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Theorem modaddmodlo 11763
Description: The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
Assertion
Ref Expression
modaddmodlo  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  ( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M
) ) )

Proof of Theorem modaddmodlo
StepHypRef Expression
1 elfzoelz 11553 . . . . . . . 8  |-  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  B  e.  ZZ )
21zred 10747 . . . . . . 7  |-  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  B  e.  RR )
32adantr 465 . . . . . 6  |-  ( ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  ->  B  e.  RR )
4 zmodcl 11727 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( A  mod  M
)  e.  NN0 )
54nn0red 10637 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( A  mod  M
)  e.  RR )
65adantl 466 . . . . . 6  |-  ( ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( A  mod  M
)  e.  RR )
73, 6readdcld 9413 . . . . 5  |-  ( ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  /\  ( A  e.  ZZ  /\  M  e.  NN ) )  -> 
( B  +  ( A  mod  M ) )  e.  RR )
87ancoms 453 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( B  +  ( A  mod  M ) )  e.  RR )
9 nnrp 11000 . . . . 5  |-  ( M  e.  NN  ->  M  e.  RR+ )
109ad2antlr 726 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  M  e.  RR+ )
112adantl 466 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  B  e.  RR )
125adantr 465 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( A  mod  M
)  e.  RR )
13 elfzole1 11560 . . . . . 6  |-  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  0  <_  B )
1413adantl 466 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
0  <_  B )
154nn0ge0d 10639 . . . . . 6  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  0  <_  ( A  mod  M ) )
1615adantr 465 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
0  <_  ( A  mod  M ) )
1711, 12, 14, 16addge0d 9915 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
0  <_  ( B  +  ( A  mod  M ) ) )
18 elfzolt2 11561 . . . . . 6  |-  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  B  <  ( M  -  ( A  mod  M ) ) )
1918adantl 466 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  B  <  ( M  -  ( A  mod  M ) ) )
20 nnre 10329 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  RR )
2120ad2antlr 726 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  M  e.  RR )
2211, 12, 21ltaddsubd 9939 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( ( B  +  ( A  mod  M ) )  <  M  <->  B  <  ( M  -  ( A  mod  M ) ) ) )
2319, 22mpbird 232 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( B  +  ( A  mod  M ) )  <  M )
24 modid 11732 . . . 4  |-  ( ( ( ( B  +  ( A  mod  M ) )  e.  RR  /\  M  e.  RR+ )  /\  ( 0  <_  ( B  +  ( A  mod  M ) )  /\  ( B  +  ( A  mod  M ) )  <  M ) )  ->  ( ( B  +  ( A  mod  M ) )  mod  M
)  =  ( B  +  ( A  mod  M ) ) )
258, 10, 17, 23, 24syl22anc 1219 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( ( B  +  ( A  mod  M ) )  mod  M )  =  ( B  +  ( A  mod  M ) ) )
26 zre 10650 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
2726adantr 465 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  A  e.  RR )
2827adantr 465 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  A  e.  RR )
29 modadd2mod 11749 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  (
( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod 
M ) )
3028, 11, 10, 29syl3anc 1218 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod 
M ) )
3125, 30eqtr3d 2477 . 2  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M ) )
3231ex 434 1  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  ( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4292  (class class class)co 6091   RRcr 9281   0cc0 9282    + caddc 9285    < clt 9418    <_ cle 9419    - cmin 9595   NNcn 10322   ZZcz 10646   RR+crp 10991  ..^cfzo 11548    mod cmo 11708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709
This theorem is referenced by:  cshwidxmod  12440
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