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Theorem znege1 12181
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 9310 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  <->  ( A  +  1 )  <_  B ) )
213adant3 1017 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  ( A  <  B  <->  ( A  +  1 )  <_  B ) )
32biimpa 296 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( A  +  1 )  <_  B )
4 simpl1 1000 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  e.  ZZ )
54zred 9378 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  e.  RR )
6 1red 7975 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  e.  RR )
7 simpl2 1001 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  B  e.  ZZ )
87zred 9378 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  B  e.  RR )
95, 6, 8leaddsub2d 8507 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( ( A  + 
1 )  <_  B  <->  1  <_  ( B  -  A ) ) )
103, 9mpbid 147 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  <_  ( B  -  A ) )
11 simpr 110 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  <  B )
125, 8, 11ltled 8079 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  ->  A  <_  B )
135, 8, 12abssuble0d 11189 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
( abs `  ( A  -  B )
)  =  ( B  -  A ) )
1410, 13breqtrrd 4033 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  <  B )  -> 
1  <_  ( abs `  ( A  -  B
) ) )
15 simpr 110 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  A  =  B )
16 simpl3 1002 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  A  =/=  B )
1715, 16pm2.21ddne 2430 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  A  =  B )  ->  1  <_  ( abs `  ( A  -  B
) ) )
18 simpr 110 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  <  A )
19 simpl2 1001 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  e.  ZZ )
20 simpl1 1000 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  A  e.  ZZ )
21 zltp1le 9310 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  <  A  <->  ( B  +  1 )  <_  A ) )
2219, 20, 21syl2anc 411 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( B  <  A  <->  ( B  +  1 )  <_  A ) )
2318, 22mpbid 147 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( B  +  1 )  <_  A )
2419zred 9378 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  e.  RR )
25 1red 7975 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  e.  RR )
2620zred 9378 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  A  e.  RR )
2724, 25, 26leaddsub2d 8507 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( ( B  + 
1 )  <_  A  <->  1  <_  ( A  -  B ) ) )
2823, 27mpbid 147 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  <_  ( A  -  B ) )
2924, 26, 18ltled 8079 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  ->  B  <_  A )
3024, 26, 29abssubge0d 11188 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
( abs `  ( A  -  B )
)  =  ( A  -  B ) )
3128, 30breqtrrd 4033 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  /\  B  <  A )  -> 
1  <_  ( abs `  ( A  -  B
) ) )
32 ztri3or 9299 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
33323adant3 1017 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
3414, 17, 31, 33mpjao3dan 1307 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  =/=  B )  ->  1  <_  ( abs `  ( A  -  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ w3o 977    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   1c1 7815    + caddc 7817    < clt 7995    <_ cle 7996    - cmin 8131   ZZcz 9256   abscabs 11009
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-n0 9180  df-z 9257  df-uz 9532  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011
This theorem is referenced by: (None)
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