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Theorem modifeq2int 10801
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 1028 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. NN0 )
2 nn0z 9643 . . . . . . 7 |- ( A e. NN0 -> A e. ZZ )
3 zq 10005 . . . . . . 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 10012 . . . . . 6 |- ( B e. NN -> B e. QQ )
873ad2ant2 1050 . . . . 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 9602 . . . . 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 10764 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
146, 9, 11, 12, 13syl22anc 1279 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( A mod B ) = A )
15 iftrue 3642 . . . . 5 |- ( A < B -> if ( A < B , A , ( A - B ) ) = A )
1615eqcomd 2244 . . . 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 2271 . 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 1029 . . . . . 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 9327 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> 0 < B )
2421nnred 9296 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. RR )
251nn0red 9600 . . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. RR )
2624, 25lenltd 8434 . . . . 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 1033 . . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A < ( 2 x. B ) )
29 q2submod 10800 . . . 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 1286 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = ( A - B ) )
31 iffalse 3645 . . . . 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 2244 . . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A - B ) = if ( A < B , A , ( A - B ) ) )
3430, 33eqtrd 2271 . 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 9746 . . 3 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. ZZ )
37 zdclt 9701 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B )
38 exmiddc 848 . . . 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 415 . 2 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A < B \/ -. A < B ) )
4118, 34, 40mpjaodan 810 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 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   ifcif 3635   class class class wbr 4125  (class class class)co 6075   0cc0 8169   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   NNcn 9283   2c2 9334   NN0cn0 9542   ZZcz 9623   QQcq 9998   mod cmo 10737
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-q 9999  df-rp 10034  df-fl 10683  df-mod 10738
This theorem is referenced by: (None)
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