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Theorem modqltm1p1mod 9846
Description: If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.)
Assertion
Ref Expression
modqltm1p1mod  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  + 
1 )  mod  M
)  =  ( ( A  mod  M )  +  1 ) )

Proof of Theorem modqltm1p1mod
StepHypRef Expression
1 simpll 497 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  A  e.  QQ )
2 1z 8839 . . . 4  |-  1  e.  ZZ
3 zq 9174 . . . 4  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
42, 3mp1i 10 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
1  e.  QQ )
5 simprl 499 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  M  e.  QQ )
6 simprr 500 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  <  M )
7 modqaddmod 9833 . . 3  |-  ( ( ( A  e.  QQ  /\  1  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  ( ( A  +  1 )  mod  M ) )
81, 4, 5, 6, 7syl22anc 1176 . 2  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  ( ( A  +  1 )  mod  M ) )
91, 5, 6modqcld 9798 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( A  mod  M
)  e.  QQ )
10 qaddcl 9183 . . . 4  |-  ( ( ( A  mod  M
)  e.  QQ  /\  1  e.  QQ )  ->  ( ( A  mod  M )  +  1 )  e.  QQ )
119, 4, 10syl2anc 404 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  mod  M )  +  1 )  e.  QQ )
12 0red 7552 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  e.  RR )
13 qre 9173 . . . . 5  |-  ( ( A  mod  M )  e.  QQ  ->  ( A  mod  M )  e.  RR )
149, 13syl 14 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( A  mod  M
)  e.  RR )
15 1red 7566 . . . . 5  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
1  e.  RR )
1614, 15readdcld 7580 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  mod  M )  +  1 )  e.  RR )
17 modqge0 9802 . . . . 5  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( A  mod  M
) )
181, 5, 6, 17syl3anc 1175 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  <_  ( A  mod  M ) )
1914lep1d 8455 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( A  mod  M
)  <_  ( ( A  mod  M )  +  1 ) )
2012, 14, 16, 18, 19letrd 7670 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  <_  ( ( A  mod  M )  +  1 ) )
21 simplr 498 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( A  mod  M
)  <  ( M  -  1 ) )
22 qre 9173 . . . . . 6  |-  ( M  e.  QQ  ->  M  e.  RR )
235, 22syl 14 . . . . 5  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  M  e.  RR )
2414, 15, 23ltaddsubd 8085 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( A  mod  M )  +  1 )  <  M  <->  ( A  mod  M )  <  ( M  - 
1 ) ) )
2521, 24mpbird 166 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  mod  M )  +  1 )  <  M )
26 modqid 9819 . . 3  |-  ( ( ( ( ( A  mod  M )  +  1 )  e.  QQ  /\  M  e.  QQ )  /\  ( 0  <_ 
( ( A  mod  M )  +  1 )  /\  ( ( A  mod  M )  +  1 )  <  M
) )  ->  (
( ( A  mod  M )  +  1 )  mod  M )  =  ( ( A  mod  M )  +  1 ) )
2711, 5, 20, 25, 26syl22anc 1176 . 2  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  ( ( A  mod  M )  +  1 ) )
288, 27eqtr3d 2123 1  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  + 
1 )  mod  M
)  =  ( ( A  mod  M )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   class class class wbr 3853  (class class class)co 5668   RRcr 7412   0cc0 7413   1c1 7414    + caddc 7416    < clt 7585    <_ cle 7586    - cmin 7716   ZZcz 8813   QQcq 9167    mod cmo 9792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526  ax-arch 7527
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-po 4134  df-iso 4135  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-n0 8737  df-z 8814  df-q 9168  df-rp 9198  df-fl 9740  df-mod 9793
This theorem is referenced by: (None)
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