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Theorem difmodm1lt 40378
Description: The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd  A and  N  =  2, since  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) ) would be  ( 1  -  0 )  =  1 which is not less than  ( N  -  1 )  =  1. (Contributed by AV, 6-Jun-2012.)
Assertion
Ref Expression
difmodm1lt  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  mod  N
)  -  ( ( A  -  1 )  mod  N ) )  <  ( N  - 
1 ) )

Proof of Theorem difmodm1lt
StepHypRef Expression
1 simpl 459 . . . 4  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( A  mod  N )  =  1 )
2 zre 10941 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
323ad2ant1 1029 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  A  e.  RR )
4 nnre 10616 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
543ad2ant2 1030 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  N  e.  RR )
6 1lt2 10776 . . . . . . . . . . 11  |-  1  <  2
7 1red 9658 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  1  e.  RR )
8 2re 10679 . . . . . . . . . . . . . 14  |-  2  e.  RR
98a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  2  e.  RR )
107, 9, 43jca 1188 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
1  e.  RR  /\  2  e.  RR  /\  N  e.  RR ) )
11 lttr 9710 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  2  e.  RR  /\  N  e.  RR )  ->  (
( 1  <  2  /\  2  <  N )  ->  1  <  N
) )
1210, 11syl 17 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( 1  <  2  /\  2  <  N )  ->  1  <  N
) )
136, 12mpani 682 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
2  <  N  ->  1  <  N ) )
1413a1i 11 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( N  e.  NN  ->  ( 2  <  N  -> 
1  <  N )
) )
15143imp 1202 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  1  <  N )
163, 5, 153jca 1188 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  e.  RR  /\  N  e.  RR  /\  1  < 
N ) )
1716adantl 468 . . . . . 6  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( A  e.  RR  /\  N  e.  RR  /\  1  < 
N ) )
18 m1mod0mod1 38723 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  RR  /\  1  <  N )  ->  (
( ( A  - 
1 )  mod  N
)  =  0  <->  ( A  mod  N )  =  1 ) )
1917, 18syl 17 . . . . 5  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( ( A  -  1 )  mod  N )  =  0  <->  ( A  mod  N )  =  1 ) )
201, 19mpbird 236 . . . 4  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  -  1 )  mod 
N )  =  0 )
211, 20oveq12d 6308 . . 3  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) )  =  ( 1  -  0 ) )
22 df-2 10668 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
2322breq1i 4409 . . . . . . . . 9  |-  ( 2  <  N  <->  ( 1  +  1 )  < 
N )
2423biimpi 198 . . . . . . . 8  |-  ( 2  <  N  ->  (
1  +  1 )  <  N )
2524adantl 468 . . . . . . 7  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
( 1  +  1 )  <  N )
26 1red 9658 . . . . . . . 8  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
1  e.  RR )
274adantr 467 . . . . . . . 8  |-  ( ( N  e.  NN  /\  2  <  N )  ->  N  e.  RR )
2826, 26, 27ltaddsub2d 10214 . . . . . . 7  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
( ( 1  +  1 )  <  N  <->  1  <  ( N  - 
1 ) ) )
2925, 28mpbid 214 . . . . . 6  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
1  <  ( N  -  1 ) )
30 1m0e1 10720 . . . . . . 7  |-  ( 1  -  0 )  =  1
3130breq1i 4409 . . . . . 6  |-  ( ( 1  -  0 )  <  ( N  - 
1 )  <->  1  <  ( N  -  1 ) )
