Theorem lcmval 11939

Description: Value of the lcm operator. ( M lcm N ) is the least common multiple of M and N. If either M or N is 0, the result is defined conventionally as 0. Contrast with df-gcd 11829 and gcdval 11842. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmval	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )

Distinct variable groups:   n, M   n, N

Proof of Theorem lcmval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lcm 11937	. . 3 |- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )
2	1	a1i 9	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) ) )
3		eqeq1 2164	. . . . . 6 |- ( x = M -> ( x = 0 <-> M = 0 ) )
4	3	orbi1d 781	. . . . 5 |- ( x = M -> ( ( x = 0 \/ y = 0 ) <-> ( M = 0 \/ y = 0 ) ) )
5		breq1 3968	. . . . . . . 8 |- ( x = M -> ( x || n <-> M || n ) )
6	5	anbi1d 461	. . . . . . 7 |- ( x = M -> ( ( x || n /\ y || n ) <-> ( M || n /\ y || n ) ) )
7	6	rabbidv 2701	. . . . . 6 |- ( x = M -> { n e. NN | ( x || n /\ y || n ) } = { n e. NN | ( M || n /\ y || n ) } )
8	7	infeq1d 6956	. . . . 5 |- ( x = M -> inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) = inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) )
9	4, 8	ifbieq2d 3529	. . . 4 |- ( x = M -> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) ) )
10		eqeq1 2164	. . . . . 6 |- ( y = N -> ( y = 0 <-> N = 0 ) )
11	10	orbi2d 780	. . . . 5 |- ( y = N -> ( ( M = 0 \/ y = 0 ) <-> ( M = 0 \/ N = 0 ) ) )
12		breq1 3968	. . . . . . . 8 |- ( y = N -> ( y || n <-> N || n ) )
13	12	anbi2d 460	. . . . . . 7 |- ( y = N -> ( ( M || n /\ y || n ) <-> ( M || n /\ N || n ) ) )
14	13	rabbidv 2701	. . . . . 6 |- ( y = N -> { n e. NN | ( M || n /\ y || n ) } = { n e. NN | ( M || n /\ N || n ) } )
15	14	infeq1d 6956	. . . . 5 |- ( y = N -> inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
16	11, 15	ifbieq2d 3529	. . . 4 |- ( y = N -> if ( ( M = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
17	9, 16	sylan9eq 2210	. . 3 |- ( ( x = M /\ y = N ) -> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
18	17	adantl 275	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( x = M /\ y = N ) ) -> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
19		simpl 108	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
20		simpr 109	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
21		c0ex 7872	. . . 4 |- 0 e. _V
22	21	a1i 9	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> 0 e. _V )
23		1zzd 9194	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> 1 e. ZZ )
24		nnuz 9474	. . . . . 6 |- NN = ( ZZ>= ` 1 )
25		rabeq 2704	. . . . . 6 |- ( NN = ( ZZ>= ` 1 ) -> { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) } )
26	24, 25	ax-mp 5	. . . . 5 |- { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) }
27		dvdsmul1 11708	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) )
28	27	adantr 274	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M || ( M x. N ) )
29		simpll 519	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. ZZ )
30		simplr 520	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
31	29, 30	zmulcld 9292	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) e. ZZ )
32		dvdsabsb 11705	. . . . . . . 8 |- ( ( M e. ZZ /\ ( M x. N ) e. ZZ ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
33	29, 31, 32	syl2anc 409	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
34	28, 33	mpbid 146	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M || ( abs ` ( M x. N ) ) )
35		dvdsmul2 11709	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M x. N ) )
36	35	adantr 274	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N || ( M x. N ) )
37		dvdsabsb 11705	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M x. N ) e. ZZ ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
38	30, 31, 37	syl2anc 409	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
39	36, 38	mpbid 146	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N || ( abs ` ( M x. N ) ) )
40	29	zcnd 9287	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. CC )
