Theorem lcmass 12017

Description: Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmass	|- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) lcm P ) = ( N lcm ( M lcm P ) ) )

Proof of Theorem lcmass

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orass 757	. . 3 |- ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) <-> ( N = 0 \/ ( M = 0 \/ P = 0 ) ) )
2		anass 399	. . . . . 6 |- ( ( ( N || x /\ M || x ) /\ P || x ) <-> ( N || x /\ ( M || x /\ P || x ) ) )
3	2	a1i 9	. . . . 5 |- ( x e. NN -> ( ( ( N || x /\ M || x ) /\ P || x ) <-> ( N || x /\ ( M || x /\ P || x ) ) ) )
4	3	rabbiia 2711	. . . 4 |- { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } = { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) }
5	4	infeq1i 6978	. . 3 |- inf ( { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } , RR , < ) = inf ( { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) } , RR , < )
6	1, 5	ifbieq2i 3543	. 2 |- if ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) , 0 , inf ( { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } , RR , < ) ) = if ( ( N = 0 \/ ( M = 0 \/ P = 0 ) ) , 0 , inf ( { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) } , RR , < ) )
7		lcmcl 12004	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N lcm M ) e. NN0 )
8	7	3adant3 1007	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm M ) e. NN0 )
9	8	nn0zd 9311	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm M ) e. ZZ )
10		simp3 989	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> P e. ZZ )
11		lcmval 11995	. . . 4 |- ( ( ( N lcm M ) e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) lcm P ) = if ( ( ( N lcm M ) = 0 \/ P = 0 ) , 0 , inf ( { x e. NN | ( ( N lcm M ) || x /\ P || x ) } , RR , < ) ) )
12	9, 10, 11	syl2anc 409	. . 3 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) lcm P ) = if ( ( ( N lcm M ) = 0 \/ P = 0 ) , 0 , inf ( { x e. NN | ( ( N lcm M ) || x /\ P || x ) } , RR , < ) ) )
13		lcmeq0 12003	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N lcm M ) = 0 <-> ( N = 0 \/ M = 0 ) ) )
14	13	3adant3 1007	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) = 0 <-> ( N = 0 \/ M = 0 ) ) )
15	14	orbi1d 781	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( ( N lcm M ) = 0 \/ P = 0 ) <-> ( ( N = 0 \/ M = 0 ) \/ P = 0 ) ) )
16	15	bicomd 140	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) <-> ( ( N lcm M ) = 0 \/ P = 0 ) ) )
17		nnz 9210	. . . . . . . . 9 |- ( x e. NN -> x e. ZZ )
18	17	adantl 275	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> x e. ZZ )
19		simp1 987	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> N e. ZZ )
20	19	adantr 274	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> N e. ZZ )
21		simpl2 991	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> M e. ZZ )
22		lcmdvdsb 12016	. . . . . . . 8 |- ( ( x e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( N || x /\ M || x ) <-> ( N lcm M ) || x ) )
23	18, 20, 21, 22	syl3anc 1228	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> ( ( N || x /\ M || x ) <-> ( N lcm M ) || x ) )
24	23	anbi1d 461	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> ( ( ( N || x /\ M || x ) /\ P || x ) <-> ( ( N lcm M ) || x /\ P || x ) ) )
25	24	rabbidva 2714	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } = { x e. NN | ( ( N lcm M ) || x /\ P || x ) } )
26	25	infeq1d 6977	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> inf ( { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } , RR , < ) = inf ( { x e. NN | ( ( N lcm M ) || x /\ P || x ) } , RR , < ) )
27	16, 26	ifbieq2d 3544	. . 3 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> if ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) , 0 , inf ( { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } , RR , < ) ) = if ( ( ( N lcm M ) = 0 \/ P = 0 ) , 0 , inf ( { x e. NN | ( ( N lcm M ) || x /\ P || x ) } , RR , < ) ) )
28	12, 27	eqtr4d 2201	. 2 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) lcm P ) = if ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) , 0 , inf ( { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } , RR , < ) ) )
29		lcmcl 12004	. . . . . 6 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. NN0 )
30	29	3adant1 1005	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. NN0 )
31	30	nn0zd 9311	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. ZZ )
32		lcmval 11995	. . . 4 |- ( ( N e. ZZ /\ ( M lcm P ) e. ZZ ) -> ( N lcm ( M lcm P ) ) = if ( ( N = 0 \/ ( M lcm P ) = 0 ) , 0 , inf ( { x e. NN | ( N || x /\ ( M lcm P ) || x ) } , RR , < ) ) )
33	19, 31, 32	syl2anc 409	. . 3 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm ( M lcm P ) ) = if ( ( N = 0 \/ ( M lcm P ) = 0 ) , 0 , inf ( { x e. NN | ( N || x /\ ( M lcm P ) || x ) } , RR , < ) ) )
34		lcmeq0 12003	. . . . . . 7 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( ( M lcm P ) = 0 <-> ( M = 0 \/ P = 0 ) ) )
35	34	3adant1 1005	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M lcm P ) = 0 <-> ( M = 0 \/ P = 0 ) ) )
36	35	orbi2d 780	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N = 0 \/ ( M lcm P ) = 0 ) <-> ( N = 0 \/ ( M = 0 \/ P = 0 ) ) ) )
37	36	bicomd 140	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N = 0 \/ ( M = 0 \/ P = 0 ) ) <-> ( N = 0 \/ ( M lcm P ) = 0 ) ) )
38	10	adantr 274	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> P e. ZZ )
39		lcmdvdsb 12016	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M || x /\ P || x ) <-> ( M lcm P ) || x ) )
40	18, 21, 38, 39	syl3anc 1228	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> ( ( M || x /\ P || x ) <-> ( M lcm P ) || x ) )
41	40	anbi2d 460	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> ( ( N || x /\ ( M || x /\ P || x ) ) <-> ( N || x /\ ( M lcm P ) || x ) ) )
42	41	rabbidva 2714	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) } = { x e. NN | ( N || x /\ ( M lcm P ) || x ) } )
43	42	infeq1d 6977	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> inf ( { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) } , RR , < ) = inf ( { x e. NN | ( N || x /\ ( M lcm P ) || x ) } , RR , < ) )
44	37, 43	ifbieq2d 3544	. . 3 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> if ( ( N = 0 \/ ( M = 0 \/ P = 0 ) ) , 0 , inf ( { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) } , RR , < ) ) = if ( ( N = 0 \/ ( M lcm P ) = 0 ) , 0 , inf ( { x e. NN | ( N || x /\ ( M lcm P ) || x ) } , RR , < ) ) )
45	33, 44	eqtr4d 2201	. 2 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm ( M lcm P ) ) = if ( ( N = 0 \/ ( M = 0 \/ P = 0 ) ) , 0 , inf ( { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) } , RR , < ) ) )
46	6, 28, 45	3eqtr4a 2225	1 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) lcm P ) = ( N lcm ( M lcm P ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   {crab 2448   ifcif 3520   class class class wbr 3982  (class class class)co 5842  infcinf 6948   RRcr 7752   0cc0 7753   < clt 7933   NNcn 8857   NN0cn0 9114   ZZcz 9191   || cdvds 11727   lcm clcm 11992

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876  df-lcm 11993

This theorem is referenced by: (None)