Theorem modqlt 10555

Description: The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)

Assertion

Ref	Expression
modqlt	|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )

Proof of Theorem modqlt

Step	Hyp	Ref	Expression
1		qcn 9829	. . . . . 6 |- ( A e. QQ -> A e. CC )
2	1	3ad2ant1 1042	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. CC )
3		qcn 9829	. . . . . 6 |- ( B e. QQ -> B e. CC )
4	3	3ad2ant2 1043	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
5		qre 9820	. . . . . . 7 |- ( B e. QQ -> B e. RR )
6	5	3ad2ant2 1043	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
7		simp3 1023	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
8	6, 7	gt0ap0d 8776	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B # 0 )
9	2, 4, 8	divcanap2d 8939	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( A / B ) ) = A )
10	9	oveq1d 6016	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( B x. ( A / B ) ) - ( B x. ( |_ ` ( A / B ) ) ) ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
11	7	gt0ne0d 8659	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
12		qdivcl 9838	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
13	11, 12	syld3an3 1316	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
14		qcn 9829	. . . . 5 |- ( ( A / B ) e. QQ -> ( A / B ) e. CC )
15	13, 14	syl 14	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. CC )
16	13	flqcld 10497	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
17	16	zcnd 9570	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. CC )
18	4, 15, 17	subdid 8560	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) = ( ( B x. ( A / B ) ) - ( B x. ( |_ ` ( A / B ) ) ) ) )
19		modqval 10546	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
20	10, 18, 19	3eqtr4rd 2273	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) )
21		qfraclt1 10500	. . . . 5 |- ( ( A / B ) e. QQ -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < 1 )
22	13, 21	syl 14	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < 1 )
23	4, 8	dividapd 8933	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B / B ) = 1 )
24	22, 23	breqtrrd 4111	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) )
25		qre 9820	. . . . . 6 |- ( ( A / B ) e. QQ -> ( A / B ) e. RR )
26	13, 25	syl 14	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. RR )
27	16	zred 9569	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
28	26, 27	resubcld 8527	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) e. RR )
29		ltmuldiv2 9022	. . . 4 |- ( ( ( ( A / B ) - ( |_ ` ( A / B ) ) ) e. RR /\ B e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) < B <-> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) ) )
30	28, 6, 6, 7, 29	syl112anc 1275	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) < B <-> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) ) )
31	24, 30	mpbird 167	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) < B )
32	20, 31	eqbrtrd 4105	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   e. wcel 2200   =/= wne 2400   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   x. cmul 8004   < clt 8181   - cmin 8317   / cdiv 8819   QQcq 9814   |_cfl 10488   mod cmo 10544

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-n0 9370  df-z 9447  df-q 9815  df-rp 9850  df-fl 10490  df-mod 10545

This theorem is referenced by:  modqelico  10556  zmodfz  10568  modqid2  10573  modqabs  10579  modqmuladdim  10589  modaddmodup  10609  modqsubdir  10615  divalglemnn  12429  divalgmod  12438  bitsmod  12467  bitsinv1lem  12472  bezoutlemnewy  12517  bezoutlemstep  12518  eucalglt  12579  odzdvds  12768  fldivp1  12871  4sqlem6  12906  4sqlem12  12925  lgseisenlem1  15749