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Theorem mulp1mod1 10333
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 9510 . . . . . . . . 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 9347 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  CC )
52, 4mulcomd 7953 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( N  x.  A
)  =  ( A  x.  N ) )
65oveq1d 5880 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( N  x.  A )  mod  N
)  =  ( ( A  x.  N )  mod  N ) )
7 eluzelz 9508 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  ZZ )
8 zq 9597 . . . . . . . . 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 7933 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  e.  RR )
12 2re 8960 . . . . . . . . 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 9346 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  RR )
16 2pos 8981 . . . . . . . . 9  |-  0  <  2
1716a1i 9 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  <  2 )
18 eluzle 9511 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  2  <_  N )
1918adantl 277 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
2  <_  N )
2011, 13, 15, 17, 19ltletrd 8354 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
0  <  N )
21 mulqmod0 10298 . . . . . . 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 2208 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( N  x.  A )  mod  N
)  =  0 )
2423oveq1d 5880 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( N  x.  A )  mod 
N )  +  1 )  =  ( 0  +  1 ) )
25 0p1e1 9004 . . . 4  |-  ( 0  +  1 )  =  1
2624, 25eqtrdi 2224 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( N  x.  A )  mod 
N )  +  1 )  =  1 )
2726oveq1d 5880 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( ( ( N  x.  A )  mod  N )  +  1 )  mod  N
)  =  ( 1  mod  N ) )
28 zq 9597 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  QQ )
293, 28syl 14 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  QQ )
30 qmulcl 9608 . . . 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 9250 . . . 4  |-  1  e.  ZZ
33 zq 9597 . . . 4  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
3432, 33mp1i 10 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
1  e.  QQ )
35 modqaddmod 10331 . . 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 9573 . . . 4  |-  ( N  e.  ( ZZ>= `  2
)  ->  1  <  N )
3837adantl 277 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
1  <  N )
39 q1mod 10324 . . 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 2216 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 2146   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786   1c1 7787    + caddc 7789    x. cmul 7791    < clt 7966    <_ cle 7967   2c2 8941   ZZcz 9224   ZZ>=cuz 9499   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-2 8949  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fl 10238  df-mod 10291
This theorem is referenced by: (None)
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