ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvdsabseq Unicode version

Theorem dvdsabseq 10392
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 10345 . . 3  |-  ( M 
||  N  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2 simpr 108 . . . . . . 7  |-  ( ( M  ||  N  /\  N  ||  M )  ->  N  ||  M )
3 breq1 3796 . . . . . . . . 9  |-  ( N  =  0  ->  ( N  ||  M  <->  0  ||  M ) )
4 0dvds 10360 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  (
0  ||  M  <->  M  = 
0 ) )
54adantr 270 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  M  <->  M  =  0 ) )
6 zcn 8437 . . . . . . . . . . . . 13  |-  ( M  e.  ZZ  ->  M  e.  CC )
76abs00ad 10089 . . . . . . . . . . . 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 5209 . . . . . . . . . . 11  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
13 abs0 10082 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1412, 13syl6eq 2130 . . . . . . . . . 10  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
1514adantr 270 . . . . . . . . 9  |-  ( ( N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  N )  =  0 )
1615eqeq2d 2093 . . . . . . . 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 2254 . . . . . . . 8  |-  ( -.  N  =  0  ->  N  =/=  0 )
2524adantr 270 . . . . . . 7  |-  ( ( -.  N  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  N  =/=  0 )
26 dvdsleabs2 10391 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  ( M  ||  N  ->  ( abs `  M )  <_ 
( abs `  N
) ) )
2721, 23, 25, 26syl3anc 1170 . . . . . 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 3796 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
30 0dvds 10360 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
31 eqcom 2084 . . . . . . . . . . . . . . . . . 18  |-  ( ( abs `  N )  =  0  <->  0  =  ( abs `  N ) )
32 zcn 8437 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ZZ  ->  N  e.  CC )
3332abs00ad 10089 . . . . . . . . . . . . . . . . . 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 5209 . . . . . . . . . . . . . . . . 17  |-  ( M  =  0  ->  ( abs `  M )  =  ( abs `  0
) )
3938, 13syl6eq 2130 . . . . . . . . . . . . . . . 16  |-  ( M  =  0  ->  ( abs `  M )  =  0 )
4039adantr 270 . . . . . . . . . . . . . . 15  |-  ( ( M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( abs `  M )  =  0 )
4140eqeq1d 2090 . . . . . . . . . . . . . 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 1371 . . . . . . . . . 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 2254 . . . . . . . . . . . . . 14  |-  ( -.  M  =  0  ->  M  =/=  0 )
5049adantr 270 . . . . . . . . . . . . 13  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  M  =/=  0 )
51 dvdsleabs2 10391 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  M  =/=  0 )  ->  ( N  ||  M  ->  ( abs `  N )  <_ 
( abs `  M
) ) )
5247, 48, 50, 51syl3anc 1170 . . . . . . . . . . . 12  |-  ( ( -.  M  =  0  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( N  ||  M  ->  ( abs `  N
)  <_  ( abs `  M ) ) )
53 eqcom 2084 . . . . . . . . . . . . . . . 16  |-  ( ( abs `  M )  =  ( abs `  N
)  <->  ( abs `  N
)  =  ( abs `  M ) )
5432abscld 10205 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
556abscld 10205 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  ZZ  ->  ( abs `  M )  e.  RR )
56 letri3 7259 . . . . . . . . . . . . . . . . 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 8443 . . . . . . . . . . . 12  |-  0  e.  ZZ
66 zdceq 8504 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  -> DECID  M  =  0 )
6765, 66mpan2 416 . . . . . . . . . . 11  |-  ( M  e.  ZZ  -> DECID  M  =  0
)
68 exmiddc 778 . . . . . . . . . . 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 671 . . . . . . . 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 8504 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
7765, 76mpan2 416 . . . . . 6  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
78 exmiddc 778 . . . . . 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 671 . . 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 662  DECID wdc 776    = wceq 1285    e. wcel 1434    =/= wne 2246   class class class wbr 3793   ` cfv 4932   RRcr 7042   0cc0 7043    <_ cle 7216   ZZcz 8432   abscabs 10021    || cdvds 10340
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157  ax-caucvg 7158
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356  df-z 8433  df-uz 8701  df-q 8786  df-rp 8816  df-iseq 9522  df-iexp 9573  df-cj 9867  df-re 9868  df-im 9869  df-rsqrt 10022  df-abs 10023  df-dvds 10341
This theorem is referenced by:  dvdseq  10393
  Copyright terms: Public domain W3C validator