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Theorem mulp1mod1 9499
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 8763 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  CC )
21adantl 271 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  CC )
3 simpl 107 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  ZZ )
43zcnd 8603 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  CC )
52, 4mulcomd 7254 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( N  x.  A
)  =  ( A  x.  N ) )
65oveq1d 5578 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( N  x.  A )  mod  N
)  =  ( ( A  x.  N )  mod  N ) )
7 eluzelz 8761 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  ZZ )
8 zq 8844 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  QQ )
97, 8syl 14 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  QQ )
109adantl 271 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  QQ )
11 0red 7234 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  e.  RR )
12 2re 8228 . . . . . . . . 9  |-  2  e.  RR
1312a1i 9 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
2  e.  RR )
147adantl 271 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  ZZ )
1514zred 8602 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  RR )
16 2pos 8249 . . . . . . . . 9  |-  0  <  2
1716a1i 9 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  <  2 )
18 eluzle 8764 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  2  <_  N )
1918adantl 271 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
2  <_  N )
2011, 13, 15, 17, 19ltletrd 7646 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  <  N )
21 mulqmod0 9464 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( A  x.  N
)  mod  N )  =  0 )
223, 10, 20, 21syl3anc 1170 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A  x.  N )  mod  N
)  =  0 )
236, 22eqtrd 2115 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( N  x.  A )  mod  N
)  =  0 )
2423oveq1d 5578 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( N  x.  A )  mod 
N )  +  1 )  =  ( 0  +  1 ) )
25 0p1e1 8272 . . . 4  |-  ( 0  +  1 )  =  1
2624, 25syl6eq 2131 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( N  x.  A )  mod 
N )  +  1 )  =  1 )
2726oveq1d 5578 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( ( N  x.  A )  mod  N )  +  1 )  mod  N
)  =  ( 1  mod  N ) )
28 zq 8844 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  QQ )
293, 28syl 14 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  QQ )
30 qmulcl 8855 . . . 4  |-  ( ( N  e.  QQ  /\  A  e.  QQ )  ->  ( N  x.  A
)  e.  QQ )
3110, 29, 30syl2anc 403 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( N  x.  A
)  e.  QQ )
32 1z 8510 . . . 4  |-  1  e.  ZZ
33 zq 8844 . . . 4  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
3432, 33mp1i 10 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
1  e.  QQ )
35 modqaddmod 9497 . . 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 1171 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( ( N  x.  A )  mod  N )  +  1 )  mod  N
)  =  ( ( ( N  x.  A
)  +  1 )  mod  N ) )
37 eluz2gt1 8822 . . . 4  |-  ( N  e.  ( ZZ>= `  2
)  ->  1  <  N )
3837adantl 271 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
1  <  N )
39 q1mod 9490 . . 3  |-  ( ( N  e.  QQ  /\  1  <  N )  -> 
( 1  mod  N
)  =  1 )
4010, 38, 39syl2anc 403 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( 1  mod  N
)  =  1 )
4127, 36, 403eqtr3d 2123 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 102    = wceq 1285    e. wcel 1434   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   CCcc 7093   RRcr 7094   0cc0 7095   1c1 7096    + caddc 7098    x. cmul 7100    < clt 7267    <_ cle 7268   2c2 8208   ZZcz 8484   ZZ>=cuz 8752   QQcq 8837    mod cmo 9456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fl 9404  df-mod 9457
This theorem is referenced by: (None)
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