Theorem lcmass 12407

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 769	. . 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 2757	. . . 4 |- { x e. NN | ( ( N || x /\ M || x ) /\ P || x ) } = { x e. NN | ( N || x /\ ( M || x /\ P || x ) ) }
5	4	infeq1i 7115	. . 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 3594	. 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 12394	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N lcm M ) e. NN0 )
8	7	3adant3 1020	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm M ) e. NN0 )
9	8	nn0zd 9493	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N lcm M ) e. ZZ )
10		simp3 1002	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> P e. ZZ )
11		lcmval 12385	. . . 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 12393	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N lcm M ) = 0 <-> ( N = 0 \/ M = 0 ) ) )
14	13	3adant3 1020	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N lcm M ) = 0 <-> ( N = 0 \/ M = 0 ) ) )
15	14	orbi1d 793	. . . . 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 9391	. . . . . . . . 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 1000	. . . . . . . . 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 1004	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. NN ) -> M e. ZZ )
22		lcmdvdsb 12406	. . . . . . . 8 |- ( ( x e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( N || x /\ M || x ) <-> ( N lcm M ) || x ) )
23	18, 20, 21, 22	syl3anc 1250	. . . . . . 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 2760	. . . . 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 7114	. . . 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 3595	. . 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 2241	. 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 12394	. . . . . 6 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. NN0 )
30	29	3adant1 1018	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. NN0 )
31	30	nn0zd 9493	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M lcm P ) e. ZZ )
32		lcmval 12385	. . . 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 12393	. . . . . . 7 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( ( M lcm P ) = 0 <-> ( M = 0 \/ P = 0 ) ) )
35	34	3adant1 1018	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M lcm P ) = 0 <-> ( M = 0 \/ P = 0 ) ) )
36	35	orbi2d 792	. . . . 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 12406	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M || x /\ P || x ) <-> ( M lcm P ) || x ) )
40	18, 21, 38, 39	syl3anc 1250	. . . . . . 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 2760	. . . . 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 7114	. . . 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 3595	. . 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 2241	. 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 2264	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 710   /\ w3a 981   = wceq 1373   e. wcel 2176   {crab 2488   ifcif 3571   class class class wbr 4044  (class class class)co 5944  infcinf 7085   RRcr 7924   0cc0 7925   < clt 8107   NNcn 9036   NN0cn0 9295   ZZcz 9372   || cdvds 12098   lcm clcm 12382

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-sup 7086  df-inf 7087  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-fl 10413  df-mod 10468  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-dvds 12099  df-gcd 12275  df-lcm 12383

This theorem is referenced by: (None)