Theorem lcmgcdeq 12276

Description: Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

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

Proof of Theorem lcmgcdeq

Step	Hyp	Ref	Expression
1		dvdslcm 12262	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
2	1	simpld 112	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M lcm N ) )
3	2	adantr 276	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> M || ( M lcm N ) )
4		gcddvds 12155	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
5	4	simprd 114	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || N )
6		breq1 4037	. . . . . . 7 |- ( ( M lcm N ) = ( M gcd N ) -> ( ( M lcm N ) || N <-> ( M gcd N ) || N ) )
7	5, 6	syl5ibrcom 157	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = ( M gcd N ) -> ( M lcm N ) || N ) )
8	7	imp 124	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( M lcm N ) || N )
9		lcmcl 12265	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
10	9	nn0zd 9463	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. ZZ )
11		dvdstr 12010	. . . . . . . . . 10 |- ( ( M e. ZZ /\ ( M lcm N ) e. ZZ /\ N e. ZZ ) -> ( ( M || ( M lcm N ) /\ ( M lcm N ) || N ) -> M || N ) )
12	10, 11	syl3an2 1283	. . . . . . . . 9 |- ( ( M e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) /\ N e. ZZ ) -> ( ( M || ( M lcm N ) /\ ( M lcm N ) || N ) -> M || N ) )
13	12	3com12 1209	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || ( M lcm N ) /\ ( M lcm N ) || N ) -> M || N ) )
14	13	3expb 1206	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( M || ( M lcm N ) /\ ( M lcm N ) || N ) -> M || N ) )
15	14	anidms 397	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M || ( M lcm N ) /\ ( M lcm N ) || N ) -> M || N ) )
16	15	adantr 276	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( ( M || ( M lcm N ) /\ ( M lcm N ) || N ) -> M || N ) )
17	3, 8, 16	mp2and 433	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> M || N )
18		absdvdsb 11991	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || N ) )
19		zabscl 11268	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
20		dvdsabsb 11992	. . . . . . 7 |- ( ( ( abs ` M ) e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) || N <-> ( abs ` M ) || ( abs ` N ) ) )
21	19, 20	sylan 283	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) || N <-> ( abs ` M ) || ( abs ` N ) ) )
22	18, 21	bitrd 188	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || ( abs ` N ) ) )
23	22	adantr 276	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( M || N <-> ( abs ` M ) || ( abs ` N ) ) )
24	17, 23	mpbid 147	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( abs ` M ) || ( abs ` N ) )
25	1	simprd 114	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M lcm N ) )
26	25	adantr 276	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> N || ( M lcm N ) )
27	4	simpld 112	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) || M )
28		breq1 4037	. . . . . . 7 |- ( ( M lcm N ) = ( M gcd N ) -> ( ( M lcm N ) || M <-> ( M gcd N ) || M ) )
29	27, 28	syl5ibrcom 157	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = ( M gcd N ) -> ( M lcm N ) || M ) )
30	29	imp 124	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( M lcm N ) || M )
31		dvdstr 12010	. . . . . . . . . 10 |- ( ( N e. ZZ /\ ( M lcm N ) e. ZZ /\ M e. ZZ ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || M ) -> N || M ) )
32	10, 31	syl3an2 1283	. . . . . . . . 9 |- ( ( N e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) /\ M e. ZZ ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || M ) -> N || M ) )
33	32	3coml 1212	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M e. ZZ /\ N e. ZZ ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || M ) -> N || M ) )
34	33	3expb 1206	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || M ) -> N || M ) )
35	34	anidms 397	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || M ) -> N || M ) )
36	35	adantr 276	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( ( N || ( M lcm N ) /\ ( M lcm N ) || M ) -> N || M ) )
37	26, 30, 36	mp2and 433	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> N || M )
38		absdvdsb 11991	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N || M <-> ( abs ` N ) || M ) )
