Theorem nzin 36737

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 14387	. . . . . . . . 9 |- ( M || n -> ( M e. ZZ /\ n e. ZZ ) )
2		dvdszrcl 14387	. . . . . . . . 9 |- ( N || n -> ( N e. ZZ /\ n e. ZZ ) )
3	1, 2	anim12i 576	. . . . . . . 8 |- ( ( M || n /\ N || n ) -> ( ( M e. ZZ /\ n e. ZZ ) /\ ( N e. ZZ /\ n e. ZZ ) ) )
4		anandir 845	. . . . . . . 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 217	. . . . . . 7 |- ( ( M || n /\ N || n ) -> ( ( M e. ZZ /\ N e. ZZ ) /\ n e. ZZ ) )
6	5	ancomd 458	. . . . . 6 |- ( ( M || n /\ N || n ) -> ( n e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) )
7		lcmdvds 14652	. . . . . . 7 |- ( ( n e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( M || n /\ N || n ) -> ( M lcm N ) || n ) )
8	7	3expb 1232	. . . . . 6 |- ( ( n e. ZZ /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( M || n /\ N || n ) -> ( M lcm N ) || n ) )
9	6, 8	mpcom 36	. . . . 5 |- ( ( M || n /\ N || n ) -> ( M lcm N ) || n )
10		elin 3608	. . . . . 6 |- ( n e. ( ( || " { M } ) i^i ( || " { N } ) ) <-> ( n e. ( || " { M } ) /\ n e. ( || " { N } ) ) )
11		reldvds 36734	. . . . . . . 8 |- Rel ||
12		elrelimasn 5198	. . . . . . . 8 |- ( Rel || -> ( n e. ( || " { M } ) <-> M || n ) )
13	11, 12	ax-mp 5	. . . . . . 7 |- ( n e. ( || " { M } ) <-> M || n )
14		elrelimasn 5198	. . . . . . . 8 |- ( Rel || -> ( n e. ( || " { N } ) <-> N || n ) )
15	11, 14	ax-mp 5	. . . . . . 7 |- ( n e. ( || " { N } ) <-> N || n )
16	13, 15	anbi12i 711	. . . . . 6 |- ( ( n e. ( || " { M } ) /\ n e. ( || " { N } ) ) <-> ( M || n /\ N || n ) )
17	10, 16	bitri 257	. . . . 5 |- ( n e. ( ( || " { M } ) i^i ( || " { N } ) ) <-> ( M || n /\ N || n ) )
18		elrelimasn 5198	. . . . . 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 274	. . . 4 |- ( n e. ( ( || " { M } ) i^i ( || " { N } ) ) -> n e. ( || " { ( M lcm N ) } ) )
21	20	ssriv 3422	. . 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 14642	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
26	23, 24, 25	syl2anc 673	. . . . 5 |- ( ph -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
27	26	simpld 466	. . . 4 |- ( ph -> M || ( M lcm N ) )
28		lcmcl 14645	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) e. NN0 )
29	23, 24, 28	syl2anc 673	. . . . . 6 |- ( ph -> ( M lcm N ) e. NN0 )
30	29	nn0zd 11061	. . . . 5 |- ( ph -> ( M lcm N ) e. ZZ )
31	30, 23	nzss 36736	. . . 4 |- ( ph -> ( ( || " { ( M lcm N ) } ) C_ ( || " { M } ) <-> M || ( M lcm N ) ) )
32	27, 31	mpbird 240	. . 3 |- ( ph -> ( || " { ( M lcm N ) } ) C_ ( || " { M } ) )
33	26	simprd 470	. . . 4 |- ( ph -> N || ( M lcm N ) )
34	30, 24	nzss 36736	. . . 4 |- ( ph -> ( ( || " { ( M lcm N ) } ) C_ ( || " { N } ) <-> N || ( M lcm N ) ) )
35	33, 34	mpbird 240	. . 3 |- ( ph -> ( || " { ( M lcm N ) } ) C_ ( || " { N } ) )
36	32, 35	ssind 3647	. 2 |- ( ph -> ( || " { ( M lcm N ) } ) C_ ( ( || " { M } ) i^i ( || " { N } ) ) )
37	22, 36	eqssd 3435	1 |- ( ph -> ( ( || " { M } ) i^i ( || " { N } ) ) = ( || " { ( M lcm N ) } ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   i^i cin 3389   C_ wss 3390   {csn 3959   class class class wbr 4395   "cima 4842   Rel wrel 4844  (class class class)co 6308   NN0cn0 10893   ZZcz 10961   || cdvds 14382   lcm clcm 14626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-lcm 14630

This theorem is referenced by:  nzprmdif  36738