Theorem lcmass 12223

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 768	. . 3 |- ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) <-> ( N = 0 \/ ( M = 0 \/ P = 0 ) ) )
2		anass 401	. . . . . 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 2745	. . . 4 |- { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } = { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) }
5	4	infeq1i 7072	. . 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 3580	. 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 12210	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N lcm M ) e. NN0 )
8	7	3adant3 1019	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm M ) e. NN0 )
9	8	nn0zd 9437	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm M ) e. ZZ )
10		simp3 1001	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> P e. ZZ )
11		lcmval 12201	. . . 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 411	. . 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 12209	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N lcm M ) = 0 <-> ( N = 0 \/ M = 0 ) ) )
14	13	3adant3 1019	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) = 0 <-> ( N = 0 \/ M = 0 ) ) )
15	14	orbi1d 792	. . . . 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 141	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) <-> ( ( N lcm M ) = 0 \/ P = 0 ) ) )
17		nnz 9336	. . . . . . . . 9 |- ( x e. NN -> x e. ZZ )
18	17	adantl 277	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> x e. ZZ )
19		simp1 999	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> N e. ZZ )
20	19	adantr 276	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> N e. ZZ )
21		simpl2 1003	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> M e. ZZ )
22		lcmdvdsb 12222	. . . . . . . 8 |- ( ( x e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( N || x /\ M || x ) <-> ( N lcm M ) || x ) )
23	18, 20, 21, 22	syl3anc 1249	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> ( ( N || x /\ M || x ) <-> ( N lcm M ) || x ) )
24	23	anbi1d 465	. . . . . 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 2748	. . . . 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 7071	. . . 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 3581	. . 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 2229	. 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 12210	. . . . . 6 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. NN0 )
30	29	3adant1 1017	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. NN0 )
31	30	nn0zd 9437	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. ZZ )
32		lcmval 12201	. . . 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 411	. . 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 12209	. . . . . . 7 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( ( M lcm P ) = 0 <-> ( M = 0 \/ P = 0 ) ) )
35	34	3adant1 1017	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M lcm P ) = 0 <-> ( M = 0 \/ P = 0 ) ) )
36	35	orbi2d 791	. . . . 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 141	. . . 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 276	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> P e. ZZ )
39		lcmdvdsb 12222	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M || x /\ P || x ) <-> ( M lcm P ) || x ) )
40	18, 21, 38, 39	syl3anc 1249	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> ( ( M || x /\ P || x ) <-> ( M lcm P ) || x ) )
41	40	anbi2d 464	. . . . . 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 2748	. . . . 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 7071	. . . 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 3581	. . 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 2229	. 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 2252	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 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   {crab 2476   ifcif 3557   class class class wbr 4029  (class class class)co 5918  infcinf 7042   RRcr 7871   0cc0 7872   < clt 8054   NNcn 8982   NN0cn0 9240   ZZcz 9317   || 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-stab 832  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-gcd 12080  df-lcm 12199

This theorem is referenced by: (None)