39		zabscl 11268	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
40		dvdsabsb 11992	. . . . . . . 8 |- ( ( ( abs ` N ) e. ZZ /\ M e. ZZ ) -> ( ( abs ` N ) || M <-> ( abs ` N ) || ( abs ` M ) ) )
41	39, 40	sylan 283	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( abs ` N ) || M <-> ( abs ` N ) || ( abs ` M ) ) )
42	38, 41	bitrd 188	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N || M <-> ( abs ` N ) || ( abs ` M ) ) )
43	42	ancoms 268	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N || M <-> ( abs ` N ) || ( abs ` M ) ) )
44	43	adantr 276	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( N || M <-> ( abs ` N ) || ( abs ` M ) ) )
45	37, 44	mpbid 147	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( abs ` N ) || ( abs ` M ) )
46		nn0abscl 11267	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
47		nn0abscl 11267	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) e. NN0 )
48	46, 47	anim12i 338	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) e. NN0 /\ ( abs ` N ) e. NN0 ) )
49		dvdseq 12030	. . . . . 6 |- ( ( ( ( abs ` M ) e. NN0 /\ ( abs ` N ) e. NN0 ) /\ ( ( abs ` M ) || ( abs ` N ) /\ ( abs ` N ) || ( abs ` M ) ) ) -> ( abs ` M ) = ( abs ` N ) )
50	48, 49	sylan 283	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( ( abs ` M ) || ( abs ` N ) /\ ( abs ` N ) || ( abs ` M ) ) ) -> ( abs ` M ) = ( abs ` N ) )
51	50	ex 115	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) || ( abs ` N ) /\ ( abs ` N ) || ( abs ` M ) ) -> ( abs ` M ) = ( abs ` N ) ) )
52	51	adantr 276	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( ( ( abs ` M ) || ( abs ` N ) /\ ( abs ` N ) || ( abs ` M ) ) -> ( abs ` M ) = ( abs ` N ) ) )
53	24, 45, 52	mp2and 433	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M lcm N ) = ( M gcd N ) ) -> ( abs ` M ) = ( abs ` N ) )
54		lcmid 12273	. . . . . . . 8 |- ( ( abs ` M ) e. ZZ -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( abs ` ( abs ` M ) ) )
55	19, 54	syl 14	. . . . . . 7 |- ( M e. ZZ -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( abs ` ( abs ` M ) ) )
56		gcdid 12178	. . . . . . . 8 |- ( ( abs ` M ) e. ZZ -> ( ( abs ` M ) gcd ( abs ` M ) ) = ( abs ` ( abs ` M ) ) )
57	19, 56	syl 14	. . . . . . 7 |- ( M e. ZZ -> ( ( abs ` M ) gcd ( abs ` M ) ) = ( abs ` ( abs ` M ) ) )
58	55, 57	eqtr4d 2232	. . . . . 6 |- ( M e. ZZ -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` M ) ) )
59		oveq2 5933	. . . . . . 7 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) lcm ( abs ` N ) ) )
60		oveq2 5933	. . . . . . 7 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) gcd ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) )
61	59, 60	eqeq12d 2211	. . . . . 6 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` M ) ) <-> ( ( abs ` M ) lcm ( abs ` N ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) ) )
62	58, 61	syl5ibcom 155	. . . . 5 |- ( M e. ZZ -> ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) ) )
63	62	imp 124	. . . 4 |- ( ( M e. ZZ /\ ( abs ` M ) = ( abs ` N ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) )
64	63	adantlr 477	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( abs ` M ) = ( abs ` N ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) )
65		lcmabs 12269	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
66		gcdabs 12180	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
67	65, 66	eqeq12d 2211	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) <-> ( M lcm N ) = ( M gcd N ) ) )
68	67	adantr 276	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( abs ` M ) = ( abs ` N ) ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) <-> ( M lcm N ) = ( M gcd N ) ) )
69	64, 68	mpbid 147	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( abs ` M ) = ( abs ` N ) ) -> ( M lcm N ) = ( M gcd N ) )
70	53, 69	impbida 596	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M lcm N ) = ( M gcd N ) <-> ( abs ` M ) = ( abs ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   NN0cn0 9266   ZZcz 9343   abscabs 11179   || cdvds 11969   gcd cgcd 12145   lcm clcm 12253

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146  df-lcm 12254

This theorem is referenced by: (None)