Theorem lcmdvds 15321

Description: The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

Ref	Expression
lcmdvds	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )

Proof of Theorem lcmdvds

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 |- ( 0 || K -> 0 || K )
2		breq1 4656	. . . . . . . 8 |- ( M = 0 -> ( M || K <-> 0 || K ) )
3	2	adantl 482	. . . . . . 7 |- ( ( N e. ZZ /\ M = 0 ) -> ( M || K <-> 0 || K ) )
4		oveq1 6657	. . . . . . . . 9 |- ( M = 0 -> ( M lcm N ) = ( 0 lcm N ) )
5		0z 11388	. . . . . . . . . . 11 |- 0 e. ZZ
6		lcmcom 15306	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 lcm N ) = ( N lcm 0 ) )
7	5, 6	mpan 706	. . . . . . . . . 10 |- ( N e. ZZ -> ( 0 lcm N ) = ( N lcm 0 ) )
8		lcm0val 15307	. . . . . . . . . 10 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
9	7, 8	eqtrd 2656	. . . . . . . . 9 |- ( N e. ZZ -> ( 0 lcm N ) = 0 )
10	4, 9	sylan9eqr 2678	. . . . . . . 8 |- ( ( N e. ZZ /\ M = 0 ) -> ( M lcm N ) = 0 )
11	10	breq1d 4663	. . . . . . 7 |- ( ( N e. ZZ /\ M = 0 ) -> ( ( M lcm N ) || K <-> 0 || K ) )
12	3, 11	imbi12d 334	. . . . . 6 |- ( ( N e. ZZ /\ M = 0 ) -> ( ( M || K -> ( M lcm N ) || K ) <-> ( 0 || K -> 0 || K ) ) )
13	1, 12	mpbiri 248	. . . . 5 |- ( ( N e. ZZ /\ M = 0 ) -> ( M || K -> ( M lcm N ) || K ) )
14	13	3ad2antl3 1225	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M || K -> ( M lcm N ) || K ) )
15	14	adantrd 484	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
16	15	ex 450	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( M = 0 -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
17		breq1 4656	. . . . . . . 8 |- ( N = 0 -> ( N || K <-> 0 || K ) )
18	17	adantl 482	. . . . . . 7 |- ( ( M e. ZZ /\ N = 0 ) -> ( N || K <-> 0 || K ) )
19		oveq2 6658	. . . . . . . . 9 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
20		lcm0val 15307	. . . . . . . . 9 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
21	19, 20	sylan9eqr 2678	. . . . . . . 8 |- ( ( M e. ZZ /\ N = 0 ) -> ( M lcm N ) = 0 )
22	21	breq1d 4663	. . . . . . 7 |- ( ( M e. ZZ /\ N = 0 ) -> ( ( M lcm N ) || K <-> 0 || K ) )
23	18, 22	imbi12d 334	. . . . . 6 |- ( ( M e. ZZ /\ N = 0 ) -> ( ( N || K -> ( M lcm N ) || K ) <-> ( 0 || K -> 0 || K ) ) )
24	1, 23	mpbiri 248	. . . . 5 |- ( ( M e. ZZ /\ N = 0 ) -> ( N || K -> ( M lcm N ) || K ) )
25	24	3ad2antl2 1224	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( N || K -> ( M lcm N ) || K ) )
26	25	adantld 483	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
27	26	ex 450	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( N = 0 -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
28		neanior 2886	. . . . . 6 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
29		lcmcl 15314	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
30	29	nn0zd 11480	. . . . . . . . . . . . . . . . 17 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. ZZ )
31		dvds0 14997	. . . . . . . . . . . . . . . . 17 |- ( ( M lcm N ) e. ZZ -> ( M lcm N ) || 0 )
32	30, 31	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) || 0 )
33	32	a1d 25	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M || 0 /\ N || 0 ) -> ( M lcm N ) || 0 ) )
34	33	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K = 0 ) -> ( ( M || 0 /\ N || 0 ) -> ( M lcm N ) || 0 ) )
35		breq2 4657	. . . . . . . . . . . . . . . . 17 |- ( K = 0 -> ( M || K <-> M || 0 ) )
36		breq2 4657	. . . . . . . . . . . . . . . . 17 |- ( K = 0 -> ( N || K <-> N || 0 ) )
37	35, 36	anbi12d 747	. . . . . . . . . . . . . . . 16 |- ( K = 0 -> ( ( M || K /\ N || K ) <-> ( M || 0 /\ N || 0 ) ) )
38		breq2 4657	. . . . . . . . . . . . . . . 16 |- ( K = 0 -> ( ( M lcm N ) || K <-> ( M lcm N ) || 0 ) )
