 11624	. . . . . . 7 |- ( M e. CC -> ( abs ` ( abs ` M ) ) = ( abs ` M ) )
53		absidm 11624	. . . . . . 7 |- ( N e. CC -> ( abs ` ( abs ` N ) ) = ( abs ` N ) )
54	52, 53	oveqan12d 6026	. . . . . 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 11611	. . . . . . . 8 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
57	56	nn0cnd 9435	. . . . . . 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 11611	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. NN0 )
60	59	nn0cnd 9435	. . . . . . 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 11720	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( abs ` ( ( abs ` M ) x. ( abs ` N ) ) ) = ( ( abs ` ( abs ` M ) ) x. ( abs ` ( abs ` N ) ) ) )
63	27, 12	absmuld 11720	. . . . 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 12599	. . 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 4083   ` cfv 5318  (class class class)co 6007   CCcc 8008   0cc0 8010   x. cmul 8015   NNcn 9121   ZZcz 9457   abscabs 11523   || cdvds 12313   gcd cgcd 12489   lcm clcm 12597

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-dvds 12314  df-gcd 12490  df-lcm 12598

This theorem is referenced by:  lcmid  12617  lcm1  12618  lcmgcdnn  12619