41	30	zcnd 9287	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. CC )
42	40, 41	absmuld 11094	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
43		simpr 109	. . . . . . . . . . . . 13 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ N = 0 ) )
44		ioran 742	. . . . . . . . . . . . 13 |- ( -. ( M = 0 \/ N = 0 ) <-> ( -. M = 0 /\ -. N = 0 ) )
45	43, 44	sylib 121	. . . . . . . . . . . 12 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( -. M = 0 /\ -. N = 0 ) )
46	45	simpld 111	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. M = 0 )
47	46	neqned 2334	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M =/= 0 )
48		nnabscl 11000	. . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
49	29, 47, 48	syl2anc 409	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` M ) e. NN )
50	45	simprd 113	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. N = 0 )
51	50	neqned 2334	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N =/= 0 )
52		nnabscl 11000	. . . . . . . . . 10 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
53	30, 51, 52	syl2anc 409	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` N ) e. NN )
54	49, 53	nnmulcld 8882	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( abs ` M ) x. ( abs ` N ) ) e. NN )
55	42, 54	eqeltrd 2234	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. NN )
56		breq2 3969	. . . . . . . . 9 |- ( n = ( abs ` ( M x. N ) ) -> ( M || n <-> M || ( abs ` ( M x. N ) ) ) )
57		breq2 3969	. . . . . . . . 9 |- ( n = ( abs ` ( M x. N ) ) -> ( N || n <-> N || ( abs ` ( M x. N ) ) ) )
58	56, 57	anbi12d 465	. . . . . . . 8 |- ( n = ( abs ` ( M x. N ) ) -> ( ( M || n /\ N || n ) <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
59	58	elrab3 2869	. . . . . . 7 |- ( ( abs ` ( M x. N ) ) e. NN -> ( ( abs ` ( M x. N ) ) e. { n e. NN | ( M || n /\ N || n ) } <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
60	55, 59	syl 14	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( abs ` ( M x. N ) ) e. { n e. NN | ( M || n /\ N || n ) } <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
61	34, 39, 60	mpbir2and 929	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. { n e. NN | ( M || n /\ N || n ) } )
62		elfzelz 9928	. . . . . . 7 |- ( n e. ( 1 ... ( abs ` ( M x. N ) ) ) -> n e. ZZ )
63		zdvdsdc 11707	. . . . . . 7 |- ( ( M e. ZZ /\ n e. ZZ ) -> DECID M || n )
64	29, 62, 63	syl2an 287	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID M || n )
65		zdvdsdc 11707	. . . . . . 7 |- ( ( N e. ZZ /\ n e. ZZ ) -> DECID N || n )
66	30, 62, 65	syl2an 287	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID N || n )
67		dcan 919	. . . . . 6 |- (DECID M || n -> (DECID N || n -> DECID ( M || n /\ N || n ) ) )
68	64, 66, 67	sylc 62	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID ( M || n /\ N || n ) )
69	23, 26, 61, 68	infssuzcldc 11837	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) e. { n e. NN | ( M || n /\ N || n ) } )
70	69	elexd 2725	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) e. _V )
71		lcmmndc 11938	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
72	22, 70, 71	ifcldadc 3534	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) e. _V )
73	2, 18, 19, 20, 72	ovmpod 5948	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 820   = wceq 1335   e. wcel 2128   =/= wne 2327   {crab 2439   _Vcvv 2712   ifcif 3505   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   e. cmpo 5826  infcinf 6927   RRcr 7731   0cc0 7732   1c1 7733   x. cmul 7737   < clt 7912   NNcn 8833   ZZcz 9167   ZZ>=cuz 9439   ...cfz 9912   abscabs 10897   || cdvds 11683   lcm clcm 11936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-sup 6928  df-inf 6929  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fzo 10042  df-fl 10169  df-mod 10222  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-dvds 11684  df-lcm 11937

This theorem is referenced by:  lcmcom  11940  lcm0val  11941  lcmn0val  11942  lcmass  11961