39	37, 38	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( K = 0 -> ( ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) <-> ( ( M || 0 /\ N || 0 ) -> ( M lcm N ) || 0 ) ) )
40	39	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K = 0 ) -> ( ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) <-> ( ( M || 0 /\ N || 0 ) -> ( M lcm N ) || 0 ) ) )
41	34, 40	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K = 0 ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
42	41	adantrl 752	. . . . . . . . . . . 12 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ K = 0 ) ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
43	42	adantllr 755	. . . . . . . . . . 11 |- ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ N e. ZZ ) /\ ( K e. ZZ /\ K = 0 ) ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
44	43	adantlrr 757	. . . . . . . . . 10 |- ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( K e. ZZ /\ K = 0 ) ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
45	44	anassrs 680	. . . . . . . . 9 |- ( ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ K e. ZZ ) /\ K = 0 ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
46		nnabscl 14065	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
47		nnabscl 14065	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
48		nnabscl 14065	. . . . . . . . . . . . . 14 |- ( ( K e. ZZ /\ K =/= 0 ) -> ( abs ` K ) e. NN )
49		lcmgcdlem 15319	. . . . . . . . . . . . . . 15 |- ( ( ( 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 ) ) ) /\ ( ( ( abs ` K ) e. NN /\ ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) ) )
50	49	simprd 479	. . . . . . . . . . . . . 14 |- ( ( ( abs ` M ) e. NN /\ ( abs ` N ) e. NN ) -> ( ( ( abs ` K ) e. NN /\ ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) )
51	48, 50	sylani 686	. . . . . . . . . . . . 13 |- ( ( ( abs ` M ) e. NN /\ ( abs ` N ) e. NN ) -> ( ( ( K e. ZZ /\ K =/= 0 ) /\ ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) )
52	46, 47, 51	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( ( K e. ZZ /\ K =/= 0 ) /\ ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) )
53	52	expdimp 453	. . . . . . . . . . 11 |- ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) )
54		dvdsabsb 15001	. . . . . . . . . . . . . . . . . . 19 |- ( ( M e. ZZ /\ K e. ZZ ) -> ( M || K <-> M || ( abs ` K ) ) )
55		zabscl 14053	. . . . . . . . . . . . . . . . . . . 20 |- ( K e. ZZ -> ( abs ` K ) e. ZZ )
56		absdvdsb 15000	. . . . . . . . . . . . . . . . . . . 20 |- ( ( M e. ZZ /\ ( abs ` K ) e. ZZ ) -> ( M || ( abs ` K ) <-> ( abs ` M ) || ( abs ` K ) ) )
57	55, 56	sylan2 491	. . . . . . . . . . . . . . . . . . 19 |- ( ( M e. ZZ /\ K e. ZZ ) -> ( M || ( abs ` K ) <-> ( abs ` M ) || ( abs ` K ) ) )
58	54, 57	bitrd 268	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. ZZ /\ K e. ZZ ) -> ( M || K <-> ( abs ` M ) || ( abs ` K ) ) )
59	58	adantlr 751	. . . . . . . . . . . . . . . . 17 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( M || K <-> ( abs ` M ) || ( abs ` K ) ) )
60		dvdsabsb 15001	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N || K <-> N || ( abs ` K ) ) )
61		absdvdsb 15000	. . . . . . . . . . . . . . . . . . . 20 |- ( ( N e. ZZ /\ ( abs ` K ) e. ZZ ) -> ( N || ( abs ` K ) <-> ( abs ` N ) || ( abs ` K ) ) )
62	55, 61	sylan2 491	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N || ( abs ` K ) <-> ( abs ` N ) || ( abs ` K ) ) )
63	60, 62	bitrd 268	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. ZZ /\ K e. ZZ ) -> ( N || K <-> ( abs ` N ) || ( abs ` K ) ) )
64	63	adantll 750	. . . . . . . . . . . . . . . . 17 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( N || K <-> ( abs ` N ) || ( abs ` K ) ) )
