Theorem lcmgcdeq 12654

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 12640	. . . . . . 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 12533	. . . . . . . 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 4091	. . . . . . 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 12643	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
10	9	nn0zd 9599	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. ZZ )
11		dvdstr 12388	. . . . . . . . . 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 1307	. . . . . . . . 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 1233	. . . . . . . 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 1230	. . . . . . 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 12369	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || N ) )
19		zabscl 11646	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. ZZ )
20		dvdsabsb 12370	. . . . . . 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 4091	. . . . . . 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 12388	. . . . . . . . . 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 1307	. . . . . . . . 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 1236	. . . . . . . 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 1230	. . . . . . 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 12369	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N || M <-> ( abs ` N ) || M ) )
39		zabscl 11646	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
40		dvdsabsb 12370	. . . . . . . 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 11645	. . . . . . 7 |- ( M e. ZZ -> ( abs ` M ) e. NN0 )
47		nn0abscl 11645	. . . . . . 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 12408	. . . . . 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 12651	. . . . . . . 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 12556	. . . . . . . 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 2267	. . . . . 6 |- ( M e. ZZ -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` M ) ) )
59		oveq2 6025	. . . . . . 7 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) lcm ( abs ` M ) ) = ( ( abs ` M ) lcm ( abs ` N ) ) )
60		oveq2 6025	. . . . . . 7 |- ( ( abs ` M ) = ( abs ` N ) -> ( ( abs ` M ) gcd ( abs ` M ) ) = ( ( abs ` M ) gcd ( abs ` N ) ) )
61	59, 60	eqeq12d 2246	. . . . . 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 12647	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
66		gcdabs 12558	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) gcd ( abs ` N ) ) = ( M gcd N ) )
67	65, 66	eqeq12d 2246	. . . 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 600	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 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   NN0cn0 9401   ZZcz 9478   abscabs 11557   || cdvds 12347   gcd cgcd 12523   lcm clcm 12631

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524  df-lcm 12632

This theorem is referenced by: (None)