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Theorem dvdseq 13583
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 10584 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simprl 755 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  ||  N )
3 nn0z 10672 . . . . . . . . 9  |-  ( M  e.  NN0  ->  M  e.  ZZ )
43ad2antrr 725 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  e.  ZZ )
5 simplr 754 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  e.  NN )
6 dvdsle 13581 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( M  ||  N  ->  M  <_  N )
)
74, 5, 6syl2anc 661 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  ||  N  ->  M  <_  N ) )
82, 7mpd 15 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  <_  N )
9 simprr 756 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  ||  M )
10 nnz 10671 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
1110ad2antlr 726 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  e.  ZZ )
12 nnne0 10357 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  =/=  0 )
1312ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  =/=  0 )
14 breq1 4298 . . . . . . . . . . . . . 14  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
1514biimpcd 224 . . . . . . . . . . . . 13  |-  ( M 
||  N  ->  ( M  =  0  ->  0 
||  N ) )
1615ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  =  0  ->  0 
||  N ) )
17 0dvds 13556 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
1811, 17syl 16 . . . . . . . . . . . 12  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  (
0  ||  N  <->  N  = 
0 ) )
1916, 18sylibd 214 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  =  0  ->  N  =  0 ) )
2019necon3ad 2647 . . . . . . . . . 10  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( N  =/=  0  ->  -.  M  =  0 ) )
2113, 20mpd 15 . . . . . . . . 9  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  -.  M  =  0 )
22 simpll 753 . . . . . . . . . . 11  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  e.  NN0 )
23 elnn0 10584 . . . . . . . . . . 11  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
2422, 23sylib 196 . . . . . . . . . 10  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  e.  NN  \/  M  =  0 ) )
2524ord 377 . . . . . . . . 9  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( -.  M  e.  NN  ->  M  =  0 ) )
2621, 25mt3d 125 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  e.  NN )
27 dvdsle 13581 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  NN )  ->  ( N  ||  M  ->  N  <_  M )
)
2811, 26, 27syl2anc 661 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( N  ||  M  ->  N  <_  M ) )
299, 28mpd 15 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  N  <_  M )
30 nn0re 10591 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  RR )
31 nnre 10332 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  RR )
32 letri3 9463 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
3330, 31, 32syl2an 477 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN )  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
3433adantr 465 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  ( M  =  N  <->  ( M  <_  N  /\  N  <_  M ) ) )
358, 29, 34mpbir2and 913 . . . . 5  |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( M  ||  N  /\  N  ||  M
) )  ->  M  =  N )
3635ex 434 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN )  ->  ( ( M  ||  N  /\  N  ||  M
)  ->  M  =  N ) )
37 simpr 461 . . . . 5  |-  ( ( M  ||  N  /\  N  ||  M )  ->  N  ||  M )
38 breq1 4298 . . . . . . 7  |-  ( N  =  0  ->  ( N  ||  M  <->  0  ||  M ) )
39 0dvds 13556 . . . . . . . 8  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
403, 39syl 16 . . . . . . 7  |-  ( M  e.  NN0  ->  ( 0 
||  M  <->  M  = 
0 ) )
4138, 40sylan9bbr 700 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  ( N  ||  M 
<->  M  =  0 ) )
42 simpr 461 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  N  =  0 )
4342eqeq2d 2454 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  ( M  =  N  <->  M  =  0
) )
4441, 43bitr4d 256 . . . . 5  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  ( N  ||  M 
<->  M  =  N ) )
4537, 44syl5ib 219 . . . 4  |-  ( ( M  e.  NN0  /\  N  =  0 )  ->  ( ( M 
||  N  /\  N  ||  M )  ->  M  =  N ) )
4636, 45jaodan 783 . . 3  |-  ( ( M  e.  NN0  /\  ( N  e.  NN  \/  N  =  0
) )  ->  (
( M  ||  N  /\  N  ||  M )  ->  M  =  N ) )
471, 46sylan2b 475 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  ||  N  /\  N  ||  M
)  ->  M  =  N ) )
4847imp 429 1  |-  ( ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  ||  N  /\  N  ||  M ) )  ->  M  =  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2609   class class class wbr 4295   RRcr 9284   0cc0 9285    <_ cle 9422   NNcn 10325   NN0cn0 10582   ZZcz 10649    || cdivides 13538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-n0 10583  df-z 10650  df-dvds 13539
This theorem is referenced by:  dvds1  13584  dvdsext  13587  mulgcd  13733  rpmulgcd2  13794  isprm6  13798  pc11  13949  pcprmpw2  13951  odeq  16056  odadd  16335  gexexlem  16337  lt6abl  16374  cyggex2  16376  ablfacrp2  16571  ablfac1c  16575  ablfac1eu  16577  znidomb  17997  dvdsmulf1o  22537
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