65	59, 64	anbi12d 747	. . . . . . . . . . . . . . . 16 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( M || K /\ N || K ) <-> ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) ) )
66	65	bicomd 213	. . . . . . . . . . . . . . 15 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) <-> ( M || K /\ N || K ) ) )
67		lcmabs 15318	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) lcm ( abs ` N ) ) = ( M lcm N ) )
68	67	breq1d 4663	. . . . . . . . . . . . . . . . 17 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) <-> ( M lcm N ) || ( abs ` K ) ) )
69	68	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) <-> ( M lcm N ) || ( abs ` K ) ) )
70		dvdsabsb 15001	. . . . . . . . . . . . . . . . 17 |- ( ( ( M lcm N ) e. ZZ /\ K e. ZZ ) -> ( ( M lcm N ) || K <-> ( M lcm N ) || ( abs ` K ) ) )
71	30, 70	sylan 488	. . . . . . . . . . . . . . . 16 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( M lcm N ) || K <-> ( M lcm N ) || ( abs ` K ) ) )
72	69, 71	bitr4d 271	. . . . . . . . . . . . . . 15 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) <-> ( M lcm N ) || K ) )
73	66, 72	imbi12d 334	. . . . . . . . . . . . . 14 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) <-> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
74	73	adantrr 753	. . . . . . . . . . . . 13 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) <-> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
75	74	adantllr 755	. . . . . . . . . . . 12 |- ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ N e. ZZ ) /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) <-> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
76	75	adantlrr 757	. . . . . . . . . . 11 |- ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( ( ( abs ` M ) || ( abs ` K ) /\ ( abs ` N ) || ( abs ` K ) ) -> ( ( abs ` M ) lcm ( abs ` N ) ) || ( abs ` K ) ) <-> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
77	53, 76	mpbid 222	. . . . . . . . . 10 |- ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ ( K e. ZZ /\ K =/= 0 ) ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
78	77	anassrs 680	. . . . . . . . 9 |- ( ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ K e. ZZ ) /\ K =/= 0 ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
79	45, 78	pm2.61dane 2881	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) /\ K e. ZZ ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )
80	79	ex 450	. . . . . . 7 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( K e. ZZ -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
81	80	an4s 869	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( K e. ZZ -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
82	28, 81	sylan2br 493	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( K e. ZZ -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
83	82	impancom 456	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( -. ( M = 0 \/ N = 0 ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
84	83	3impa 1259	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( -. ( M = 0 \/ N = 0 ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
85	84	3comr 1273	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 \/ N = 0 ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) ) )
86	16, 27, 85	ecase3d 984	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || K /\ N || K ) -> ( M lcm N ) || K ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   x. cmul 9941   NNcn 11020   ZZcz 11377   abscabs 13974   || cdvds 14983   gcd cgcd 15216   lcm clcm 15301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-lcm 15303

This theorem is referenced by:  lcmdvdsb  15326  lcmftp  15349  lcmfunsnlem1  15350  lcmfunsnlem2lem1  15351  nzin  38517