Theorem modifeq2int 10694

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

Step	Hyp	Ref	Expression
1		simp1 1024	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. NN0 )
2		nn0z 9543	. . . . . . 7 |- ( A e. NN0 -> A e. ZZ )
3		zq 9904	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
4	2, 3	syl 14	. . . . . 6 |- ( A e. NN0 -> A e. QQ )
5	1, 4	syl 14	. . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. QQ )
6	5	adantr 276	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> A e. QQ )
7		nnq 9911	. . . . . 6 |- ( B e. NN -> B e. QQ )
8	7	3ad2ant2 1046	. . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. QQ )
9	8	adantr 276	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> B e. QQ )
10	1	nn0ge0d 9502	. . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> 0 <_ A )
11	10	adantr 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 10657	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
14	6, 9, 11, 12, 13	syl22anc 1275	. . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( A mod B ) = A )
15		iftrue 3614	. . . . 5 |- ( A < B -> if ( A < B , A , ( A - B ) ) = A )
16	15	eqcomd 2237	. . . 4 |- ( A < B -> A = if ( A < B , A , ( A - B ) ) )
17	16	adantl 277	. . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> A = if ( A < B , A , ( A - B ) ) )
18	14, 17	eqtrd 2264	. 2 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )
19	5	adantr 276	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A e. QQ )
20	8	adantr 276	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B e. QQ )
21		simp2 1025	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. NN )
22	21	adantr 276	. . . . 5 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B e. NN )
23	22	nngt0d 9229	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> 0 < B )
24	21	nnred 9198	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. RR )
25	1	nn0red 9500	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. RR )
26	24, 25	lenltd 8339	. . . . 5 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( B <_ A <-> -. A < B ) )
27	26	biimpar 297	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> B <_ A )
28		simpl3 1029	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A < ( 2 x. B ) )
29		q2submod 10693	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = ( A - B ) )
30	19, 20, 23, 27, 28, 29	syl32anc 1282	. . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = ( A - B ) )
31		iffalse 3617	. . . . 5 |- ( -. A < B -> if ( A < B , A , ( A - B ) ) = ( A - B ) )
32	31	adantl 277	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> if ( A < B , A , ( A - B ) ) = ( A - B ) )
33	32	eqcomd 2237	. . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A - B ) = if ( A < B , A , ( A - B ) ) )
34	30, 33	eqtrd 2264	. 2 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )
35	1, 2	syl 14	. . 3 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. ZZ )
36	21	nnzd 9645	. . 3 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. ZZ )
37		zdclt 9601	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B )
38		exmiddc 844	. . . 4 |- (DECID A < B -> ( A < B \/ -. A < B ) )
39	37, 38	syl 14	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B \/ -. A < B ) )
40	35, 36, 39	syl2anc 411	. 2 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A < B \/ -. A < B ) )
41	18, 34, 40	mpjaodan 806	1 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> ( A mod B ) = if ( A < B , A , ( A - B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2202   ifcif 3607   class class class wbr 4093  (class class class)co 6028   0cc0 8075   x. cmul 8080   < clt 8256   <_ cle 8257   - cmin 8392   NNcn 9185   2c2 9236   NN0cn0 9444   ZZcz 9523   QQcq 9897   mod cmo 10630

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-n0 9445  df-z 9524  df-q 9898  df-rp 9933  df-fl 10576  df-mod 10631

This theorem is referenced by: (None)