Theorem nzin 38517

Description: The intersection of the set of multiples of m, mℤ, and those of n, nℤ, is the set of multiples of their least common multiple. Roughly Lemma 2.1(c) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5 and Problem 1(b) of https://people.math.binghamton.edu/mazur/teach/40107/40107h16sol.pdf p. 1, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Hypotheses

Ref	Expression
nzin.m	|- ( ph -> M e. ZZ )
nzin.n	|- ( ph -> N e. ZZ )

Assertion

Ref	Expression
nzin	|- ( ph -> ( ( || " { M } ) i^i ( || " { N } ) ) = ( || " { ( M lcm N ) } ) )

Proof of Theorem nzin

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdszrcl 14988	. . . . . . . . 9 |- ( M || n -> ( M e. ZZ /\ n e. ZZ ) )
2		dvdszrcl 14988	. . . . . . . . 9 |- ( N || n -> ( N e. ZZ /\ n e. ZZ ) )
3	1, 2	anim12i 590	. . . . . . . 8 |- ( ( M || n /\ N || n ) -> ( ( M e. ZZ /\ n e. ZZ ) /\ ( N e. ZZ /\ n e. ZZ ) ) )
4		anandir 872	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ n e. ZZ ) <-> ( ( M e. ZZ /\ n e. ZZ ) /\ ( N e. ZZ /\ n e. ZZ ) ) )
5	3, 4	sylibr 224	. . . . . . 7 |- ( ( M || n /\ N || n ) -> ( ( M e. ZZ /\ N e. ZZ ) /\ n e. ZZ ) )
6	5	ancomd 467	. . . . . 6 |- ( ( M || n /\ N || n ) -> ( n e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) )
7		lcmdvds 15321	. . . . . . 7 |- ( ( n e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || n /\ N || n ) -> ( M lcm N ) || n ) )
8	7	3expb 1266	. . . . . 6 |- ( ( n e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( M || n /\ N || n ) -> ( M lcm N ) || n ) )
9	6, 8	mpcom 38	. . . . 5 |- ( ( M || n /\ N || n ) -> ( M lcm N ) || n )
10		elin 3796	. . . . . 6 |- ( n e. ( ( || " { M } ) i^i ( || " { N } ) ) <-> ( n e. ( || " { M } ) /\ n e. ( || " { N } ) ) )
11		reldvds 38514	. . . . . . . 8 |- Rel ||
12		elrelimasn 5489	. . . . . . . 8 |- ( Rel || -> ( n e. ( || " { M } ) <-> M || n ) )
13	11, 12	ax-mp 5	. . . . . . 7 |- ( n e. ( || " { M } ) <-> M || n )
14		elrelimasn 5489	. . . . . . . 8 |- ( Rel || -> ( n e. ( || " { N } ) <-> N || n ) )
15	11, 14	ax-mp 5	. . . . . . 7 |- ( n e. ( || " { N } ) <-> N || n )
16	13, 15	anbi12i 733	. . . . . 6 |- ( ( n e. ( || " { M } ) /\ n e. ( || " { N } ) ) <-> ( M || n /\ N || n ) )
17	10, 16	bitri 264	. . . . 5 |- ( n e. ( ( || " { M } ) i^i ( || " { N } ) ) <-> ( M || n /\ N || n ) )
18		elrelimasn 5489	. . . . . 6 |- ( Rel || -> ( n e. ( || " { ( M lcm N ) } ) <-> ( M lcm N ) || n ) )
19	11, 18	ax-mp 5	. . . . 5 |- ( n e. ( || " { ( M lcm N ) } ) <-> ( M lcm N ) || n )
20	9, 17, 19	3imtr4i 281	. . . 4 |- ( n e. ( ( || " { M } ) i^i ( || " { N } ) ) -> n e. ( || " { ( M lcm N ) } ) )
21	20	ssriv 3607	. . 3 |- ( ( || " { M } ) i^i ( || " { N } ) ) C_ ( || " { ( M lcm N ) } )
22	21	a1i 11	. 2 |- ( ph -> ( ( || " { M } ) i^i ( || " { N } ) ) C_ ( || " { ( M lcm N ) } ) )
23		nzin.m	. . . . . 6 |- ( ph -> M e. ZZ )
24		nzin.n	. . . . . 6 |- ( ph -> N e. ZZ )
25		dvdslcm 15311	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
26	23, 24, 25	syl2anc 693	. . . . 5 |- ( ph -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
27	26	simpld 475	. . . 4 |- ( ph -> M || ( M lcm N ) )
28		lcmcl 15314	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
29	23, 24, 28	syl2anc 693	. . . . . 6 |- ( ph -> ( M lcm N ) e. NN0 )
30	29	nn0zd 11480	. . . . 5 |- ( ph -> ( M lcm N ) e. ZZ )
31	30, 23	nzss 38516	. . . 4 |- ( ph -> ( ( || " { ( M lcm N ) } ) C_ ( || " { M } ) <-> M || ( M lcm N ) ) )
32	27, 31	mpbird 247	. . 3 |- ( ph -> ( || " { ( M lcm N ) } ) C_ ( || " { M } ) )
33	26	simprd 479	. . . 4 |- ( ph -> N || ( M lcm N ) )
34	30, 24	nzss 38516	. . . 4 |- ( ph -> ( ( || " { ( M lcm N ) } ) C_ ( || " { N } ) <-> N || ( M lcm N ) ) )
35	33, 34	mpbird 247	. . 3 |- ( ph -> ( || " { ( M lcm N ) } ) C_ ( || " { N } ) )
36	32, 35	ssind 3837	. 2 |- ( ph -> ( || " { ( M lcm N ) } ) C_ ( ( || " { M } ) i^i ( || " { N } ) ) )
37	22, 36	eqssd 3620	1 |- ( ph -> ( ( || " { M } ) i^i ( || " { N } ) ) = ( || " { ( M lcm N ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   "cima 5117   Rel wrel 5119  (class class class)co 6650   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:  nzprmdif  38518