Theorem modqval 10585

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 10532 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 9858	. . 3 |- ( A e. QQ -> A e. RR )
2	1	3ad2ant1 1044	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. RR )
3		qre 9858	. . . 4 |- ( B e. QQ -> B e. RR )
4	3	3ad2ant2 1045	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
5		simp3 1025	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
6	4, 5	elrpd 9927	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR+ )
7	5	gt0ne0d 8691	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
8		qdivcl 9876	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
9	7, 8	syld3an3 1318	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
10	9	flqcld 10536	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
11	10	zred 9601	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
12	4, 11	remulcld 8209	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( |_ ` ( A / B ) ) ) e. RR )
13	2, 12	resubcld 8559	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) e. RR )
14		oveq1 6024	. . . . . 6 |- ( x = A -> ( x / y ) = ( A / y ) )
15	14	fveq2d 5643	. . . . 5 |- ( x = A -> ( |_ ` ( x / y ) ) = ( |_ ` ( A / y ) ) )
16	15	oveq2d 6033	. . . 4 |- ( x = A -> ( y x. ( |_ ` ( x / y ) ) ) = ( y x. ( |_ ` ( A / y ) ) ) )
17		oveq12 6026	. . . 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 6025	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
20	19	fveq2d 5643	. . . . 5 |- ( y = B -> ( |_ ` ( A / y ) ) = ( |_ ` ( A / B ) ) )
21		oveq12 6026	. . . . 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 6033	. . 3 |- ( y = B -> ( A - ( y x. ( |_ ` ( A / y ) ) ) ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
24		df-mod 10584	. . 3 |- mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
25	18, 23, 24	ovmpog 6155	. 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 1273	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   RRcr 8030   0cc0 8031   x. cmul 8036   < clt 8213   - cmin 8349   / cdiv 8851   QQcq 9852   RR+crp 9887   |_cfl 10527   mod cmo 10583

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-n0 9402  df-z 9479  df-q 9853  df-rp 9888  df-fl 10529  df-mod 10584

This theorem is referenced by:  modqvalr  10586  modqcl  10587  modq0  10590  modqge0  10593  modqlt  10594  modqdiffl  10596  modqfrac  10598  modqmulnn  10603  zmodcl  10605  modqid  10610  modqcyc  10620  modqadd1  10622  modqmul1  10638  modqdi  10653  modqsubdir  10654  iexpcyc  10905  dvdsmod  12422  divalgmod  12487  modgcd  12561  prmdiv  12806  odzdvds  12817  fldivp1  12920  mulgmodid  13747  lgseisenlem4  15801