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Theorem znege1 10781
Description: The absolute value of the difference between two unequal integers is at least one. (Contributed by Jim Kingdon, 31-Jan-2022.)
Assertion
Ref Expression
znege1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  1  <_  ( abs `  ( A  -  B )
) )

Proof of Theorem znege1
StepHypRef Expression
1 zltp1le 8556 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  <->  ( A  +  1 )  <_  B ) )
213adant3 959 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  ( A  <  B  <->  ( A  +  1 )  <_  B ) )
32biimpa 290 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( A  +  1 )  <_  B )
4 simpl1 942 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  e.  ZZ )
54zred 8620 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  e.  RR )
6 1red 7266 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  e.  RR )
7 simpl2 943 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  B  e.  ZZ )
87zred 8620 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  B  e.  RR )
95, 6, 8leaddsub2d 7784 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( ( A  + 
1 )  <_  B  <->  1  <_  ( B  -  A ) ) )
103, 9mpbid 145 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  <_  ( B  -  A ) )
11 simpr 108 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  <  B )
125, 8, 11ltled 7365 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  <_  B )
135, 8, 12abssuble0d 10282 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( abs `  ( A  -  B )
)  =  ( B  -  A ) )
1410, 13breqtrrd 3831 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  <_  ( abs `  ( A  -  B
) ) )
15 simpr 108 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  A  =  B )
16 simpl3 944 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  A  =/=  B )
1715, 16pm2.21ddne 2332 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  1  <_  ( abs `  ( A  -  B
) ) )
18 simpr 108 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  <  A )
19 simpl2 943 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  e.  ZZ )
20 simpl1 942 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  A  e.  ZZ )
21 zltp1le 8556 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  <  A  <->  ( B  +  1 )  <_  A ) )
2219, 20, 21syl2anc 403 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( B  <  A  <->  ( B  +  1 )  <_  A ) )
2318, 22mpbid 145 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( B  +  1 )  <_  A )
2419zred 8620 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  e.  RR )
25 1red 7266 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  e.  RR )
2620zred 8620 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  A  e.  RR )
2724, 25, 26leaddsub2d 7784 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( ( B  + 
1 )  <_  A  <->  1  <_  ( A  -  B ) ) )
2823, 27mpbid 145 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  <_  ( A  -  B ) )
2924, 26, 18ltled 7365 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  <_  A )
3024, 26, 29abssubge0d 10281 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( abs `  ( A  -  B )
)  =  ( A  -  B ) )
3128, 30breqtrrd 3831 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  <_  ( abs `  ( A  -  B
) ) )
32 ztri3or 8545 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
33323adant3 959 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
3414, 17, 31, 33mpjao3dan 1239 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  1  <_  ( abs `  ( A  -  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ w3o 919    /\ w3a 920    = wceq 1285    e. wcel 1434    =/= wne 2249   class class class wbr 3805   ` cfv 4952  (class class class)co 5564   1c1 7114    + caddc 7116    < clt 7285    <_ cle 7286    - cmin 7416   ZZcz 8502   abscabs 10102
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 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226
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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-2 8235  df-n0 8426  df-z 8503  df-uz 8771  df-iseq 9592  df-iexp 9643  df-cj 9948  df-re 9949  df-im 9950  df-rsqrt 10103  df-abs 10104
This theorem is referenced by: (None)
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