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Theorem modifeq2int 11859
Description: If a nonnegative integer is less than the double of a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
Assertion
Ref Expression
modifeq2int  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  mod  B )  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )

Proof of Theorem modifeq2int
StepHypRef Expression
1 nn0re 10686 . . . . . . . 8  |-  ( A  e.  NN0  ->  A  e.  RR )
2 nnrp 11098 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  RR+ )
31, 2anim12i 566 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  e.  RR  /\  B  e.  RR+ )
)
433adant3 1008 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  e.  RR  /\  B  e.  RR+ ) )
54adantl 466 . . . . 5  |-  ( ( A  <  B  /\  ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  e.  RR  /\  B  e.  RR+ ) )
6 nn0ge0 10703 . . . . . . . 8  |-  ( A  e.  NN0  ->  0  <_  A )
763ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  0  <_  A )
87anim1i 568 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( 0  <_  A  /\  A  <  B ) )
98ancoms 453 . . . . 5  |-  ( ( A  <  B  /\  ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) ) )  ->  ( 0  <_  A  /\  A  <  B
) )
10 modid 11830 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( A  mod  B )  =  A )
115, 9, 10syl2anc 661 . . . 4  |-  ( ( A  <  B  /\  ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  A )
12 iftrue 3892 . . . . . 6  |-  ( A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  A )
1312eqcomd 2458 . . . . 5  |-  ( A  <  B  ->  A  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )
1413adantr 465 . . . 4  |-  ( ( A  <  B  /\  ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) ) )  ->  A  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
1511, 14eqtrd 2491 . . 3  |-  ( ( A  <  B  /\  ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
1615ex 434 . 2  |-  ( A  <  B  ->  (
( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  -> 
( A  mod  B
)  =  if ( A  <  B ,  A ,  ( A  -  B ) ) ) )
174adantr 465 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  e.  RR  /\  B  e.  RR+ ) )
18 nnre 10427 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  RR )
19 lenlt 9551 . . . . . . . 8  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <_  A  <->  -.  A  <  B ) )
2018, 1, 19syl2anr 478 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( B  <_  A  <->  -.  A  <  B ) )
21203adant3 1008 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( B  <_  A  <->  -.  A  <  B ) )
2221biimpar 485 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  <_  A
)
23 simpl3 993 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  <  (
2  x.  B ) )
24 2submod 11858 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
2517, 22, 23, 24syl12anc 1217 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  mod  B )  =  ( A  -  B ) )
26 iffalse 3894 . . . . . 6  |-  ( -.  A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B ) )
2726adantl 466 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B
) )
2827eqcomd 2458 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  -  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
2925, 28eqtrd 2491 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  mod  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
3029expcom 435 . 2  |-  ( -.  A  <  B  -> 
( ( A  e. 
NN0  /\  B  e.  NN  /\  A  <  (
2  x.  B ) )  ->  ( A  mod  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) ) )
3116, 30pm2.61i 164 1  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  mod  B )  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ifcif 3886   class class class wbr 4387  (class class class)co 6187   RRcr 9379   0cc0 9380    x. cmul 9385    < clt 9516    <_ cle 9517    - cmin 9693   NNcn 10420   2c2 10469   NN0cn0 10677   RR+crp 11089    mod cmo 11806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-pre-sup 9458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-recs 6929  df-rdg 6963  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-sup 7789  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-n0 10678  df-z 10745  df-uz 10960  df-rp 11090  df-fl 11740  df-mod 11807
This theorem is referenced by: (None)
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