ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  modaddmodup Unicode version

Theorem modaddmodup 10128
Description: The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
Assertion
Ref Expression
modaddmodup  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  (
( B  +  ( A  mod  M ) )  -  M )  =  ( ( B  +  A )  mod 
M ) ) )

Proof of Theorem modaddmodup
StepHypRef Expression
1 elfzoelz 9892 . . . . . . 7  |-  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  B  e.  ZZ )
21adantr 274 . . . . . 6  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  B  e.  ZZ )
3 zmodcl 10085 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( A  mod  M
)  e.  NN0 )
43adantl 275 . . . . . . 7  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( A  mod  M
)  e.  NN0 )
54nn0zd 9139 . . . . . 6  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( A  mod  M
)  e.  ZZ )
62, 5zaddcld 9145 . . . . 5  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( B  +  ( A  mod  M ) )  e.  ZZ )
7 zq 9386 . . . . 5  |-  ( ( B  +  ( A  mod  M ) )  e.  ZZ  ->  ( B  +  ( A  mod  M ) )  e.  QQ )
86, 7syl 14 . . . 4  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( B  +  ( A  mod  M ) )  e.  QQ )
9 simprr 506 . . . . 5  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  M  e.  NN )
10 nnq 9393 . . . . 5  |-  ( M  e.  NN  ->  M  e.  QQ )
119, 10syl 14 . . . 4  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  M  e.  QQ )
129nngt0d 8732 . . . 4  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
0  <  M )
13 elfzole1 9900 . . . . . 6  |-  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  ( M  -  ( A  mod  M ) )  <_  B )
1413adantr 274 . . . . 5  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( M  -  ( A  mod  M ) )  <_  B )
159nnred 8701 . . . . . 6  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  M  e.  RR )
163nn0red 8999 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( A  mod  M
)  e.  RR )
1716adantl 275 . . . . . 6  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( A  mod  M
)  e.  RR )
181zred 9141 . . . . . . 7  |-  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  B  e.  RR )
1918adantr 274 . . . . . 6  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  B  e.  RR )
2015, 17, 19lesubaddd 8272 . . . . 5  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( ( M  -  ( A  mod  M ) )  <_  B  <->  M  <_  ( B  +  ( A  mod  M ) ) ) )
2114, 20mpbid 146 . . . 4  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  M  <_  ( B  +  ( A  mod  M ) ) )
22 elfzolt2 9901 . . . . . . 7  |-  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  B  <  M
)
2322adantr 274 . . . . . 6  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  B  <  M )
24 zq 9386 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  QQ )
2524ad2antrl 481 . . . . . . 7  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  A  e.  QQ )
26 modqlt 10074 . . . . . . 7  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  ( A  mod  M )  < 
M )
2725, 11, 12, 26syl3anc 1201 . . . . . 6  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( A  mod  M
)  <  M )
2819, 17, 15, 15, 23, 27lt2addd 8297 . . . . 5  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( B  +  ( A  mod  M ) )  <  ( M  +  M ) )
299nncnd 8702 . . . . . 6  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  M  e.  CC )
30292timesd 8930 . . . . 5  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( 2  x.  M
)  =  ( M  +  M ) )
3128, 30breqtrrd 3926 . . . 4  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( B  +  ( A  mod  M ) )  <  ( 2  x.  M ) )
32 q2submod 10126 . . . 4  |-  ( ( ( ( B  +  ( A  mod  M ) )  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( M  <_  ( B  +  ( A  mod  M ) )  /\  ( B  +  ( A  mod  M ) )  <  (
2  x.  M ) ) )  ->  (
( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  ( A  mod  M ) )  -  M
) )
338, 11, 12, 21, 31, 32syl32anc 1209 . . 3  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  ( A  mod  M ) )  -  M
) )
34 zq 9386 . . . . 5  |-  ( B  e.  ZZ  ->  B  e.  QQ )
352, 34syl 14 . . . 4  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  B  e.  QQ )
36 modqadd2mod 10115 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod 
M ) )
3725, 35, 11, 12, 36syl22anc 1202 . . 3  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod 
M ) )
3833, 37eqtr3d 2152 . 2  |-  ( ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( ( B  +  ( A  mod  M ) )  -  M )  =  ( ( B  +  A )  mod 
M ) )
3938expcom 115 1  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  (
( B  +  ( A  mod  M ) )  -  M )  =  ( ( B  +  A )  mod 
M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588    + caddc 7591    x. cmul 7593    < clt 7768    <_ cle 7769    - cmin 7901   NNcn 8688   2c2 8739   NN0cn0 8945   ZZcz 9022   QQcq 9379  ..^cfzo 9887    mod cmo 10063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator