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Theorem p1modz1 11690
Description: If a number greater than 1 divides another number, the second number increased by 1 is 1 modulo the first number. (Contributed by AV, 19-Mar-2022.)
Assertion
Ref Expression
p1modz1  |-  ( ( M  ||  A  /\  1  <  M )  -> 
( ( A  + 
1 )  mod  M
)  =  1 )

Proof of Theorem p1modz1
StepHypRef Expression
1 dvdszrcl 11688 . . 3  |-  ( M 
||  A  ->  ( M  e.  ZZ  /\  A  e.  ZZ ) )
2 0red 7879 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  1  <  M )  -> 
0  e.  RR )
3 1red 7893 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  1  <  M )  -> 
1  e.  RR )
4 zre 9171 . . . . . . . . . . . . . . 15  |-  ( M  e.  ZZ  ->  M  e.  RR )
54adantr 274 . . . . . . . . . . . . . 14  |-  ( ( M  e.  ZZ  /\  1  <  M )  ->  M  e.  RR )
62, 3, 53jca 1162 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  1  <  M )  -> 
( 0  e.  RR  /\  1  e.  RR  /\  M  e.  RR )
)
7 0lt1 8002 . . . . . . . . . . . . . . 15  |-  0  <  1
87a1i 9 . . . . . . . . . . . . . 14  |-  ( M  e.  ZZ  ->  0  <  1 )
98anim1i 338 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  1  <  M )  -> 
( 0  <  1  /\  1  <  M ) )
10 lttr 7951 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  M  e.  RR )  ->  (
( 0  <  1  /\  1  <  M )  ->  0  <  M
) )
116, 9, 10sylc 62 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  1  <  M )  -> 
0  <  M )
1211ex 114 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  (
1  <  M  ->  0  <  M ) )
13 elnnz 9177 . . . . . . . . . . . 12  |-  ( M  e.  NN  <->  ( M  e.  ZZ  /\  0  < 
M ) )
1413simplbi2 383 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  (
0  <  M  ->  M  e.  NN ) )
1512, 14syld 45 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  (
1  <  M  ->  M  e.  NN ) )
1615adantr 274 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  A  e.  ZZ )  ->  ( 1  <  M  ->  M  e.  NN ) )
1716imp 123 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M
)  ->  M  e.  NN )
18 dvdsmod0 11689 . . . . . . . 8  |-  ( ( M  e.  NN  /\  M  ||  A )  -> 
( A  mod  M
)  =  0 )
1917, 18sylan 281 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  M  ||  A )  ->  ( A  mod  M )  =  0 )
2019ex 114 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M
)  ->  ( M  ||  A  ->  ( A  mod  M )  =  0 ) )
21 oveq1 5831 . . . . . . . . . . 11  |-  ( ( A  mod  M )  =  0  ->  (
( A  mod  M
)  +  1 )  =  ( 0  +  1 ) )
22 0p1e1 8947 . . . . . . . . . . 11  |-  ( 0  +  1 )  =  1
2321, 22eqtrdi 2206 . . . . . . . . . 10  |-  ( ( A  mod  M )  =  0  ->  (
( A  mod  M
)  +  1 )  =  1 )
2423oveq1d 5839 . . . . . . . . 9  |-  ( ( A  mod  M )  =  0  ->  (
( ( A  mod  M )  +  1 )  mod  M )  =  ( 1  mod  M
) )
2524adantl 275 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  ( A  mod  M )  =  0 )  ->  (
( ( A  mod  M )  +  1 )  mod  M )  =  ( 1  mod  M
) )
26 zq 9535 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  A  e.  QQ )
2726ad3antlr 485 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  ( A  mod  M )  =  0 )  ->  A  e.  QQ )
28 1z 9193 . . . . . . . . . 10  |-  1  e.  ZZ
29 zq 9535 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
3028, 29mp1i 10 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  ( A  mod  M )  =  0 )  ->  1  e.  QQ )
31 zq 9535 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  QQ )
3231ad3antrrr 484 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  ( A  mod  M )  =  0 )  ->  M  e.  QQ )
3311ad4ant13 505 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  ( A  mod  M )  =  0 )  ->  0  <  M )
34 modqaddmod 10262 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  1  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( ( A  mod  M )  +  1 )  mod  M
)  =  ( ( A  +  1 )  mod  M ) )
3527, 30, 32, 33, 34syl22anc 1221 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  ( A  mod  M )  =  0 )  ->  (
( ( A  mod  M )  +  1 )  mod  M )  =  ( ( A  + 
1 )  mod  M
) )
3631adantr 274 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  A  e.  ZZ )  ->  M  e.  QQ )
37 q1mod 10255 . . . . . . . . . 10  |-  ( ( M  e.  QQ  /\  1  <  M )  -> 
( 1  mod  M
)  =  1 )
3836, 37sylan 281 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M
)  ->  ( 1  mod  M )  =  1 )
3938adantr 274 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  ( A  mod  M )  =  0 )  ->  (
1  mod  M )  =  1 )
4025, 35, 393eqtr3d 2198 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M )  /\  ( A  mod  M )  =  0 )  ->  (
( A  +  1 )  mod  M )  =  1 )
4140ex 114 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M
)  ->  ( ( A  mod  M )  =  0  ->  ( ( A  +  1 )  mod  M )  =  1 ) )
4220, 41syld 45 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  A  e.  ZZ )  /\  1  <  M
)  ->  ( M  ||  A  ->  ( ( A  +  1 )  mod  M )  =  1 ) )
4342ex 114 . . . 4  |-  ( ( M  e.  ZZ  /\  A  e.  ZZ )  ->  ( 1  <  M  ->  ( M  ||  A  ->  ( ( A  + 
1 )  mod  M
)  =  1 ) ) )
4443com23 78 . . 3  |-  ( ( M  e.  ZZ  /\  A  e.  ZZ )  ->  ( M  ||  A  ->  ( 1  <  M  ->  ( ( A  + 
1 )  mod  M
)  =  1 ) ) )
451, 44mpcom 36 . 2  |-  ( M 
||  A  ->  (
1  <  M  ->  ( ( A  +  1 )  mod  M )  =  1 ) )
4645imp 123 1  |-  ( ( M  ||  A  /\  1  <  M )  -> 
( ( A  + 
1 )  mod  M
)  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1335    e. wcel 2128   class class class wbr 3965  (class class class)co 5824   RRcr 7731   0cc0 7732   1c1 7733    + caddc 7735    < clt 7912   NNcn 8833   ZZcz 9167   QQcq 9528    mod cmo 10221    || cdvds 11683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-n0 9091  df-z 9168  df-q 9529  df-rp 9561  df-fl 10169  df-mod 10222  df-dvds 11684
This theorem is referenced by: (None)
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