3229, 31sylibr 216 . . . . 5  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
( 1  -  0 )  <  ( N  -  1 ) )
33323adant1 1026 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
1  -  0 )  <  ( N  - 
1 ) )
3433adantl 468 . . 3  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( 1  -  0 )  <  ( N  -  1 ) )
3521, 34eqbrtrd 4423 . 2  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) )  <  ( N  -  1 ) )
36 zmodfz 12118 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  ( A  mod  N
)  e.  ( 0 ... ( N  - 
1 ) ) )
37363adant3 1028 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  mod  N )  e.  ( 0 ... ( N  -  1 ) ) )
38 elfzle2 11803 . . . . . 6  |-  ( ( A  mod  N )  e.  ( 0 ... ( N  -  1 ) )  ->  ( A  mod  N )  <_ 
( N  -  1 ) )
3937, 38syl 17 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  mod  N )  <_ 
( N  -  1 ) )
4039adantl 468 . . . 4  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( A  mod  N )  <_  ( N  -  1 ) )
41 nnrp 11311 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR+ )
42413ad2ant2 1030 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  N  e.  RR+ )
433, 42modcld 12102 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  mod  N )  e.  RR )
44 peano2rem 9941 . . . . . . . . 9  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
454, 44syl 17 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  RR )
46453ad2ant2 1030 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( N  -  1 )  e.  RR )
47 peano2zm 10980 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  ( A  -  1 )  e.  ZZ )
4847zred 11040 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( A  -  1 )  e.  RR )
49483ad2ant1 1029 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  -  1 )  e.  RR )
5049, 42modcld 12102 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  -  1 )  mod  N )  e.  RR )
5143, 46, 503jca 1188 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  mod  N
)  e.  RR  /\  ( N  -  1
)  e.  RR  /\  ( ( A  - 
1 )  mod  N
)  e.  RR ) )
5251adantl 468 . . . . 5  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  mod  N )  e.  RR  /\  ( N  -  1 )  e.  RR  /\  ( ( A  -  1 )  mod  N )  e.  RR ) )
53 lesub1 10108 . . . . 5  |-  ( ( ( A  mod  N
)  e.  RR  /\  ( N  -  1
)  e.  RR  /\  ( ( A  - 
1 )  mod  N
)  e.  RR )  ->  ( ( A  mod  N )  <_ 
( N  -  1 )  <->  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) )  <_  (
( N  -  1 )  -  ( ( A  -  1 )  mod  N ) ) ) )
5452, 53syl 17 . . . 4  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  mod  N )  <_ 
( N  -  1 )  <->  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) )  <_  (
( N  -  1 )  -  ( ( A  -  1 )  mod  N ) ) ) )
5540, 54mpbid 214 . . 3  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) )  <_  (
( N  -  1 )  -  ( ( A  -  1 )  mod  N ) ) )
5649, 42jca 535 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  -  1 )  e.  RR  /\  N  e.  RR+ ) )
5756adantl 468 . . . . . . 7  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  -  1 )  e.  RR  /\  N  e.  RR+ ) )
58 modge0 12106 . . . . . . 7  |-  ( ( ( A  -  1 )  e.  RR  /\  N  e.  RR+ )  -> 
0  <_  ( ( A  -  1 )  mod  N ) )
5957, 58syl 17 . . . . . 6  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  0  <_  (
( A  -  1 )  mod  N ) )
6016, 18syl 17 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( ( A  - 
1 )  mod  N
)  =  0  <->  ( A  mod  N )  =  1 ) )
6160bicomd 205 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  mod  N
)  =  1  <->  (
( A  -  1 )  mod  N )  =  0 ) )
