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Theorem modqltm1p1mod 10344
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 527 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  A  e.  QQ )
2 1z 9250 . . . 4  |-  1  e.  ZZ
3 zq 9597 . . . 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 529 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  ->  M  e.  QQ )
6 simprr 531 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  <  M )
7 modqaddmod 10331 . . 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 1239 . 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 10296 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( A  mod  M
)  e.  QQ )
10 qaddcl 9606 . . . 4  |-  ( ( ( A  mod  M
)  e.  QQ  /\  1  e.  QQ )  ->  ( ( A  mod  M )  +  1 )  e.  QQ )
119, 4, 10syl2anc 411 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  mod  M )  +  1 )  e.  QQ )
12 0red 7933 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  e.  RR )
13 qre 9596 . . . . 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 7947 . . . . 5  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
1  e.  RR )
1614, 15readdcld 7961 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  mod  M )  +  1 )  e.  RR )
17 modqge0 10300 . . . . 5  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( A  mod  M
) )
181, 5, 6, 17syl3anc 1238 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  <_  ( A  mod  M ) )
1914lep1d 8859 . . . 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 8055 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
0  <_  ( ( A  mod  M )  +  1 ) )
21 simplr 528 . . . 4  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( A  mod  M
)  <  ( M  -  1 ) )
22 qre 9596 . . . . . 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 8476 . . . 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 167 . . 3  |-  ( ( ( A  e.  QQ  /\  ( A  mod  M
)  <  ( M  -  1 ) )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  mod  M )  +  1 )  <  M )
26 modqid 10317 . . 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 1239 . 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 2210 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 104    = wceq 1353    e. wcel 2146   class class class wbr 3998  (class class class)co 5865   RRcr 7785   0cc0 7786   1c1 7787    + caddc 7789    < clt 7966    <_ cle 7967    - cmin 8102   ZZcz 9224   QQcq 9590    mod cmo 10290
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-n0 9148  df-z 9225  df-q 9591  df-rp 9623  df-fl 10238  df-mod 10291
This theorem is referenced by: (None)
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