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Theorem modqval 9274
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 9225 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.
StepHypRef Expression
1 qre 8657 . . 3  |-  ( A  e.  QQ  ->  A  e.  RR )
213ad2ant1 936 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  A  e.  RR )
3 qre 8657 . . . 4  |-  ( B  e.  QQ  ->  B  e.  RR )
433ad2ant2 937 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  RR )
5 simp3 917 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  0  <  B )
64, 5elrpd 8718 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  RR+ )
75gt0ne0d 7578 . . . . . . 7  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  =/=  0 )
8 qdivcl 8675 . . . . . . 7  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  B  =/=  0 )  ->  ( A  /  B )  e.  QQ )
97, 8syld3an3 1191 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  /  B )  e.  QQ )
109flqcld 9227 . . . . 5  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  ZZ )
1110zred 8419 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( |_ `  ( A  /  B ) )  e.  RR )
124, 11remulcld 7115 . . 3  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( B  x.  ( |_ `  ( A  /  B
) ) )  e.  RR )
132, 12resubcld 7451 . 2  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  e.  RR )
14 oveq1 5547 . . . . . 6  |-  ( x  =  A  ->  (
x  /  y )  =  ( A  / 
y ) )
1514fveq2d 5210 . . . . 5  |-  ( x  =  A  ->  ( |_ `  ( x  / 
y ) )  =  ( |_ `  ( A  /  y ) ) )
1615oveq2d 5556 . . . 4  |-  ( x  =  A  ->  (
y  x.  ( |_
`  ( x  / 
y ) ) )  =  ( y  x.  ( |_ `  ( A  /  y ) ) ) )
17 oveq12 5549 . . . 4  |-  ( ( x  =  A  /\  ( y  x.  ( |_ `  ( x  / 
y ) ) )  =  ( y  x.  ( |_ `  ( A  /  y ) ) ) )  ->  (
x  -  ( y  x.  ( |_ `  ( x  /  y
) ) ) )  =  ( A  -  ( y  x.  ( |_ `  ( A  / 
y ) ) ) ) )
1816, 17mpdan 406 . . 3  |-  ( x  =  A  ->  (
x  -  ( y  x.  ( |_ `  ( x  /  y
) ) ) )  =  ( A  -  ( y  x.  ( |_ `  ( A  / 
y ) ) ) ) )
19 oveq2 5548 . . . . . 6  |-  ( y  =  B  ->  ( A  /  y )  =  ( A  /  B
) )
2019fveq2d 5210 . . . . 5  |-  ( y  =  B  ->  ( |_ `  ( A  / 
y ) )  =  ( |_ `  ( A  /  B ) ) )
21 oveq12 5549 . . . . 5  |-  ( ( y  =  B  /\  ( |_ `  ( A  /  y ) )  =  ( |_ `  ( A  /  B
) ) )  -> 
( y  x.  ( |_ `  ( A  / 
y ) ) )  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) )
2220, 21mpdan 406 . . . 4  |-  ( y  =  B  ->  (
y  x.  ( |_
`  ( A  / 
y ) ) )  =  ( B  x.  ( |_ `  ( A  /  B ) ) ) )
2322oveq2d 5556 . . 3  |-  ( y  =  B  ->  ( A  -  ( y  x.  ( |_ `  ( A  /  y ) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
24 df-mod 9273 . . 3  |-  mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  (
x  /  y ) ) ) ) )
2518, 23, 24ovmpt2g 5663 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) )  e.  RR )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
262, 6, 13, 25syl3anc 1146 1  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 896    = wceq 1259    e. wcel 1409    =/= wne 2220   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   RRcr 6946   0cc0 6947    x. cmul 6952    < clt 7119    - cmin 7245    / cdiv 7725   QQcq 8651   RR+crp 8681   |_cfl 9220    mod cmo 9272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060  ax-arch 7061
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-n0 8240  df-z 8303  df-q 8652  df-rp 8682  df-fl 9222  df-mod 9273
This theorem is referenced by:  modqvalr  9275  modqcl  9276  modq0  9279  modqge0  9282  modqlt  9283  modqdiffl  9285  modqfrac  9287  modqmulnn  9292  zmodcl  9294  modqid  9299  modqcyc  9309  modqadd1  9311  modqmul1  9327  modqdi  9342  modqsubdir  9343  iexpcyc  9523  dvdsmod  10174  divalgmod  10239
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