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Theorem dvdsabseq 11126
Description: If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.)
Assertion
Ref Expression
dvdsabseq  |-  ( ( M  ||  N  /\  N  ||  M )  -> 
( abs `  M
)  =  ( abs `  N ) )

Proof of Theorem dvdsabseq
StepHypRef Expression
1 dvdszrcl 11079 . . 3  |-  ( M 
||  N  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 simpr 108 . . . . . . 7  |-  ( ( M  ||  N  /\  N  ||  M )  ->  N  ||  M )
3 breq1 3848 . . . . . . . . 9  |-  ( N  =  0  ->  ( N  ||  M  <->  0  ||  M ) )
4 0dvds 11094 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
54adantr 270 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  M  <->  M  =  0 ) )
6 zcn 8755 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  M  e.  CC )
76abs00ad 10498 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  (
( abs `  M
)  =  0  <->  M  =  0 ) )
87bicomd 139 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  ( M  =  0  <->  ( abs `  M )  =  0 ) )
98adantr 270 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  <-> 
( abs `  M
)  =  0 ) )
105, 9bitrd 186 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  M  <->  ( abs `  M )  =  0 ) )
113, 10sylan9bb 450 . . . . . . . 8  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( N  ||  M  <->  ( abs `  M
)  =  0 ) )
12 fveq2 5305 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
13 abs0 10491 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1412, 13syl6eq 2136 . . . . . . . . . 10  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
1514adantr 270 . . . . . . . . 9  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  N )  =  0 )
1615eqeq2d 2099 . . . . . . . 8  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( abs `  M )  =  ( abs `  N
)  <->  ( abs `  M
)  =  0 ) )
1711, 16bitr4d 189 . . . . . . 7  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( N  ||  M  <->  ( abs `  M
)  =  ( abs `  N ) ) )
182, 17syl5ib 152 . . . . . 6  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( M  ||  N  /\  N  ||  M )  ->  ( abs `  M )  =  ( abs `  N
) ) )
1918expd 254 . . . . 5  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N ) ) ) )
2019expcom 114 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M
)  =  ( abs `  N ) ) ) ) )
21 simprl 498 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  e.  ZZ )
22 simpr 108 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2322adantl 271 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  e.  ZZ )
24 neqne 2263 . . . . . . . 8  |-  ( -.  N  =  0  ->  N  =/=  0 )
2524adantr 270 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  =/=  0 )
26 dvdsleabs2 11125 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( M  ||  N  ->  ( abs `  M )  <_ 
( abs `  N
) ) )
2721, 23, 25, 26syl3anc 1174 . . . . . 6  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  ||  N  ->  ( abs `  M
)  <_  ( abs `  N ) ) )
28 simpr 108 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  M  ||  N )  ->  M  ||  N )
29 breq1 3848 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
30 0dvds 11094 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
31 eqcom 2090 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  N )  =  0  <->  0  =  ( abs `  N ) )
32 zcn 8755 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ZZ  ->  N  e.  CC )
3332abs00ad 10498 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  ZZ  ->  (
( abs `  N
)  =  0  <->  N  =  0 ) )
3431, 33syl5rbbr 193 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  ( N  =  0  <->  0  =  ( abs `  N ) ) )
3530, 34bitrd 186 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  0  =  ( abs `  N ) ) )
3635adantl 271 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  N  <->  0  =  ( abs `  N
) ) )
3729, 36sylan9bb 450 . . . . . . . . . . . . . 14  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  <->  0  =  ( abs `  N ) ) )
38 fveq2 5305 . . . . . . . . . . . . . . . . 17  |-  ( M  =  0  ->  ( abs `  M )  =  ( abs `  0
) )
3938, 13syl6eq 2136 . . . . . . . . . . . . . . . 16  |-  ( M  =  0  ->  ( abs `  M )  =  0 )
4039adantr 270 . . . . . . . . . . . . . . 15  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  M )  =  0 )
4140eqeq1d 2096 . . . . . . . . . . . . . 14  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( abs `  M )  =  ( abs `  N
)  <->  0  =  ( abs `  N ) ) )
4237, 41bitr4d 189 . . . . . . . . . . . . 13  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  <->  ( abs `  M
)  =  ( abs `  N ) ) )
4328, 42syl5ib 152 . . . . . . . . . . . 12  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( N  ||  M  /\  M  ||  N )  ->  ( abs `  M )  =  ( abs `  N
) ) )
4443a1dd 47 . . . . . . . . . . 11  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( N  ||  M  /\  M  ||  N )  ->  (
( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) )
4544expcomd 1375 . . . . . . . . . 10  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_ 
