Theorem modqval 10326

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 10275 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 9627	. . 3 |- ( A e. QQ -> A e. RR )
2	1	3ad2ant1 1018	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. RR )
3		qre 9627	. . . 4 |- ( B e. QQ -> B e. RR )
4	3	3ad2ant2 1019	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
5		simp3 999	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
6	4, 5	elrpd 9695	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR+ )
7	5	gt0ne0d 8471	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
8		qdivcl 9645	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
9	7, 8	syld3an3 1283	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
10	9	flqcld 10279	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
11	10	zred 9377	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
12	4, 11	remulcld 7990	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( |_ ` ( A / B ) ) ) e. RR )
13	2, 12	resubcld 8340	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) e. RR )
14		oveq1 5884	. . . . . 6 |- ( x = A -> ( x / y ) = ( A / y ) )
15	14	fveq2d 5521	. . . . 5 |- ( x = A -> ( |_ ` ( x / y ) ) = ( |_ ` ( A / y ) ) )
16	15	oveq2d 5893	. . . 4 |- ( x = A -> ( y x. ( |_ ` ( x / y ) ) ) = ( y x. ( |_ ` ( A / y ) ) ) )
17		oveq12 5886	. . . 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 5885	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
20	19	fveq2d 5521	. . . . 5 |- ( y = B -> ( |_ ` ( A / y ) ) = ( |_ ` ( A / B ) ) )
21		oveq12 5886	. . . . 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 5893	. . 3 |- ( y = B -> ( A - ( y x. ( |_ ` ( A / y ) ) ) ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
24		df-mod 10325	. . 3 |- mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
25	18, 23, 24	ovmpog 6011	. 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 1238	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )

Syntax hints:   -> wi 4   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   RRcr 7812   0cc0 7813   x. cmul 7818   < clt 7994   - cmin 8130   / cdiv 8631   QQcq 9621   RR+crp 9655   |_cfl 10270   mod cmo 10324

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-n0 9179  df-z 9256  df-q 9622  df-rp 9656  df-fl 10272  df-mod 10325

This theorem is referenced by:  modqvalr  10327  modqcl  10328  modq0  10331  modqge0  10334  modqlt  10335  modqdiffl  10337  modqfrac  10339  modqmulnn  10344  zmodcl  10346  modqid  10351  modqcyc  10361  modqadd1  10363  modqmul1  10379  modqdi  10394  modqsubdir  10395  iexpcyc  10627  dvdsmod  11870  divalgmod  11934  modgcd  11994  prmdiv  12237  odzdvds  12247  fldivp1  12348  mulgmodid  13027