Theorem lcmval 12201

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 12080 and gcdval 12096. (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 12199	. . 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 2200	. . . . . 6 |- ( x = M -> ( x = 0 <-> M = 0 ) )
4	3	orbi1d 792	. . . . 5 |- ( x = M -> ( ( x = 0 \/ y = 0 ) <-> ( M = 0 \/ y = 0 ) ) )
5		breq1 4032	. . . . . . . 8 |- ( x = M -> ( x || n <-> M || n ) )
6	5	anbi1d 465	. . . . . . 7 |- ( x = M -> ( ( x || n /\ y || n ) <-> ( M || n /\ y || n ) ) )
7	6	rabbidv 2749	. . . . . 6 |- ( x = M -> { n e. NN | ( x || n /\ y || n ) } = { n e. NN | ( M || n /\ y || n ) } )
8	7	infeq1d 7071	. . . . 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 3581	. . . 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 2200	. . . . . 6 |- ( y = N -> ( y = 0 <-> N = 0 ) )
11	10	orbi2d 791	. . . . 5 |- ( y = N -> ( ( M = 0 \/ y = 0 ) <-> ( M = 0 \/ N = 0 ) ) )
12		breq1 4032	. . . . . . . 8 |- ( y = N -> ( y || n <-> N || n ) )
13	12	anbi2d 464	. . . . . . 7 |- ( y = N -> ( ( M || n /\ y || n ) <-> ( M || n /\ N || n ) ) )
14	13	rabbidv 2749	. . . . . 6 |- ( y = N -> { n e. NN | ( M || n /\ y || n ) } = { n e. NN | ( M || n /\ N || n ) } )
15	14	infeq1d 7071	. . . . 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 3581	. . . 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 2246	. . 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 277	. 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 109	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
20		simpr 110	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
21		c0ex 8013	. . . 4 |- 0 e. _V
22	21	a1i 9	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> 0 e. _V )
23		1zzd 9344	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> 1 e. ZZ )
24		nnuz 9628	. . . . . 6 |- NN = ( ZZ>= ` 1 )
25	24	rabeqi 2753	. . . . 5 |- { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) }
26		dvdsmul1 11956	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) )
27	26	adantr 276	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M || ( M x. N ) )
28		simpll 527	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. ZZ )
29		simplr 528	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
30	28, 29	zmulcld 9445	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) e. ZZ )
31		dvdsabsb 11953	. . . . . . . 8 |- ( ( M e. ZZ /\ ( M x. N ) e. ZZ ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
32	28, 30, 31	syl2anc 411	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
33	27, 32	mpbid 147	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M || ( abs ` ( M x. N ) ) )
34		dvdsmul2 11957	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M x. N ) )
35	34	adantr 276	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N || ( M x. N ) )
36		dvdsabsb 11953	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M x. N ) e. ZZ ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
37	29, 30, 36	syl2anc 411	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
38	35, 37	mpbid 147	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N || ( abs ` ( M x. N ) ) )
39	28	zcnd 9440	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. CC )
40	29	zcnd 9440	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. CC )
41	39, 40	absmuld 11338	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
42		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ N = 0 ) )
43		ioran 753	. . . . . . . . . . . . 13 |- ( -. ( M = 0 \/ N = 0 ) <-> ( -. M = 0 /\ -. N = 0 ) )
44	42, 43	sylib 122	. . . . . . . . . . . 12 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( -. M = 0 /\ -. N = 0 ) )
45	44	simpld 112	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. M = 0 )
46	45	neneqad 2443	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M =/= 0 )
47		nnabscl 11244	. . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
48	28, 46, 47	syl2anc 411	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` M ) e. NN )
49	44	simprd 114	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. N = 0 )
50	49	neneqad 2443	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N =/= 0 )
51		nnabscl 11244	. . . . . . . . . 10 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
52	29, 50, 51	syl2anc 411	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` N ) e. NN )
53	48, 52	nnmulcld 9031	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( abs ` M ) x. ( abs ` N ) ) e. NN )
54	41, 53	eqeltrd 2270	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. NN )
55		breq2 4033	. . . . . . . . 9 |- ( n = ( abs ` ( M x. N ) ) -> ( M || n <-> M || ( abs ` ( M x. N ) ) ) )
56		breq2 4033	. . . . . . . . 9 |- ( n = ( abs ` ( M x. N ) ) -> ( N || n <-> N || ( abs ` ( M x. N ) ) ) )
57	55, 56	anbi12d 473	. . . . . . . 8 |- ( n = ( abs ` ( M x. N ) ) -> ( ( M || n /\ N || n ) <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
58	57	elrab3 2917	. . . . . . 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 ) ) ) ) )
59	54, 58	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 ) ) ) ) )
60	33, 38, 59	mpbir2and 946	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. { n e. NN | ( M || n /\ N || n ) } )
61		elfzelz 10091	. . . . . . 7 |- ( n e. ( 1 ... ( abs ` ( M x. N ) ) ) -> n e. ZZ )
62		zdvdsdc 11955	. . . . . . 7 |- ( ( M e. ZZ /\ n e. ZZ ) -> DECID M || n )
63	28, 61, 62	syl2an 289	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID M || n )
64		zdvdsdc 11955	. . . . . . 7 |- ( ( N e. ZZ /\ n e. ZZ ) -> DECID N || n )
65	29, 61, 64	syl2an 289	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID N || n )
66	63, 65	dcand 934	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID ( M || n /\ N || n ) )
67	23, 25, 60, 66	infssuzcldc 12088	. . . 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 ) } )
68	67	elexd 2773	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) e. _V )
69		lcmmndc 12200	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
70	22, 68, 69	ifcldadc 3586	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) e. _V )
71	2, 18, 19, 20, 70	ovmpod 6046	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 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   =/= wne 2364   {crab 2476   _Vcvv 2760   ifcif 3557   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   e. cmpo 5920  infcinf 7042   RRcr 7871   0cc0 7872   1c1 7873   x. cmul 7877   < clt 8054   NNcn 8982   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   abscabs 11141   || cdvds 11930   lcm clcm 12198

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-lcm 12199

This theorem is referenced by:  lcmcom  12202  lcm0val  12203  lcmn0val  12204  lcmass  12223