Theorem modqval 10541

Description: The value of the modulo operation. The modulo congruence notation of number theory, J == K (modulo N), can be expressed in our notation as ( J mod N ) = ( K mod N ). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive numbers to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) As with flqcl 10488 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)

Assertion

Ref	Expression
modqval	|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )

Proof of Theorem modqval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qre 9816	. . 3 |- ( A e. QQ -> A e. RR )
2	1	3ad2ant1 1042	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. RR )
3		qre 9816	. . . 4 |- ( B e. QQ -> B e. RR )
4	3	3ad2ant2 1043	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
5		simp3 1023	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
6	4, 5	elrpd 9885	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR+ )
7	5	gt0ne0d 8655	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
8		qdivcl 9834	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
9	7, 8	syld3an3 1316	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
10	9	flqcld 10492	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
11	10	zred 9565	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
12	4, 11	remulcld 8173	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( |_ ` ( A / B ) ) ) e. RR )
13	2, 12	resubcld 8523	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) e. RR )
14		oveq1 6007	. . . . . 6 |- ( x = A -> ( x / y ) = ( A / y ) )
15	14	fveq2d 5630	. . . . 5 |- ( x = A -> ( |_ ` ( x / y ) ) = ( |_ ` ( A / y ) ) )
16	15	oveq2d 6016	. . . 4 |- ( x = A -> ( y x. ( |_ ` ( x / y ) ) ) = ( y x. ( |_ ` ( A / y ) ) ) )
17		oveq12 6009	. . . 4 |- ( ( x = A /\ ( y x. ( |_ ` ( x / y ) ) ) = ( y x. ( |_ ` ( A / y ) ) ) ) -> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) = ( A - ( y x. ( |_ ` ( A / y ) ) ) ) )
18	16, 17	mpdan 421	. . 3 |- ( x = A -> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) = ( A - ( y x. ( |_ ` ( A / y ) ) ) ) )
19		oveq2 6008	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
20	19	fveq2d 5630	. . . . 5 |- ( y = B -> ( |_ ` ( A / y ) ) = ( |_ ` ( A / B ) ) )
21		oveq12 6009	. . . . 5 |- ( ( y = B /\ ( |_ ` ( A / y ) ) = ( |_ ` ( A / B ) ) ) -> ( y x. ( |_ ` ( A / y ) ) ) = ( B x. ( |_ ` ( A / B ) ) ) )
22	20, 21	mpdan 421	. . . 4 |- ( y = B -> ( y x. ( |_ ` ( A / y ) ) ) = ( B x. ( |_ ` ( A / B ) ) ) )
23	22	oveq2d 6016	. . 3 |- ( y = B -> ( A - ( y x. ( |_ ` ( A / y ) ) ) ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
24		df-mod 10540	. . 3 |- mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
25	18, 23, 24	ovmpog 6138	. 2 |- ( ( A e. RR /\ B e. RR+ /\ ( A - ( B x. ( |_ ` ( A / B ) ) ) ) e. RR ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
26	2, 6, 13, 25	syl3anc 1271	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   RRcr 7994   0cc0 7995   x. cmul 8000   < clt 8177   - cmin 8313   / cdiv 8815   QQcq 9810   RR+crp 9845   |_cfl 10483   mod cmo 10539

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-q 9811  df-rp 9846  df-fl 10485  df-mod 10540

This theorem is referenced by:  modqvalr  10542  modqcl  10543  modq0  10546  modqge0  10549  modqlt  10550  modqdiffl  10552  modqfrac  10554  modqmulnn  10559  zmodcl  10561  modqid  10566  modqcyc  10576  modqadd1  10578  modqmul1  10594  modqdi  10609  modqsubdir  10610  iexpcyc  10861  dvdsmod  12368  divalgmod  12433  modgcd  12507  prmdiv  12752  odzdvds  12763  fldivp1  12866  mulgmodid  13693  lgseisenlem4  15746