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Theorem modifeq2int 10321
Description: If a nonnegative integer is less than twice 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 simp1 987 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. NN0 )
2 nn0z 9211 . . . . . . 7 |- ( A e. NN0 -> A e. ZZ )
3 zq 9564 . . . . . . 7 |- ( A e. ZZ -> A e. QQ )
42, 3syl 14 . . . . . 6 |- ( A e. NN0 -> A e. QQ )
51, 4syl 14 . . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. QQ )
65adantr 274 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> A e. QQ )
7 nnq 9571 . . . . . 6 |- ( B e. NN -> B e. QQ )
873ad2ant2 1009 . . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. QQ )
98adantr 274 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> B e. QQ )
101nn0ge0d 9170 . . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> 0 <_ A )
1110adantr 274 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> 0 <_ A )
12 simpr 109 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> A < B )
13 modqid 10284 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
146, 9, 11, 12, 13syl22anc 1229 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( A mod B ) = A )
15 iftrue 3525 . . . . 5 |- ( A < B -> if ( A < B , A , ( A - B ) ) = A )
1615eqcomd 2171 . . . 4 |- ( A < B -> A = if ( A < B , A , ( A - B ) ) )
1716adantl 275 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> A = if ( A < B , A , ( A - B ) ) )
1814, 17eqtrd 2198 . 2 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )
195adantr 274 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A e. QQ )
208adantr 274 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B e. QQ )
21 simp2 988 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. NN )
2221adantr 274 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B e. NN )
2322nngt0d 8901 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> 0 < B )
2421nnred 8870 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. RR )
251nn0red 9168 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. RR )
2624, 25lenltd 8016 . . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( B <_ A <-> -. A < B ) )
2726biimpar 295 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B <_ A )
28 simpl3 992 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A < ( 2 x. B ) )
29 q2submod 10320 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = ( A - B ) )
3019, 20, 23, 27, 28, 29syl32anc 1236 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = ( A - B ) )
31 iffalse 3528 . . . . 5 |- ( -. A < B -> if ( A < B , A , ( A - B ) ) = ( A - B ) )
3231adantl 275 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> if ( A < B , A , ( A - B ) ) = ( A - B ) )
3332eqcomd 2171 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A - B ) = if ( A < B , A , ( A - B ) ) )
3430, 33eqtrd 2198 . 2 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )
351, 2syl 14 . . 3 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. ZZ )
3621nnzd 9312 . . 3 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. ZZ )
37 zdclt 9268 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B )
38 exmiddc 826 . . . 4 |- (DECID A < B -> ( A < B \/ -. A < B ) )
3937, 38syl 14 . . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B \/ -. A < B ) )
4035, 36, 39syl2anc 409 . 2 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A < B \/ -. A < B ) )
4118, 34, 40mpjaodan 788 1 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   ifcif 3520   class class class wbr 3982  (class class class)co 5842   0cc0 7753   x. cmul 7758   < clt 7933   <_ cle 7934   - cmin 8069   NNcn 8857   2c2 8908   NN0cn0 9114   ZZcz 9191   QQcq 9557   mod cmo 10257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-q 9558  df-rp 9590  df-fl 10205  df-mod 10258
This theorem is referenced by: (None)
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