Theorem lcmgcdeq 12605

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 12591	. . . . . . 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 12484	. . . . . . . 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 4086	. . . . . . 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 12594	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
10	9	nn0zd 9567	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. ZZ )
11		dvdstr 12339	. . . . . . . . . 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 1305	. . . . . . . . 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 1231	. . . . . . . 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 1228	. . . . . . 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 12320	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || N ) )
19		zabscl 11597	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
20		dvdsabsb 12321	. . . . . . 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 4086	. . . . . . 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 12339	. . . . . . . . . 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 1305	. . . . . . . . 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 1234	. . . . . . . 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 1228	. . . . . . 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 12320	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N || M <-> ( abs ` N ) || M ) )
39		zabscl 11597	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
40		dvdsabsb 12321	. . . . . . . 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 11596	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
47		nn0abscl 11596	. . . . . . 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 12359	. . . . . 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 12602	. . . . . . . 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 12507	. . . . . . . 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 2265	. . . . . 6 |- ( M e. ZZ -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` M ) ) )
59		oveq2 6009	. . . . . . 7 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) lcm ( abs ` N ) ) )
60		oveq2 6009	. . . . . . 7 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) gcd ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) )
61	59, 60	eqeq12d 2244	. . . . . 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 12598	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
66		gcdabs 12509	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
67	65, 66	eqeq12d 2244	. . . 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 598	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 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   NN0cn0 9369   ZZcz 9446   abscabs 11508   || cdvds 12298   gcd cgcd 12474   lcm clcm 12582

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 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  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 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475  df-lcm 12583

This theorem is referenced by: (None)