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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.
StepHypRef 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 ) )
31, 2anim12i 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 ) ) )
53, 4sylibr 217 . . . . . . 7  |-  ( ( M  ||  n  /\  N  ||  n )  -> 
( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  n  e.  ZZ ) )
65ancomd 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 )
)
873expb 1232 . . . . . 6  |-  ( ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( M  ||  n  /\  N  ||  n )  ->  ( M lcm  N )  ||  n
) )
96, 8mpcom 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
) )
1311, 12ax-mp 5 . . . . . . 7  |-  ( n  e.  (  ||  " { M } )  <->  M  ||  n
)
14 elrelimasn 5198 . . . . . . . 8  |-  ( Rel  ||  ->  ( n  e.  (  ||  " { N } )  <->  N  ||  n
) )
1511, 14ax-mp 5 . . . . . . 7  |-  ( n  e.  (  ||  " { N } )  <->  N  ||  n
)
1613, 15anbi12i 711 . . . . . 6  |-  ( ( n  e.  (  ||  " { M } )  /\  n  e.  ( 
||  " { N }
) )  <->  ( M  ||  n  /\  N  ||  n ) )
1710, 16bitri 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 ) )
1911, 18ax-mp 5 . . . . 5  |-  ( n  e.  (  ||  " {
( M lcm  N ) } )  <->  ( M lcm  N )  ||  n )
209, 17, 193imtr4i 274 . . . 4  |-  ( n  e.  ( (  ||  " { M } )  i^i  (  ||  " { N } ) )  ->  n  e.  (  ||  " { ( M lcm  N
) } ) )
2120ssriv 3422 . . 3  |-  ( ( 
||  " { M }
)  i^i  (  ||  " { N } ) )  C_  (  ||  " { ( M lcm  N
) } )
2221a1i 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 ) ) )
2623, 24, 25syl2anc 673 . . . . 5  |-  ( ph  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
2726simpld 466 . . . 4  |-  ( ph  ->  M  ||  ( M lcm 
N ) )
28 lcmcl 14645 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
2923, 24, 28syl2anc 673 . . . . . 6  |-  ( ph  ->  ( M lcm  N )  e.  NN0 )
3029nn0zd 11061 . . . . 5  |-  ( ph  ->  ( M lcm  N )  e.  ZZ )
3130, 23nzss 36736 . . . 4  |-  ( ph  ->  ( (  ||  " {
( M lcm  N ) } )  C_  (  ||  " { M }
)  <->  M  ||  ( M lcm 
N ) ) )
3227, 31mpbird 240 . . 3  |-  ( ph  ->  (  ||  " {
( M lcm  N ) } )  C_  (  ||  " { M }
) )
3326simprd 470 . . . 4  |-  ( ph  ->  N  ||  ( M lcm 
N ) )
3430, 24nzss 36736 . . . 4  |-  ( ph  ->  ( (  ||  " {
( M lcm  N ) } )  C_  (  ||  " { N }
)  <->  N  ||  ( M lcm 
N ) ) )
3533, 34mpbird 240 . . 3  |-  ( ph  ->  (  ||  " {
( M lcm  N ) } )  C_  (  ||  " { N }
) )
3632, 35ssind 3647 . 2  |-  ( ph  ->  (  ||  " {
( M lcm  N ) } )  C_  (
(  ||  " { M } )  i^i  (  ||  " { N }
) ) )
3722, 36eqssd 3435 1  |-  ( ph  ->  ( (  ||  " { M } )  i^i  (  ||  " { N }
) )  =  ( 
||  " { ( M lcm 
N ) } ) )
Colors of variables: wff setvar class
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
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