Theorem modqval 10576

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 10523 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 9849	. . 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 9849	. . . 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 9918	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR+ )
7	5	gt0ne0d 8682	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
8		qdivcl 9867	. . . . . . 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 10527	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
11	10	zred 9592	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
12	4, 11	remulcld 8200	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( |_ ` ( A / B ) ) ) e. RR )
13	2, 12	resubcld 8550	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) e. RR )
14		oveq1 6020	. . . . . 6 |- ( x = A -> ( x / y ) = ( A / y ) )
15	14	fveq2d 5639	. . . . 5 |- ( x = A -> ( |_ ` ( x / y ) ) = ( |_ ` ( A / y ) ) )
16	15	oveq2d 6029	. . . 4 |- ( x = A -> ( y x. ( |_ ` ( x / y ) ) ) = ( y x. ( |_ ` ( A / y ) ) ) )
17		oveq12 6022	. . . 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 6021	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
20	19	fveq2d 5639	. . . . 5 |- ( y = B -> ( |_ ` ( A / y ) ) = ( |_ ` ( A / B ) ) )
21		oveq12 6022	. . . . 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 6029	. . 3 |- ( y = B -> ( A - ( y x. ( |_ ` ( A / y ) ) ) ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
24		df-mod 10575	. . 3 |- mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
25	18, 23, 24	ovmpog 6151	. 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 4086   ` cfv 5324  (class class class)co 6013   RRcr 8021   0cc0 8022   x. cmul 8027   < clt 8204   - cmin 8340   / cdiv 8842   QQcq 9843   RR+crp 9878   |_cfl 10518   mod cmo 10574

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-n0 9393  df-z 9470  df-q 9844  df-rp 9879  df-fl 10520  df-mod 10575

This theorem is referenced by:  modqvalr  10577  modqcl  10578  modq0  10581  modqge0  10584  modqlt  10585  modqdiffl  10587  modqfrac  10589  modqmulnn  10594  zmodcl  10596  modqid  10601  modqcyc  10611  modqadd1  10613  modqmul1  10629  modqdi  10644  modqsubdir  10645  iexpcyc  10896  dvdsmod  12413  divalgmod  12478  modgcd  12552  prmdiv  12797  odzdvds  12808  fldivp1  12911  mulgmodid  13738  lgseisenlem4  15792