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Theorem modqm1p1mod0 10395
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 1002 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  ->  A  e.  QQ )
2 1z 9299 . . . . 5  |-  1  e.  ZZ
3 zq 9646 . . . . 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 1000 . . . . 5  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  QQ )
65adantr 276 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  ->  M  e.  QQ )
7 simpl3 1004 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
0  <  M )
8 modqaddmod 10383 . . . 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 1250 . . 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 5899 . . . . . 6  |-  ( ( A  mod  M )  =  ( M  - 
1 )  ->  (
( A  mod  M
)  +  1 )  =  ( ( M  -  1 )  +  1 ) )
1110oveq1d 5907 . . . . 5  |-  ( ( A  mod  M )  =  ( M  - 
1 )  ->  (
( ( A  mod  M )  +  1 )  mod  M )  =  ( ( ( M  -  1 )  +  1 )  mod  M
) )
1211adantl 277 . . . 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 9654 . . . . . . . 8  |-  ( M  e.  QQ  ->  M  e.  CC )
145, 13syl 14 . . . . . . 7  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  M  e.  CC )
1514adantr 276 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  ->  M  e.  CC )
16 npcan1 8355 . . . . . 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 5907 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( ( M  -  1 )  +  1 )  mod  M
)  =  ( M  mod  M ) )
19 modqid0 10370 . . . . 5  |-  ( ( M  e.  QQ  /\  0  <  M )  -> 
( M  mod  M
)  =  0 )
206, 7, 19syl2anc 411 . . . 4  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( M  mod  M
)  =  0 )
2112, 18, 203eqtrd 2226 . . 3  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  0 )
229, 21eqtr3d 2224 . 2  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  ( M  - 
1 ) )  -> 
( ( A  + 
1 )  mod  M
)  =  0 )
2322ex 115 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5892   CCcc 7829   0cc0 7831   1c1 7832    + caddc 7834    < clt 8012    - cmin 8148   ZZcz 9273   QQcq 9639    mod cmo 10342
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-mulrcl 7930  ax-addcom 7931  ax-mulcom 7932  ax-addass 7933  ax-mulass 7934  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-1rid 7938  ax-0id 7939  ax-rnegex 7940  ax-precex 7941  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-apti 7946  ax-pre-ltadd 7947  ax-pre-mulgt0 7948  ax-pre-mulext 7949  ax-arch 7950
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-reap 8552  df-ap 8559  df-div 8650  df-inn 8940  df-n0 9197  df-z 9274  df-q 9640  df-rp 9674  df-fl 10290  df-mod 10343
This theorem is referenced by: (None)
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