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Theorem modifeq2int 10512
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 999 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. NN0 )
2 nn0z 9374 . . . . . . 7 |- ( A e. NN0 -> A e. ZZ )
3 zq 9729 . . . . . . 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 276 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> A e. QQ )
7 nnq 9736 . . . . . 6 |- ( B e. NN -> B e. QQ )
873ad2ant2 1021 . . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. QQ )
98adantr 276 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> B e. QQ )
101nn0ge0d 9333 . . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> 0 <_ A )
1110adantr 276 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> 0 <_ A )
12 simpr 110 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> A < B )
13 modqid 10475 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
146, 9, 11, 12, 13syl22anc 1250 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( A mod B ) = A )
15 iftrue 3575 . . . . 5 |- ( A < B -> if ( A < B , A , ( A - B ) ) = A )
1615eqcomd 2210 . . . 4 |- ( A < B -> A = if ( A < B , A , ( A - B ) ) )
1716adantl 277 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> A = if ( A < B , A , ( A - B ) ) )
1814, 17eqtrd 2237 . 2 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )
195adantr 276 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A e. QQ )
208adantr 276 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B e. QQ )
21 simp2 1000 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. NN )
2221adantr 276 . . . . 5 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B e. NN )
2322nngt0d 9062 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> 0 < B )
2421nnred 9031 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. RR )
251nn0red 9331 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. RR )
2624, 25lenltd 8172 . . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( B <_ A <-> -. A < B ) )
2726biimpar 297 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B <_ A )
28 simpl3 1004 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A < ( 2 x. B ) )
29 q2submod 10511 . . . 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 1257 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = ( A - B ) )
31 iffalse 3578 . . . . 5 |- ( -. A < B -> if ( A < B , A , ( A - B ) ) = ( A - B ) )
3231adantl 277 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> if ( A < B , A , ( A - B ) ) = ( A - B ) )
3332eqcomd 2210 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A - B ) = if ( A < B , A , ( A - B ) ) )
3430, 33eqtrd 2237 . 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 9476 . . 3 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. ZZ )
37 zdclt 9432 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B )
38 exmiddc 837 . . . 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 411 . 2 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A < B \/ -. A < B ) )
4118, 34, 40mpjaodan 799 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 104   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1372   e. wcel 2175   ifcif 3570   class class class wbr 4043  (class class class)co 5934   0cc0 7907   x. cmul 7912   < clt 8089   <_ cle 8090   - cmin 8225   NNcn 9018   2c2 9069   NN0cn0 9277   ZZcz 9354   QQcq 9722   mod cmo 10448
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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025  ax-arch 8026
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-po 4341  df-iso 4342  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-n0 9278  df-z 9355  df-q 9723  df-rp 9758  df-fl 10394  df-mod 10449
This theorem is referenced by: (None)
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