Theorem lcmgcdeq 12085

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 12071	. . . . . . 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 11966	. . . . . . . 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 4008	. . . . . . 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 12074	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
10	9	nn0zd 9375	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. ZZ )
11		dvdstr 11837	. . . . . . . . . 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 1272	. . . . . . . . 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 1207	. . . . . . . 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 1204	. . . . . . 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 11818	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || N ) )
19		zabscl 11097	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
20		dvdsabsb 11819	. . . . . . 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 4008	. . . . . . 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 11837	. . . . . . . . . 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 1272	. . . . . . . . 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 1210	. . . . . . . 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 1204	. . . . . . 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 11818	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N || M <-> ( abs ` N ) || M ) )
39		zabscl 11097	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
40		dvdsabsb 11819	. . . . . . . 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 11096	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
47		nn0abscl 11096	. . . . . . 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 11856	. . . . . 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 12082	. . . . . . . 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 11989	. . . . . . . 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 2213	. . . . . 6 |- ( M e. ZZ -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` M ) ) )
59		oveq2 5885	. . . . . . 7 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) lcm ( abs ` N ) ) )
60		oveq2 5885	. . . . . . 7 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) gcd ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) )
61	59, 60	eqeq12d 2192	. . . . . 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 12078	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
66		gcdabs 11991	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
67	65, 66	eqeq12d 2192	. . . 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 1353   e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   NN0cn0 9178   ZZcz 9255   abscabs 11008   || cdvds 11796   gcd cgcd 11945   lcm clcm 12062

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 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  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 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946  df-lcm 12063

This theorem is referenced by: (None)