Theorem lcmass 15327

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 546	. . 3 |- ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) <-> ( N = 0 \/ ( M = 0 \/ P = 0 ) ) )
2		anass 681	. . . . . 6 |- ( ( ( N || x /\ M || x ) /\ P || x ) <-> ( N || x /\ ( M || x /\ P || x ) ) )
3	2	a1i 11	. . . . 5 |- ( x e. NN -> ( ( ( N || x /\ M || x ) /\ P || x ) <-> ( N || x /\ ( M || x /\ P || x ) ) ) )
4	3	rabbiia 3185	. . . 4 |- { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } = { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) }
5	4	infeq1i 8384	. . 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 4110	. 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 15314	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N lcm M ) e. NN0 )
8	7	3adant3 1081	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm M ) e. NN0 )
9	8	nn0zd 11480	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm M ) e. ZZ )
10		simp3 1063	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> P e. ZZ )
11		lcmval 15305	. . . 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 693	. . 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 15313	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N lcm M ) = 0 <-> ( N = 0 \/ M = 0 ) ) )
14	13	3adant3 1081	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) = 0 <-> ( N = 0 \/ M = 0 ) ) )
15	14	orbi1d 739	. . . . 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 213	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( ( N = 0 \/ M = 0 ) \/ P = 0 ) <-> ( ( N lcm M ) = 0 \/ P = 0 ) ) )
17		nnz 11399	. . . . . . . . 9 |- ( x e. NN -> x e. ZZ )
18	17	adantl 482	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> x e. ZZ )
19		simp1 1061	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> N e. ZZ )
20	19	adantr 481	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> N e. ZZ )
21		simpl2 1065	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> M e. ZZ )
22		lcmdvdsb 15326	. . . . . . . 8 |- ( ( x e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( N || x /\ M || x ) <-> ( N lcm M ) || x ) )
23	18, 20, 21, 22	syl3anc 1326	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> ( ( N || x /\ M || x ) <-> ( N lcm M ) || x ) )
24	23	anbi1d 741	. . . . . 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 3188	. . . . 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 8383	. . . 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 4111	. . 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 2659	. 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 15314	. . . . . 6 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. NN0 )
30	29	3adant1 1079	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. NN0 )
31	30	nn0zd 11480	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. ZZ )
32		lcmval 15305	. . . 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 693	. . 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 15313	. . . . . . 7 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( ( M lcm P ) = 0 <-> ( M = 0 \/ P = 0 ) ) )
35	34	3adant1 1079	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M lcm P ) = 0 <-> ( M = 0 \/ P = 0 ) ) )
36	35	orbi2d 738	. . . . 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 213	. . . 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 481	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> P e. ZZ )
39		lcmdvdsb 15326	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M || x /\ P || x ) <-> ( M lcm P ) || x ) )
40	18, 21, 38, 39	syl3anc 1326	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> ( ( M || x /\ P || x ) <-> ( M lcm P ) || x ) )
41	40	anbi2d 740	. . . . . 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 3188	. . . . 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 8383	. . . 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 4111	. . 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 2659	. 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 2682	1 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) lcm P ) = ( N lcm ( M lcm P ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   ifcif 4086   class class class wbr 4653  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   < clt 10074   NNcn 11020   NN0cn0 11292   ZZcz 11377   || cdvds 14983   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:  lcmfunsnlem2lem2  15352  lcmfun  15358