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Theorem divconjdvds 11395
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 11346 . . 3  |-  ( M 
||  N  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 simpll 501 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  M  e.  ZZ )
3 oveq1 5735 . . . . . . . . . 10  |-  ( m  =  M  ->  (
m  x.  ( N  /  M ) )  =  ( M  x.  ( N  /  M
) ) )
43eqeq1d 2123 . . . . . . . . 9  |-  ( m  =  M  ->  (
( m  x.  ( N  /  M ) )  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
54adantl 273 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  m  =  M )  ->  (
( m  x.  ( N  /  M ) )  =  N  <->  ( M  x.  ( N  /  M
) )  =  N ) )
6 zcn 8963 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
76adantl 273 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  CC )
87adantr 272 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  N  e.  CC )
9 zcn 8963 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  CC )
109adantr 272 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  CC )
1110adantr 272 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  M  e.  CC )
12 0z 8969 . . . . . . . . . . . 12  |-  0  e.  ZZ
13 zapne 9029 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  ->  ( M #  0  <->  M  =/=  0 ) )
1412, 13mpan2 419 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  ( M #  0  <->  M  =/=  0
) )
1514adantr 272 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M #  0  <->  M  =/=  0 ) )
1615biimpar 293 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  M #  0
)
178, 11, 16divcanap2d 8465 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  ( M  x.  ( N  /  M
) )  =  N )
182, 5, 17rspcedvd 2766 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  E. m  e.  ZZ  ( m  x.  ( N  /  M
) )  =  N )
1918adantr 272 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  E. m  e.  ZZ  ( m  x.  ( N  /  M
) )  =  N )
20 simpr 109 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  M  ||  N )
21 simpr 109 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  M  =/=  0 )
22 simpr 109 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2322adantr 272 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  N  e.  ZZ )
242, 21, 233jca 1144 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ ) )
2524adantr 272 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  ( M  e.  ZZ  /\  M  =/=  0  /\  N  e.  ZZ ) )
26 dvdsval2 11344 . . . . . . . . 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 146 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  ( N  /  M )  e.  ZZ )
2923adantr 272 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  N  e.  ZZ )
30 divides 11343 . . . . . . 7  |-  ( ( ( N  /  M
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( N  /  M )  ||  N  <->  E. m  e.  ZZ  (
m  x.  ( N  /  M ) )  =  N ) )
3128, 29, 30syl2anc 406 . . . . . 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 166 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0 )  /\  M  ||  N )  ->  ( N  /  M )  ||  N )
3332exp31 359 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =/=  0  ->  ( M  ||  N  ->  ( N  /  M
)  ||  N )
) )
3433com3r 79 . . 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 123 1  |-  ( ( M  ||  N  /\  M  =/=  0 )  -> 
( N  /  M
)  ||  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463    =/= wne 2282   E.wrex 2391   class class class wbr 3895  (class class class)co 5728   CCcc 7545   0cc0 7547    x. cmul 7552   # cap 8261    / cdiv 8345   ZZcz 8958    || cdvds 11341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-n0 8882  df-z 8959  df-dvds 11342
This theorem is referenced by:  dvdsdivcl  11396
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