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Theorem dvdsabseq 11867
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 11813 . . 3  |-  ( M 
||  N  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 simpr 110 . . . . . . 7  |-  ( ( M  ||  N  /\  N  ||  M )  ->  N  ||  M )
3 breq1 4018 . . . . . . . . 9  |-  ( N  =  0  ->  ( N  ||  M  <->  0  ||  M ) )
4 0dvds 11832 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
54adantr 276 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  M  <->  M  =  0 ) )
6 zcn 9272 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  M  e.  CC )
76abs00ad 11088 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  (
( abs `  M
)  =  0  <->  M  =  0 ) )
87bicomd 141 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  ( M  =  0  <->  ( abs `  M )  =  0 ) )
98adantr 276 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  <-> 
( abs `  M
)  =  0 ) )
105, 9bitrd 188 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  M  <->  ( abs `  M )  =  0 ) )
113, 10sylan9bb 462 . . . . . . . 8  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( N  ||  M  <->  ( abs `  M
)  =  0 ) )
12 fveq2 5527 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
13 abs0 11081 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1412, 13eqtrdi 2236 . . . . . . . . . 10  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
1514adantr 276 . . . . . . . . 9  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  N )  =  0 )
1615eqeq2d 2199 . . . . . . . 8  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( abs `  M )  =  ( abs `  N
)  <->  ( abs `  M
)  =  0 ) )
1711, 16bitr4d 191 . . . . . . 7  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( N  ||  M  <->  ( abs `  M
)  =  ( abs `  N ) ) )
182, 17imbitrid 154 . . . . . 6  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( M  ||  N  /\  N  ||  M )  ->  ( abs `  M )  =  ( abs `  N
) ) )
1918expd 258 . . . . 5  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N ) ) ) )
2019expcom 116 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M
)  =  ( abs `  N ) ) ) ) )
21 simprl 529 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  e.  ZZ )
22 simpr 110 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2322adantl 277 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  e.  ZZ )
24 neqne 2365 . . . . . . . 8  |-  ( -.  N  =  0  ->  N  =/=  0 )
2524adantr 276 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  =/=  0 )
26 dvdsleabs2 11866 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( M  ||  N  ->  ( abs `  M )  <_ 
( abs `  N
) ) )
2721, 23, 25, 26syl3anc 1248 . . . . . 6  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  ||  N  ->  ( abs `  M
)  <_  ( abs `  N ) ) )
28 simpr 110 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  M  ||  N )  ->  M  ||  N )
29 breq1 4018 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
30 0dvds 11832 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
31 zcn 9272 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ZZ  ->  N  e.  CC )
3231abs00ad 11088 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  ZZ  ->  (
( abs `  N
)  =  0  <->  N  =  0 ) )
33 eqcom 2189 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  N )  =  0  <->  0  =  ( abs `  N ) )
3432, 33bitr3di 195 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  ( N  =  0  <->  0  =  ( abs `  N ) ) )
3530, 34bitrd 188 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  0  =  ( abs `  N ) ) )
3635adantl 277 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  N  <->  0  =  ( abs `  N
) ) )
3729, 36sylan9bb 462 . . . . . . . . . . . . . 14  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  <->  0  =  ( abs `  N ) ) )
38 fveq2 5527 . . . . . . . . . . . . . . . . 17  |-  ( M  =  0  ->  ( abs `  M )  =  ( abs `  0
) )
3938, 13eqtrdi 2236 . . . . . . . . . . . . . . . 16  |-  ( M  =  0  ->  ( abs `  M )  =  0 )
4039adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  M )  =  0 )
4140eqeq1d 2196 . . . . . . . . . . . . . 14  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( abs `  M )  =  ( abs `  N
)  <->  0  =  ( abs `  N ) ) )
4237, 41bitr4d 191 . . . . . . . . . . . . 13  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  <->  ( abs `  M
)  =  ( abs `  N ) ) )
4328, 42imbitrid 154 . . . . . . . . . . . 12  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( N  ||  M  /\  M  ||  N )  ->  ( abs `  M )  =  ( abs `  N
) ) )
4443a1dd 48 . . . . . . . . . . 11  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( N  ||  M  /\  M  ||  N )  ->  (
( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) )
4544expcomd 1451 . . . . . . . . . 10  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_ 
( abs `  N
)  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) )
4645expcom 116 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_  ( abs `  N )  -> 
( abs `  M
)  =  ( abs `  N ) ) ) ) ) )
4722adantl 277 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  e.  ZZ )
48 simprl 529 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  e.  ZZ )
49 neqne 2365 . . . . . . . . . . . . . 14  |-  ( -.  M  =  0  ->  M  =/=  0 )
5049adantr 276 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  =/=  0 )
51 dvdsleabs2 11866 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  ( N  ||  M  ->  ( abs `  N )  <_ 
( abs `  M
) ) )
5247, 48, 50, 51syl3anc 1248 . . . . . . . . . . . 12  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( N  ||  M  ->  ( abs `  N
)  <_  ( abs `  M ) ) )
53 eqcom 2189 . . . . . . . . . . . . . . . 16  |-  ( ( abs `  M )  =  ( abs `  N
)  <->  ( abs `  N
)  =  ( abs `  M ) )
5431abscld 11204 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
556abscld 11204 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  RR )
56 letri3 8052 . . . . . . . . . . . . . . . . 17  |-  ( ( ( abs `  N
)  e.  RR  /\  ( abs `  M )  e.  RR )  -> 
( ( abs `  N
)  =  ( abs `  M )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5754, 55, 56syl2anr 290 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  N
)  =  ( abs `  M )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5853, 57bitrid 192 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  =  ( abs `  N )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5958biimprd 158 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  N )  <_  ( abs `  M )  /\  ( abs `  M )  <_  ( abs `  N
) )  ->  ( abs `  M )  =  ( abs `  N
) ) )
6059expd 258 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  N
)  <_  ( abs `  M )  ->  (
( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) )
6160adantl 277 . . . . . . . . . . . 12  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( abs `  N
)  <_  ( abs `  M )  ->  (
( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) )
6252, 61syld 45 . . . . . . . . . . 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 116 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  M  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_ 
( abs `  N
)  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) ) )
65 0z 9278 . . . . . . . . . . . 12  |-  0  e.  ZZ
66 zdceq 9342 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
6765, 66mpan2 425 . . . . . . . . . . 11  |-  ( M  e.  ZZ  -> DECID  M  =  0
)
68 exmiddc 837 . . . . . . . . . . 11  |-  (DECID  M  =  0  ->  ( M  =  0  \/  -.  M  =  0 ) )
6967, 68syl 14 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  =  0  \/  -.  M  =  0
) )
7069adantr 276 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  \/  -.  M  =  0 ) )
7146, 64, 70mpjaod 719 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) ) )
7271com34 83 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  ->  ( ( abs `  M
)  <_  ( abs `  N )  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N
) ) ) ) )
7372adantl 277 . . . . . 6  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  ||  N  ->  ( ( abs `  M
)  <_  ( abs `  N )  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N
) ) ) ) )
7427, 73mpdd 41 . . . . 5  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M
)  =  ( abs `  N ) ) ) )
7574expcom 116 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  N  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) )
76 zdceq 9342 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
7765, 76mpan2 425 . . . . . 6  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
78 exmiddc 837 . . . . . 6  |-  (DECID  N  =  0  ->  ( N  =  0  \/  -.  N  =  0 ) )
7977, 78syl 14 . . . . 5  |-  ( N  e.  ZZ  ->  ( N  =  0  \/  -.  N  =  0
) )
8079adantl 277 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  \/  -.  N  =  0 ) )
8120, 75, 80mpjaod 719 . . 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 124 1  |-  ( ( M  ||  N  /\  N  ||  M )  -> 
( abs `  M
)  =  ( abs `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1363    e. wcel 2158    =/= wne 2357   class class class wbr 4015   ` cfv 5228   RRcr 7824   0cc0 7825    <_ cle 8007   ZZcz 9267   abscabs 11020    || cdvds 11808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-dvds 11809
This theorem is referenced by:  dvdseq  11868
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