( abs `  N
)  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) )
4645expcom 114 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_  ( abs `  N )  -> 
( abs `  M
)  =  ( abs `  N ) ) ) ) ) )
4722adantl 271 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  e.  ZZ )
48 simprl 498 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  e.  ZZ )
49 neqne 2263 . . . . . . . . . . . . . 14  |-  ( -.  M  =  0  ->  M  =/=  0 )
5049adantr 270 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  =/=  0 )
51 dvdsleabs2 11125 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  ( N  ||  M  ->  ( abs `  N )  <_ 
( abs `  M
) ) )
5247, 48, 50, 51syl3anc 1174 . . . . . . . . . . . 12  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( N  ||  M  ->  ( abs `  N
)  <_  ( abs `  M ) ) )
53 eqcom 2090 . . . . . . . . . . . . . . . 16  |-  ( ( abs `  M )  =  ( abs `  N
)  <->  ( abs `  N
)  =  ( abs `  M ) )
5432abscld 10614 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
556abscld 10614 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  RR )
56 letri3 7566 . . . . . . . . . . . . . . . . 17  |-  ( ( ( abs `  N
)  e.  RR  /\  ( abs `  M )  e.  RR )  -> 
( ( abs `  N
)  =  ( abs `  M )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5754, 55, 56syl2anr 284 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  N
)  =  ( abs `  M )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5853, 57syl5bb 190 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  =  ( abs `  N )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5958biimprd 156 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  N )  <_  ( abs `  M )  /\  ( abs `  M )  <_  ( abs `  N
) )  ->  ( abs `  M )  =  ( abs `  N
) ) )
6059expd 254 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  N
)  <_  ( abs `  M )  ->  (
( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) )
6160adantl 271 . . . . . . . . . . . 12  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( abs `  N
)  <_  ( abs `  M )  ->  (
( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) )
6252, 61syld 44 . . . . . . . . . . 11  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( N  ||  M  ->  ( ( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) )
6362a1d 22 . . . . . . . . . 10  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) ) )
6463expcom 114 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  M  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_ 
( abs `  N
)  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) ) )
65 0z 8761 . . . . . . . . . . . 12  |-  0  e.  ZZ
66 zdceq 8822 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
6765, 66mpan2 416 . . . . . . . . . . 11  |-  ( M  e.  ZZ  -> DECID  M  =  0
)
68 exmiddc 782 . . . . . . . . . . 11  |-  (DECID  M  =  0  ->  ( M  =  0  \/  -.  M  =  0 ) )
6967, 68syl 14 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  =  0  \/  -.  M  =  0
) )
7069adantr 270 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  \/  -.  M  =  0 ) )
7146, 64, 70mpjaod 673 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) ) )
7271com34 82 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( ( abs `  M
)  <_  ( abs `  N )  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N
) ) ) ) )
7372adantl 271 . . . . . 6  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  ||  N  ->  ( ( abs `  M
)  <_  ( abs `  N )  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N
) ) ) ) )
7427, 73mpdd 40 . . . . 5  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M
)  =  ( abs `  N ) ) ) )
7574expcom 114 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  N  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) )
76 zdceq 8822 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
7765, 76mpan2 416 . . . . . 6  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
78 exmiddc 782 . . . . . 6  |-  (DECID  N  =  0  ->  ( N  =  0  \/  -.  N  =  0 ) )
7977, 78syl 14 . . . . 5  |-  ( N  e.  ZZ  ->  ( N  =  0  \/  -.  N  =  0
) )
8079adantl 271 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  \/  -.  N  =  0 ) )
8120, 75, 80mpjaod 673 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M
)  =  ( abs `  N ) ) ) )
821, 81mpcom 36 . 2  |-  ( M 
||  N  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N
) ) )
8382imp 122 1  |-  ( ( M  ||  N  /\  N  ||  M )  -> 
( abs `  M
)  =  ( abs `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3845   ` cfv 5015   RRcr 7349   0cc0 7350    <_ cle 7523   ZZcz 8750   abscabs 10430    || cdvds 11074
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-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  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 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-iseq 9853  df-seq3 9854  df-exp 9955  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-dvds 11075
This theorem is referenced by:  dvdseq  11127
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