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Theorem dvdsabseq 11774
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 11722 . . 3  |-  ( M 
||  N  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 simpr 109 . . . . . . 7  |-  ( ( M  ||  N  /\  N  ||  M )  ->  N  ||  M )
3 breq1 3980 . . . . . . . . 9  |-  ( N  =  0  ->  ( N  ||  M  <->  0  ||  M ) )
4 0dvds 11741 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
54adantr 274 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  M  <->  M  =  0 ) )
6 zcn 9188 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  M  e.  CC )
76abs00ad 10997 . . . . . . . . . . . 12  |-  ( M  e.  ZZ  ->  (
( abs `  M
)  =  0  <->  M  =  0 ) )
87bicomd 140 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  ( M  =  0  <->  ( abs `  M )  =  0 ) )
98adantr 274 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  <-> 
( abs `  M
)  =  0 ) )
105, 9bitrd 187 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  M  <->  ( abs `  M )  =  0 ) )
113, 10sylan9bb 458 . . . . . . . 8  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( N  ||  M  <->  ( abs `  M
)  =  0 ) )
12 fveq2 5481 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
13 abs0 10990 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1412, 13eqtrdi 2213 . . . . . . . . . 10  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
1514adantr 274 . . . . . . . . 9  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  N )  =  0 )
1615eqeq2d 2176 . . . . . . . 8  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( abs `  M )  =  ( abs `  N
)  <->  ( abs `  M
)  =  0 ) )
1711, 16bitr4d 190 . . . . . . 7  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( N  ||  M  <->  ( abs `  M
)  =  ( abs `  N ) ) )
182, 17syl5ib 153 . . . . . 6  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( M  ||  N  /\  N  ||  M )  ->  ( abs `  M )  =  ( abs `  N
) ) )
1918expd 256 . . . . 5  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N ) ) ) )
2019expcom 115 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M
)  =  ( abs `  N ) ) ) ) )
21 simprl 521 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  e.  ZZ )
22 simpr 109 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
2322adantl 275 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  e.  ZZ )
24 neqne 2342 . . . . . . . 8  |-  ( -.  N  =  0  ->  N  =/=  0 )
2524adantr 274 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  =/=  0 )
26 dvdsleabs2 11773 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( M  ||  N  ->  ( abs `  M )  <_ 
( abs `  N
) ) )
2721, 23, 25, 26syl3anc 1227 . . . . . 6  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( M  ||  N  ->  ( abs `  M
)  <_  ( abs `  N ) ) )
28 simpr 109 . . . . . . . . . . . . 13  |-  ( ( N  ||  M  /\  M  ||  N )  ->  M  ||  N )
29 breq1 3980 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
30 0dvds 11741 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
31 zcn 9188 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ZZ  ->  N  e.  CC )
3231abs00ad 10997 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  ZZ  ->  (
( abs `  N
)  =  0  <->  N  =  0 ) )
33 eqcom 2166 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  N )  =  0  <->  0  =  ( abs `  N ) )
3432, 33bitr3di 194 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  ( N  =  0  <->  0  =  ( abs `  N ) ) )
3530, 34bitrd 187 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  0  =  ( abs `  N ) ) )
3635adantl 275 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  N  <->  0  =  ( abs `  N
) ) )
3729, 36sylan9bb 458 . . . . . . . . . . . . . 14  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  <->  0  =  ( abs `  N ) ) )
38 fveq2 5481 . . . . . . . . . . . . . . . . 17  |-  ( M  =  0  ->  ( abs `  M )  =  ( abs `  0
) )
3938, 13eqtrdi 2213 . . . . . . . . . . . . . . . 16  |-  ( M  =  0  ->  ( abs `  M )  =  0 )
4039adantr 274 . . . . . . . . . . . . . . 15  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  M )  =  0 )
4140eqeq1d 2173 . . . . . . . . . . . . . 14  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( ( abs `  M )  =  ( abs `  N
)  <->  0  =  ( abs `  N ) ) )
4237, 41bitr4d 190 . . . . . . . . . . . . 13  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  <->  ( abs `  M
)  =  ( abs `  N ) ) )
4328, 42syl5ib 153 . . . . . . . . . . . 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 1428 . . . . . . . . . 10  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_ 
( abs `  N
)  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) )
4645expcom 115 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_  ( abs `  N )  -> 
( abs `  M
)  =  ( abs `  N ) ) ) ) ) )
4722adantl 275 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  e.  ZZ )
48 simprl 521 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  e.  ZZ )
