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Theorem dvdseq 12576
Description: If two integers divide each other, they must be equal, up to a difference in sign. (Contributed by Mario Carneiro, 30-May-2014.)
Assertion
Ref Expression
dvdseq  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  ||  N  /\  N  ||  M ) )  ->  M  =  N )

Proof of Theorem dvdseq
StepHypRef Expression
1 elnn0 9967 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simprl 732 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  ||  N )
3 nn0z 10046 . . . . . . . . 9  |-  ( M  e.  NN0  ->  M  e.  ZZ )
43ad2antrr 706 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  e.  ZZ )
5 simplr 731 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  e.  NN )
6 dvdsle 12574 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  ||  N  ->  M  <_  N )
)
74, 5, 6syl2anc 642 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  ||  N  ->  M  <_  N ) )
82, 7mpd 14 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  <_  N )
9 simprr 733 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  ||  M )
10 nnz 10045 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
1110ad2antlr 707 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  e.  ZZ )
12 nnne0 9778 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  =/=  0 )
1312ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  =/=  0 )
14 breq1 4026 . . . . . . . . . . . . . 14  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
1514biimpcd 215 . . . . . . . . . . . . 13  |-  ( M 
||  N  ->  ( M  =  0  ->  0 
||  N ) )
1615ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  =  0  ->  0 
||  N ) )
17 0dvds 12549 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
1811, 17syl 15 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  (
0  ||  N  <->  N  = 
0 ) )
1916, 18sylibd 205 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  =  0  ->  N  =  0 ) )
2019necon3ad 2482 . . . . . . . . . 10  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( N  =/=  0  ->  -.  M  =  0 ) )
2113, 20mpd 14 . . . . . . . . 9  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  -.  M  =  0 )
22 simpll 730 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  e.  NN0 )
23 elnn0 9967 . . . . . . . . . . 11  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
2422, 23sylib 188 . . . . . . . . . 10  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  e.  NN  \/  M  =  0 ) )
2524ord 366 . . . . . . . . 9  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( -.  M  e.  NN  ->  M  =  0 ) )
2621, 25mt3d 117 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  e.  NN )
27 dvdsle 12574 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN )  ->  ( N  ||  M  ->  N  <_  M )
)
2811, 26, 27syl2anc 642 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( N  ||  M  ->  N  <_  M ) )
299, 28mpd 14 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  <_  M )
30 nn0re 9974 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  RR )
31 nnre 9753 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  RR )
32 letri3 8907 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
3330, 31, 32syl2an 463 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN )  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
3433adantr 451 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  =  N  <->  ( M  <_  N  /\  N  <_  M ) ) )
358, 29, 34mpbir2and 888 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  =  N )
3635ex 423 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN )  ->  ( ( M  ||  N  /\  N  ||  M
)  ->  M  =  N ) )
37 simpr 447 . . . . 5  |-  ( ( M  ||  N  /\  N  ||  M )  ->  N  ||  M )
38 breq1 4026 . . . . . . 7  |-  ( N  =  0  ->  ( N  ||  M  <->  0  ||  M ) )
39 0dvds 12549 . . . . . . . 8  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
403, 39syl 15 . . . . . . 7  |-  ( M  e.  NN0  ->  ( 0 
||  M  <->  M  = 
0 ) )
4138, 40sylan9bbr 681 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  ( N  ||  M 
<->  M  =  0 ) )
42 simpr 447 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  N  =  0 )
4342eqeq2d 2294 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  ( M  =  N  <->  M  =  0
) )
4441, 43bitr4d 247 . . . . 5  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  ( N  ||  M 
<->  M  =  N ) )
4537, 44syl5ib 210 . . . 4  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  ( ( M 
||  N  /\  N  ||  M )  ->  M  =  N ) )
4636, 45jaodan 760 . . 3  |-  ( ( M  e.  NN0  /\  ( N  e.  NN  \/  N  =  0
) )  ->  (
( M  ||  N  /\  N  ||  M )  ->  M  =  N ) )
471, 46sylan2b 461 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  ||  N  /\  N  ||  M
)  ->  M  =  N ) )
4847imp 418 1  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  ||  N  /\  N  ||  M ) )  ->  M  =  N )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   RRcr 8736   0cc0 8737    <_ cle 8868   NNcn 9746   NN0cn0 9965   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  dvds1  12577  dvdsext  12579  mulgcd  12725  rpmulgcd2  12784  isprm6  12788  pc11  12932  pcprmpw2  12934  odeq  14865  odadd  15142  gexexlem  15144  lt6abl  15181  cyggex2  15183  ablfacrp2  15302  ablfac1c  15306  ablfac1eu  15308  znidomb  16515  dvdsmulf1o  20434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-dvds 12532
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