Theorem modqlt 10694

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 9965	. . . . . 6 |- ( A e. QQ -> A e. CC )
2	1	3ad2ant1 1045	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. CC )
3		qcn 9965	. . . . . 6 |- ( B e. QQ -> B e. CC )
4	3	3ad2ant2 1046	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
5		qre 9956	. . . . . . 7 |- ( B e. QQ -> B e. RR )
6	5	3ad2ant2 1046	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
7		simp3 1026	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
8	6, 7	gt0ap0d 8902	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B # 0 )
9	2, 4, 8	divcanap2d 9065	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( A / B ) ) = A )
10	9	oveq1d 6064	. . 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 8785	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
12		qdivcl 9974	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
13	11, 12	syld3an3 1319	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
14		qcn 9965	. . . . 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 10636	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
17	16	zcnd 9700	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. CC )
18	4, 15, 17	subdid 8686	. . 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 10685	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
20	10, 18, 19	3eqtr4rd 2276	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) )
21		qfraclt1 10639	. . . . 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 9059	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B / B ) = 1 )
24	22, 23	breqtrrd 4136	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) )
25		qre 9956	. . . . . 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 9699	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
28	26, 27	resubcld 8653	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) e. RR )
29		ltmuldiv2 9148	. . . 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 1278	. . 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 4130	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   e. wcel 2203   =/= wne 2412   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   CCcc 8124   RRcr 8125   0cc0 8126   1c1 8127   x. cmul 8131   < clt 8307   - cmin 8443   / cdiv 8945   QQcq 9950   |_cfl 10627   mod cmo 10683

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-n0 9496  df-z 9577  df-q 9951  df-rp 9986  df-fl 10629  df-mod 10684

This theorem is referenced by:  modqelico  10695  zmodfz  10707  modqid2  10712  modqabs  10718  modqmuladdim  10728  modaddmodup  10748  modqsubdir  10754  divalglemnn  12600  divalgmod  12609  bitsmod  12638  bitsinv1lem  12643  bezoutlemnewy  12688  bezoutlemstep  12689  eucalglt  12750  odzdvds  12939  fldivp1  13042  4sqlem6  13077  4sqlem12  13096  lgseisenlem1  15935