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Theorem modqm1p1mod0 9457
Description: If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)
Assertion
Ref Expression
modqm1p1mod0  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  ( M  -  1 )  -> 
( ( A  + 
1 )  mod  M
)  =  0 ) )

Proof of Theorem modqm1p1mod0
StepHypRef Expression
1 simpl1 942 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  ->  A  e.  QQ )
2 1z 8458 . . . . 5  |-  1  e.  ZZ
3 zq 8792 . . . . 5  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
42, 3mp1i 10 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
1  e.  QQ )
5 simp2 940 . . . . 5  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  QQ )
65adantr 270 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  ->  M  e.  QQ )
7 simpl3 944 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
0  <  M )
8 modqaddmod 9445 . . . 4  |-  ( ( ( A  e.  QQ  /\  1  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  ( ( A  +  1 )  mod  M ) )
91, 4, 6, 7, 8syl22anc 1171 . . 3  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  ( ( A  +  1 )  mod  M ) )
10 oveq1 5550 . . . . . 6  |-  ( ( A  mod  M )  =  ( M  - 
1 )  ->  (
( A  mod  M
)  +  1 )  =  ( ( M  -  1 )  +  1 ) )
1110oveq1d 5558 . . . . 5  |-  ( ( A  mod  M )  =  ( M  - 
1 )  ->  (
( ( A  mod  M )  +  1 )  mod  M )  =  ( ( ( M  -  1 )  +  1 )  mod  M
) )
1211adantl 271 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  ( ( ( M  -  1 )  +  1 )  mod  M ) )
13 qcn 8800 . . . . . . . 8  |-  ( M  e.  QQ  ->  M  e.  CC )
145, 13syl 14 . . . . . . 7  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  CC )
1514adantr 270 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  ->  M  e.  CC )
16 npcan1 7549 . . . . . 6  |-  ( M  e.  CC  ->  (
( M  -  1 )  +  1 )  =  M )
1715, 16syl 14 . . . . 5  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( M  - 
1 )  +  1 )  =  M )
1817oveq1d 5558 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( ( M  -  1 )  +  1 )  mod  M
)  =  ( M  mod  M ) )
19 modqid0 9432 . . . . 5  |-  ( ( M  e.  QQ  /\  0  <  M )  -> 
( M  mod  M
)  =  0 )
206, 7, 19syl2anc 403 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( M  mod  M
)  =  0 )
2112, 18, 203eqtrd 2118 . . 3  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  0 )
229, 21eqtr3d 2116 . 2  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( A  + 
1 )  mod  M
)  =  0 )
2322ex 113 1  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  ( M  -  1 )  -> 
( ( A  + 
1 )  mod  M
)  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046    < clt 7215    - cmin 7346   ZZcz 8432   QQcq 8785    mod cmo 9404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-q 8786  df-rp 8816  df-fl 9352  df-mod 9405
This theorem is referenced by: (None)
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