Theorem modq0 10766

Description: A mod B is zero iff A is evenly divisible by B. (Contributed by Jim Kingdon, 17-Oct-2021.)

Assertion

Ref	Expression
modq0	|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A / B ) e. ZZ ) )

Proof of Theorem modq0

Step	Hyp	Ref	Expression
1		modqval 10761	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
2	1	eqeq1d 2247	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = 0 ) )
3		qcn 10034	. . . . . 6 |- ( A e. QQ -> A e. CC )
4	3	3ad2ant1 1049	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. CC )
5		qcn 10034	. . . . . . 7 |- ( B e. QQ -> B e. CC )
6	5	3ad2ant2 1050	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
7		simp3 1030	. . . . . . . . . 10 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
8	7	gt0ne0d 8840	. . . . . . . . 9 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
9		qdivcl 10043	. . . . . . . . 9 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
10	8, 9	syld3an3 1323	. . . . . . . 8 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
11	10	flqcld 10712	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
12	11	zcnd 9769	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. CC )
13	6, 12	mulcld 8346	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( |_ ` ( A / B ) ) ) e. CC )
14	4, 13	subeq0ad 8647	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = 0 <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) )
15	2, 14	bitrd 188	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) )
16		qre 10025	. . . . . . 7 |- ( B e. QQ -> B e. RR )
17	16	3ad2ant2 1050	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
18	17, 7	gt0ap0d 8957	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B # 0 )
19	4, 12, 6, 18	divmulap2d 9154	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) = ( |_ ` ( A / B ) ) <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) )
20		eqcom 2240	. . . 4 |- ( ( A / B ) = ( |_ ` ( A / B ) ) <-> ( |_ ` ( A / B ) ) = ( A / B ) )
21	19, 20	bitr3di 195	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A = ( B x. ( |_ ` ( A / B ) ) ) <-> ( |_ ` ( A / B ) ) = ( A / B ) ) )
22	15, 21	bitrd 188	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( |_ ` ( A / B ) ) = ( A / B ) ) )
23		flqidz 10721	. . 3 |- ( ( A / B ) e. QQ -> ( ( |_ ` ( A / B ) ) = ( A / B ) <-> ( A / B ) e. ZZ ) )
24	10, 23	syl 14	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( |_ ` ( A / B ) ) = ( A / B ) <-> ( A / B ) e. ZZ ) )
25	22, 24	bitrd 188	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A / B ) e. ZZ ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   x. cmul 8184   < clt 8360   - cmin 8497   / cdiv 9002   ZZcz 9644   QQcq 10019   |_cfl 10703   mod cmo 10759

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-q 10020  df-rp 10055  df-fl 10705  df-mod 10760

This theorem is used by:  mulqmod0  10767  negqmod0  10768  modqid0  10787  q2txmodxeq0  10821  addmodlteq  10835  dvdsval3  12558