Theorem modqlt 10428

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 9711	. . . . . 6 |- ( A e. QQ -> A e. CC )
2	1	3ad2ant1 1020	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. CC )
3		qcn 9711	. . . . . 6 |- ( B e. QQ -> B e. CC )
4	3	3ad2ant2 1021	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
5		qre 9702	. . . . . . 7 |- ( B e. QQ -> B e. RR )
6	5	3ad2ant2 1021	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
7		simp3 1001	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
8	6, 7	gt0ap0d 8659	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B # 0 )
9	2, 4, 8	divcanap2d 8822	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( A / B ) ) = A )
10	9	oveq1d 5938	. . 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 8542	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
12		qdivcl 9720	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
13	11, 12	syld3an3 1294	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
14		qcn 9711	. . . . 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 10370	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
17	16	zcnd 9452	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. CC )
18	4, 15, 17	subdid 8443	. . 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 10419	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
20	10, 18, 19	3eqtr4rd 2240	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) )
21		qfraclt1 10373	. . . . 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 8816	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B / B ) = 1 )
24	22, 23	breqtrrd 4062	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) )
25		qre 9702	. . . . . 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 9451	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
28	26, 27	resubcld 8410	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) e. RR )
29		ltmuldiv2 8905	. . . 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 1253	. . 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 4056	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   e. wcel 2167   =/= wne 2367   class class class wbr 4034   ` cfv 5259  (class class class)co 5923   CCcc 7880   RRcr 7881   0cc0 7882   1c1 7883   x. cmul 7887   < clt 8064   - cmin 8200   / cdiv 8702   QQcq 9696   |_cfl 10361   mod cmo 10417

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7973  ax-resscn 7974  ax-1cn 7975  ax-1re 7976  ax-icn 7977  ax-addcl 7978  ax-addrcl 7979  ax-mulcl 7980  ax-mulrcl 7981  ax-addcom 7982  ax-mulcom 7983  ax-addass 7984  ax-mulass 7985  ax-distr 7986  ax-i2m1 7987  ax-0lt1 7988  ax-1rid 7989  ax-0id 7990  ax-rnegex 7991  ax-precex 7992  ax-cnre 7993  ax-pre-ltirr 7994  ax-pre-ltwlin 7995  ax-pre-lttrn 7996  ax-pre-apti 7997  ax-pre-ltadd 7998  ax-pre-mulgt0 7999  ax-pre-mulext 8000  ax-arch 8001

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6200  df-2nd 6201  df-pnf 8066  df-mnf 8067  df-xr 8068  df-ltxr 8069  df-le 8070  df-sub 8202  df-neg 8203  df-reap 8605  df-ap 8612  df-div 8703  df-inn 8994  df-n0 9253  df-z 9330  df-q 9697  df-rp 9732  df-fl 10363  df-mod 10418

This theorem is referenced by:  modqelico  10429  zmodfz  10441  modqid2  10446  modqabs  10452  modqmuladdim  10462  modaddmodup  10482  modqsubdir  10488  divalglemnn  12086  divalgmod  12095  bitsmod  12124  bitsinv1lem  12129  bezoutlemnewy  12174  bezoutlemstep  12175  eucalglt  12236  odzdvds  12425  fldivp1  12528  4sqlem6  12563  4sqlem12  12582  lgseisenlem1  15337