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Theorem divconjdvds 10943
Description: If a nonzero integer  M divides another integer  N, the other integer  N divided by the nonzero integer  M (i.e. the divisor conjugate of  N to  M) divides the other integer  N. Theorem 1.1(k) in [ApostolNT] p. 14. (Contributed by AV, 7-Aug-2021.)
Assertion
Ref Expression
divconjdvds  |-  ( ( M  ||  N  /\  M  =/=  0 )  -> 
( N  /  M
)  ||  N )

Proof of Theorem divconjdvds
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 dvdszrcl 10894 . . 3  |-  ( M 
||  N  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 simpll 496 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  M  e.  ZZ )
3 oveq1 5641 . . . . . . . . . 10  |-  ( m  =  M  ->  (
m  x.  ( N  /  M ) )  =  ( M  x.  ( N  /  M
) ) )
43eqeq1d 2096 . . . . . . . . 9  |-  ( m  =  M  ->  (
( m  x.  ( N  /  M ) )  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
54adantl 271 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  m  =  M )  ->  (
( m  x.  ( N  /  M ) )  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
6 zcn 8725 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
76adantl 271 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
87adantr 270 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  N  e.  CC )
9 zcn 8725 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  CC )
109adantr 270 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
1110adantr 270 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  M  e.  CC )
12 0z 8731 . . . . . . . . . . . 12  |-  0  e.  ZZ
13 zapne 8791 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  ->  ( M #  0  <->  M  =/=  0 ) )
1412, 13mpan2 416 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  ( M #  0  <->  M  =/=  0
) )
1514adantr 270 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M #  0  <->  M  =/=  0 ) )
1615biimpar 291 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  M #  0
)
178, 11, 16divcanap2d 8232 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  ( M  x.  ( N  /  M
) )  =  N )
182, 5, 17rspcedvd 2728 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  E. m  e.  ZZ  ( m  x.  ( N  /  M
) )  =  N )
1918adantr 270 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  E. m  e.  ZZ  ( m  x.  ( N  /  M
) )  =  N )
20 simpr 108 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  M  ||  N )
21 simpr 108 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  M  =/=  0 )
22 simpr 108 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2322adantr 270 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  N  e.  ZZ )
242, 21, 233jca 1123 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ ) )
2524adantr 270 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ ) )
26 dvdsval2 10892 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
2725, 26syl 14 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  ( M  ||  N  <->  ( N  /  M )  e.  ZZ ) )
2820, 27mpbid 145 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  ( N  /  M )  e.  ZZ )
2923adantr 270 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  N  e.  ZZ )
30 divides 10891 . . . . . . 7  |-  ( ( ( N  /  M
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N  /  M )  ||  N  <->  E. m  e.  ZZ  (
m  x.  ( N  /  M ) )  =  N ) )
3128, 29, 30syl2anc 403 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  (
( N  /  M
)  ||  N  <->  E. m  e.  ZZ  ( m  x.  ( N  /  M
) )  =  N ) )
3219, 31mpbird 165 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  ( N  /  M )  ||  N )
3332exp31 356 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =/=  0  ->  ( M  ||  N  ->  ( N  /  M
)  ||  N )
) )
3433com3r 78 . . 3  |-  ( M 
||  N  ->  (
( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =/=  0  ->  ( N  /  M )  ||  N
) ) )
351, 34mpd 13 . 2  |-  ( M 
||  N  ->  ( M  =/=  0  ->  ( N  /  M )  ||  N ) )
3635imp 122 1  |-  ( ( M  ||  N  /\  M  =/=  0 )  -> 
( N  /  M
)  ||  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438    =/= wne 2255   E.wrex 2360   class class class wbr 3837  (class class class)co 5634   CCcc 7327   0cc0 7329    x. cmul 7334   # cap 8034    / cdiv 8113   ZZcz 8720    || cdvds 10889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-n0 8644  df-z 8721  df-dvds 10890
This theorem is referenced by:  dvdsdivcl  10944
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