Theorem modifeq2int 10372

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 997	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. NN0 )
2		nn0z 9262	. . . . . . 7 |- ( A e. NN0 -> A e. ZZ )
3		zq 9615	. . . . . . 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 9622	. . . . . 6 |- ( B e. NN -> B e. QQ )
8	7	3ad2ant2 1019	. . . . 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 9221	. . . . 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 10335	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
14	6, 9, 11, 12, 13	syl22anc 1239	. . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ A < B ) -> ( A mod B ) = A )
15		iftrue 3539	. . . . 5 |- ( A < B -> if ( A < B , A , ( A - B ) ) = A )
16	15	eqcomd 2183	. . . 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 2210	. 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 998	. . . . . 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 8952	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> 0 < B )
24	21	nnred 8921	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. RR )
25	1	nn0red 9219	. . . . . 6 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> A e. RR )
26	24, 25	lenltd 8065	. . . . 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 1002	. . . 4 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> A < ( 2 x. B ) )
29		q2submod 10371	. . . 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 1246	. . 3 |- ( ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) /\ -. A < B ) -> ( A mod B ) = ( A - B ) )
31		iffalse 3542	. . . . 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 2183	. . 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 2210	. 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 9363	. . 3 |- ( ( A e. NN0 /\ B e. NN /\ A < ( 2 x. B ) ) -> B e. ZZ )
37		zdclt 9319	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A < B )
38		exmiddc 836	. . . 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 798	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 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2148   ifcif 3534   class class class wbr 4000  (class class class)co 5869   0cc0 7802   x. cmul 7807   < clt 7982   <_ cle 7983   - cmin 8118   NNcn 8908   2c2 8959   NN0cn0 9165   ZZcz 9242   QQcq 9608   mod cmo 10308

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309

This theorem is referenced by: (None)