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Theorem congabseq 30743
Description: If two integers are congruent, they are either equal or separated by at least the congruence base. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
congabseq  |-  ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C ) )  ->  ( ( abs `  ( B  -  C
) )  <  A  <->  B  =  C ) )

Proof of Theorem congabseq
StepHypRef Expression
1 zcn 10870 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  CC )
213ad2ant2 1018 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
32ad2antrr 725 . . 3  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  ->  B  e.  CC )
4 zcn 10870 . . . . 5  |-  ( C  e.  ZZ  ->  C  e.  CC )
543ad2ant3 1019 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
65ad2antrr 725 . . 3  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  ->  C  e.  CC )
7 zsubcl 10906 . . . . . . . . . 10  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( B  -  C
)  e.  ZZ )
873adant1 1014 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( B  -  C )  e.  ZZ )
98zcnd 10968 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( B  -  C )  e.  CC )
109abscld 13233 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( abs `  ( B  -  C ) )  e.  RR )
1110adantr 465 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C ) )  ->  ( abs `  ( B  -  C )
)  e.  RR )
12 nnre 10544 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  RR )
13123ad2ant1 1017 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  A  e.  RR )
1413adantr 465 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C ) )  ->  A  e.  RR )
1511, 14ltnled 9732 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C ) )  ->  ( ( abs `  ( B  -  C
) )  <  A  <->  -.  A  <_  ( abs `  ( B  -  C
) ) ) )
1615biimpa 484 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  ->  -.  A  <_  ( abs `  ( B  -  C )
) )
17 nnz 10887 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  ZZ )
18173ad2ant1 1017 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  A  e.  ZZ )
1918ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  /\  ( B  -  C )  =/=  0 )  ->  A  e.  ZZ )
208ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  /\  ( B  -  C )  =/=  0 )  ->  ( B  -  C )  e.  ZZ )
21 simpr 461 . . . . . . . 8  |-  ( ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  /\  ( B  -  C )  =/=  0 )  ->  ( B  -  C )  =/=  0 )
2219, 20, 213jca 1176 . . . . . . 7  |-  ( ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  /\  ( B  -  C )  =/=  0 )  ->  ( A  e.  ZZ  /\  ( B  -  C )  e.  ZZ  /\  ( B  -  C )  =/=  0 ) )
23 simpllr 758 . . . . . . 7  |-  ( ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  /\  ( B  -  C )  =/=  0 )  ->  A  ||  ( B  -  C
) )
24 dvdsleabs 13894 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( B  -  C
)  e.  ZZ  /\  ( B  -  C
)  =/=  0 )  ->  ( A  ||  ( B  -  C
)  ->  A  <_  ( abs `  ( B  -  C ) ) ) )
2522, 23, 24sylc 60 . . . . . 6  |-  ( ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  /\  ( B  -  C )  =/=  0 )  ->  A  <_  ( abs `  ( B  -  C )
) )
2625ex 434 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  ->  (
( B  -  C
)  =/=  0  ->  A  <_  ( abs `  ( B  -  C )
) ) )
2726necon1bd 2685 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  ->  ( -.  A  <_  ( abs `  ( B  -  C
) )  ->  ( B  -  C )  =  0 ) )
2816, 27mpd 15 . . 3  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  ->  ( B  -  C )  =  0 )
293, 6, 28subeq0d 9939 . 2  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  ( abs `  ( B  -  C ) )  < 
A )  ->  B  =  C )
30 oveq1 6292 . . . . . 6  |-  ( B  =  C  ->  ( B  -  C )  =  ( C  -  C ) )
3130adantl 466 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  B  =  C )  ->  ( B  -  C )  =  ( C  -  C ) )
325ad2antrr 725 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  B  =  C )  ->  C  e.  CC )
3332subidd 9919 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  B  =  C )  ->  ( C  -  C )  =  0 )
3431, 33eqtrd 2508 . . . 4  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  B  =  C )  ->  ( B  -  C )  =  0 )
3534abs00bd 13090 . . 3  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  B  =  C )  ->  ( abs `  ( B  -  C ) )  =  0 )
36 nngt0 10566 . . . . 5  |-  ( A  e.  NN  ->  0  <  A )
37363ad2ant1 1017 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  <  A )
3837ad2antrr 725 . . 3  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  B  =  C )  ->  0  <  A )
3935, 38eqbrtrd 4467 . 2  |-  ( ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C
) )  /\  B  =  C )  ->  ( abs `  ( B  -  C ) )  < 
A )
4029, 39impbida 830 1  |-  ( ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C ) )  ->  ( ( abs `  ( B  -  C
) )  <  A  <->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493    < clt 9629    <_ cle 9630    - cmin 9806   NNcn 10537   ZZcz 10865   abscabs 13033    || cdivides 13850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851
This theorem is referenced by:  acongeq  30752
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