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Theorem mulp1mod1 10351
Description: The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.)
Assertion
Ref Expression
mulp1mod1  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( N  x.  A )  +  1 )  mod  N
)  =  1 )

Proof of Theorem mulp1mod1
StepHypRef Expression
1 eluzelcn 9528 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  CC )
21adantl 277 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  CC )
3 simpl 109 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  ZZ )
43zcnd 9365 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  CC )
52, 4mulcomd 7969 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( N  x.  A
)  =  ( A  x.  N ) )
65oveq1d 5884 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( N  x.  A )  mod  N
)  =  ( ( A  x.  N )  mod  N ) )
7 eluzelz 9526 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  ZZ )
8 zq 9615 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  QQ )
97, 8syl 14 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  QQ )
109adantl 277 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  QQ )
11 0red 7949 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  e.  RR )
12 2re 8978 . . . . . . . . 9  |-  2  e.  RR
1312a1i 9 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
2  e.  RR )
147adantl 277 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  ZZ )
1514zred 9364 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  RR )
16 2pos 8999 . . . . . . . . 9  |-  0  <  2
1716a1i 9 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  <  2 )
18 eluzle 9529 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  2  <_  N )
1918adantl 277 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
2  <_  N )
2011, 13, 15, 17, 19ltletrd 8370 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  <  N )
21 mulqmod0 10316 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( A  x.  N
)  mod  N )  =  0 )
223, 10, 20, 21syl3anc 1238 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A  x.  N )  mod  N
)  =  0 )
236, 22eqtrd 2210 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( N  x.  A )  mod  N
)  =  0 )
2423oveq1d 5884 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( N  x.  A )  mod 
N )  +  1 )  =  ( 0  +  1 ) )
25 0p1e1 9022 . . . 4  |-  ( 0  +  1 )  =  1
2624, 25eqtrdi 2226 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( N  x.  A )  mod 
N )  +  1 )  =  1 )
2726oveq1d 5884 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( ( N  x.  A )  mod  N )  +  1 )  mod  N
)  =  ( 1  mod  N ) )
28 zq 9615 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  QQ )
293, 28syl 14 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  QQ )
30 qmulcl 9626 . . . 4  |-  ( ( N  e.  QQ  /\  A  e.  QQ )  ->  ( N  x.  A
)  e.  QQ )
3110, 29, 30syl2anc 411 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( N  x.  A
)  e.  QQ )
32 1z 9268 . . . 4  |-  1  e.  ZZ
33 zq 9615 . . . 4  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
3432, 33mp1i 10 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
1  e.  QQ )
35 modqaddmod 10349 . . 3  |-  ( ( ( ( N  x.  A )  e.  QQ  /\  1  e.  QQ )  /\  ( N  e.  QQ  /\  0  < 
N ) )  -> 
( ( ( ( N  x.  A )  mod  N )  +  1 )  mod  N
)  =  ( ( ( N  x.  A
)  +  1 )  mod  N ) )
3631, 34, 10, 20, 35syl22anc 1239 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( ( N  x.  A )  mod  N )  +  1 )  mod  N
)  =  ( ( ( N  x.  A
)  +  1 )  mod  N ) )
37 eluz2gt1 9591 . . . 4  |-  ( N  e.  ( ZZ>= `  2
)  ->  1  <  N )
3837adantl 277 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
1  <  N )
39 q1mod 10342 . . 3  |-  ( ( N  e.  QQ  /\  1  <  N )  -> 
( 1  mod  N
)  =  1 )
4010, 38, 39syl2anc 411 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( 1  mod  N
)  =  1 )
4127, 36, 403eqtr3d 2218 1  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( N  x.  A )  +  1 )  mod  N
)  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803    + caddc 7805    x. cmul 7807    < clt 7982    <_ cle 7983   2c2 8959   ZZcz 9242   ZZ>=cuz 9517   QQcq 9608    mod cmo 10308
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309
This theorem is referenced by: (None)
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