Theorem modqlt 10106

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 9426	. . . . . 6 |- ( A e. QQ -> A e. CC )
2	1	3ad2ant1 1002	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. CC )
3		qcn 9426	. . . . . 6 |- ( B e. QQ -> B e. CC )
4	3	3ad2ant2 1003	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
5		qre 9417	. . . . . . 7 |- ( B e. QQ -> B e. RR )
6	5	3ad2ant2 1003	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
7		simp3 983	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
8	6, 7	gt0ap0d 8391	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B # 0 )
9	2, 4, 8	divcanap2d 8552	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( A / B ) ) = A )
10	9	oveq1d 5789	. . 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 8274	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
12		qdivcl 9435	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
13	11, 12	syld3an3 1261	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
14		qcn 9426	. . . . 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 10050	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
17	16	zcnd 9174	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. CC )
18	4, 15, 17	subdid 8176	. . 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 10097	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
20	10, 18, 19	3eqtr4rd 2183	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) )
21		qfraclt1 10053	. . . . 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 8546	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B / B ) = 1 )
24	22, 23	breqtrrd 3956	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) )
25		qre 9417	. . . . . 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 9173	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
28	26, 27	resubcld 8143	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) e. RR )
29		ltmuldiv2 8633	. . . 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 1220	. . 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 166	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) < B )
32	20, 31	eqbrtrd 3950	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 962   e. wcel 1480   =/= wne 2308   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   x. cmul 7625   < clt 7800   - cmin 7933   / cdiv 8432   QQcq 9411   |_cfl 10041   mod cmo 10095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-q 9412  df-rp 9442  df-fl 10043  df-mod 10096

This theorem is referenced by:  modqelico  10107  zmodfz  10119  modqid2  10124  modqabs  10130  modqmuladdim  10140  modaddmodup  10160  modqsubdir  10166  divalglemnn  11615  divalgmod  11624  bezoutlemnewy  11684  bezoutlemstep  11685  eucalglt  11738