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Theorem znege1 11701
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 9012 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  <->  ( A  +  1 )  <_  B ) )
213adant3 984 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  ( A  <  B  <->  ( A  +  1 )  <_  B ) )
32biimpa 292 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( A  +  1 )  <_  B )
4 simpl1 967 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  e.  ZZ )
54zred 9077 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  e.  RR )
6 1red 7705 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  e.  RR )
7 simpl2 968 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  B  e.  ZZ )
87zred 9077 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  B  e.  RR )
95, 6, 8leaddsub2d 8227 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( ( A  + 
1 )  <_  B  <->  1  <_  ( B  -  A ) ) )
103, 9mpbid 146 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  <_  ( B  -  A ) )
11 simpr 109 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  <  B )
125, 8, 11ltled 7804 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  <_  B )
135, 8, 12abssuble0d 10841 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( abs `  ( A  -  B )
)  =  ( B  -  A ) )
1410, 13breqtrrd 3921 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  <_  ( abs `  ( A  -  B
) ) )
15 simpr 109 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  A  =  B )
16 simpl3 969 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  A  =/=  B )
1715, 16pm2.21ddne 2365 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  1  <_  ( abs `  ( A  -  B
) ) )
18 simpr 109 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  <  A )
19 simpl2 968 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  e.  ZZ )
20 simpl1 967 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  A  e.  ZZ )
21 zltp1le 9012 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  <  A  <->  ( B  +  1 )  <_  A ) )
2219, 20, 21syl2anc 406 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( B  <  A  <->  ( B  +  1 )  <_  A ) )
2318, 22mpbid 146 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( B  +  1 )  <_  A )
2419zred 9077 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  e.  RR )
25 1red 7705 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  e.  RR )
2620zred 9077 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  A  e.  RR )
2724, 25, 26leaddsub2d 8227 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( ( B  + 
1 )  <_  A  <->  1  <_  ( A  -  B ) ) )
2823, 27mpbid 146 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  <_  ( A  -  B ) )
2924, 26, 18ltled 7804 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  <_  A )
3024, 26, 29abssubge0d 10840 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( abs `  ( A  -  B )
)  =  ( A  -  B ) )
3128, 30breqtrrd 3921 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  <_  ( abs `  ( A  -  B
) ) )
32 ztri3or 9001 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
33323adant3 984 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
3414, 17, 31, 33mpjao3dan 1268 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  1  <_  ( abs `  ( A  -  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 944    /\ w3a 945    = wceq 1314    e. wcel 1463    =/= wne 2282   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   1c1 7548    + caddc 7550    < clt 7724    <_ cle 7725    - cmin 7856   ZZcz 8958   abscabs 10661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-n0 8882  df-z 8959  df-uz 9229  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663
This theorem is referenced by: (None)
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