49 neqne 2342 . . . . . . . . . . . . . 14  |-  ( -.  M  =  0  ->  M  =/=  0 )
5049adantr 274 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  =/=  0 )
51 dvdsleabs2 11773 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  ( N  ||  M  ->  ( abs `  N )  <_ 
( abs `  M
) ) )
5247, 48, 50, 51syl3anc 1227 . . . . . . . . . . . 12  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( N  ||  M  ->  ( abs `  N
)  <_  ( abs `  M ) ) )
53 eqcom 2166 . . . . . . . . . . . . . . . 16  |-  ( ( abs `  M )  =  ( abs `  N
)  <->  ( abs `  N
)  =  ( abs `  M ) )
5431abscld 11113 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
556abscld 11113 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  RR )
56 letri3 7971 . . . . . . . . . . . . . . . . 17  |-  ( ( ( abs `  N
)  e.  RR  /\  ( abs `  M )  e.  RR )  -> 
( ( abs `  N
)  =  ( abs `  M )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5754, 55, 56syl2anr 288 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  N
)  =  ( abs `  M )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5853, 57syl5bb 191 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  =  ( abs `  N )  <->  ( ( abs `  N )  <_ 
( abs `  M
)  /\  ( abs `  M )  <_  ( abs `  N ) ) ) )
5958biimprd 157 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  N )  <_  ( abs `  M )  /\  ( abs `  M )  <_  ( abs `  N
) )  ->  ( abs `  M )  =  ( abs `  N
) ) )
6059expd 256 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  N
)  <_  ( abs `  M )  ->  (
( abs `  M
)  <_  ( abs `  N )  ->  ( abs `  M )  =  ( abs `  N
) ) ) )
6160adantl 275 . . . . . . . . . . . 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 115 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  M  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( ( abs `  M )  <_ 
( abs `  N
)  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) ) )
65 0z 9194 . . . . . . . . . . . 12  |-  0  e.  ZZ
66 zdceq 9258 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
6765, 66mpan2 422 . . . . . . . . . . 11  |-  ( M  e.  ZZ  -> DECID  M  =  0
)
68 exmiddc 826 . . . . . . . . . . 11  |-  (DECID  M  =  0  ->  ( M  =  0  \/  -.  M  =  0 ) )
6967, 68syl 14 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M  =  0  \/  -.  M  =  0
) )
7069adantr 274 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  \/  -.  M  =  0 ) )
7146, 64, 70mpjaod 708 . . . . . . . 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 275 . . . . . 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 115 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  N  =  0  ->  ( M  ||  N  ->  ( N  ||  M  ->  ( abs `  M )  =  ( abs `  N ) ) ) ) )
76 zdceq 9258 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
7765, 76mpan2 422 . . . . . 6  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
78 exmiddc 826 . . . . . 6  |-  (DECID  N  =  0  ->  ( N  =  0  \/  -.  N  =  0 ) )
7977, 78syl 14 . . . . 5  |-  ( N  e.  ZZ  ->  ( N  =  0  \/  -.  N  =  0
) )
8079adantl 275 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  \/  -.  N  =  0 ) )
8120, 75, 80mpjaod 708 . . 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 123 1  |-  ( ( M  ||  N  /\  N  ||  M )  -> 
( abs `  M
)  =  ( abs `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698  DECID wdc 824    = wceq 1342    e. wcel 2135    =/= wne 2334   class class class wbr 3977   ` cfv 5183   RRcr 7744   0cc0 7745    <_ cle 7926   ZZcz 9183   abscabs 10929    || cdvds 11717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-iinf 4560  ax-cnex 7836  ax-resscn 7837  ax-1cn 7838  ax-1re 7839  ax-icn 7840  ax-addcl 7841  ax-addrcl 7842  ax-mulcl 7843  ax-mulrcl 7844  ax-addcom 7845  ax-mulcom 7846  ax-addass 7847  ax-mulass 7848  ax-distr 7849  ax-i2m1 7850  ax-0lt1 7851  ax-1rid 7852  ax-0id 7853  ax-rnegex 7854  ax-precex 7855  ax-cnre 7856  ax-pre-ltirr 7857  ax-pre-ltwlin 7858  ax-pre-lttrn 7859  ax-pre-apti 7860  ax-pre-ltadd 7861  ax-pre-mulgt0 7862  ax-pre-mulext 7863  ax-arch 7864  ax-caucvg 7865
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-if 3517  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-po 4269  df-iso 4270  df-iord 4339  df-on 4341  df-ilim 4342  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-riota 5793  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1st 6101  df-2nd 6102  df-recs 6265  df-frec 6351  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931  df-sub 8063  df-neg 8064  df-reap 8465  df-ap 8472  df-div 8561  df-inn 8850  df-2 8908  df-3 8909  df-4 8910  df-n0 9107  df-z 9184  df-uz 9459  df-q 9550  df-rp 9582  df-seqfrec 10372  df-exp 10446  df-cj 10774  df-re 10775  df-im 10776  df-rsqrt 10930  df-abs 10931  df-dvds 11718
This theorem is referenced by:  dvdseq  11775
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