6261notbid 296 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( -.  ( A  mod  N
)  =  1  <->  -.  ( ( A  - 
1 )  mod  N
)  =  0 ) )
6362biimpac 489 . . . . . . 7  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  -.  ( ( A  -  1 )  mod  N )  =  0 )
6463neqned 2631 . . . . . 6  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  -  1 )  mod 
N )  =/=  0
)
6559, 64jca 535 . . . . 5  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( 0  <_ 
( ( A  - 
1 )  mod  N
)  /\  ( ( A  -  1 )  mod  N )  =/=  0 ) )
66 0red 9644 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  0  e.  RR )
6766, 50jca 535 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
0  e.  RR  /\  ( ( A  - 
1 )  mod  N
)  e.  RR ) )
6867adantl 468 . . . . . 6  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( 0  e.  RR  /\  ( ( A  -  1 )  mod  N )  e.  RR ) )
69 ltlen 9735 . . . . . 6  |-  ( ( 0  e.  RR  /\  ( ( A  - 
1 )  mod  N
)  e.  RR )  ->  ( 0  < 
( ( A  - 
1 )  mod  N
)  <->  ( 0  <_ 
( ( A  - 
1 )  mod  N
)  /\  ( ( A  -  1 )  mod  N )  =/=  0 ) ) )
7068, 69syl 17 . . . . 5  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( 0  < 
( ( A  - 
1 )  mod  N
)  <->  ( 0  <_ 
( ( A  - 
1 )  mod  N
)  /\  ( ( A  -  1 )  mod  N )  =/=  0 ) ) )
7165, 70mpbird 236 . . . 4  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  0  <  (
( A  -  1 )  mod  N ) )
7250, 46jca 535 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( ( A  - 
1 )  mod  N
)  e.  RR  /\  ( N  -  1
)  e.  RR ) )
7372adantl 468 . . . . 5  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( ( A  -  1 )  mod  N )  e.  RR  /\  ( N  -  1 )  e.  RR ) )
74 ltsubpos 10106 . . . . 5  |-  ( ( ( ( A  - 
1 )  mod  N
)  e.  RR  /\  ( N  -  1
)  e.  RR )  ->  ( 0  < 
( ( A  - 
1 )  mod  N
)  <->  ( ( N  -  1 )  -  ( ( A  - 
1 )  mod  N
) )  <  ( N  -  1 ) ) )
7573, 74syl 17 . . . 4  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( 0  < 
( ( A  - 
1 )  mod  N
)  <->  ( ( N  -  1 )  -  ( ( A  - 
1 )  mod  N
) )  <  ( N  -  1 ) ) )
7671, 75mpbid 214 . . 3  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( N  -  1 )  -  ( ( A  - 
1 )  mod  N
) )  <  ( N  -  1 ) )
7743, 50resubcld 10047 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  mod  N
)  -  ( ( A  -  1 )  mod  N ) )  e.  RR )
7846, 50resubcld 10047 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( N  -  1 )  -  ( ( A  -  1 )  mod  N ) )  e.  RR )
7977, 78, 463jca 1188 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( ( A  mod  N )  -  ( ( A  -  1 )  mod  N ) )  e.  RR  /\  (
( N  -  1 )  -  ( ( A  -  1 )  mod  N ) )  e.  RR  /\  ( N  -  1 )  e.  RR ) )
8079adantl 468 . . . 4  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( ( A  mod  N )  -  ( ( A  -  1 )  mod 
N ) )  e.  RR  /\  ( ( N  -  1 )  -  ( ( A  -  1 )  mod 
N ) )  e.  RR  /\  ( N  -  1 )  e.  RR ) )
81 lelttr 9724 . . . 4  |-  ( ( ( ( A  mod  N )  -  ( ( A  -  1 )  mod  N ) )  e.  RR  /\  (
( N  -  1 )  -  ( ( A  -  1 )  mod  N ) )  e.  RR  /\  ( N  -  1 )  e.  RR )  -> 
( ( ( ( A  mod  N )  -  ( ( A  -  1 )  mod 
N ) )  <_ 
( ( N  - 
1 )  -  (
( A  -  1 )  mod  N ) )  /\  ( ( N  -  1 )  -  ( ( A  -  1 )  mod 
N ) )  < 
( N  -  1 ) )  ->  (
( A  mod  N
)  -  ( ( A  -  1 )  mod  N ) )  <  ( N  - 
1 ) ) )
8280, 81syl 17 . . 3  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( ( ( A  mod  N
)  -  ( ( A  -  1 )  mod  N ) )  <_  ( ( N  -  1 )  -  ( ( A  - 
1 )  mod  N
) )  /\  (
( N  -  1 )  -  ( ( A  -  1 )  mod  N ) )  <  ( N  - 
1 ) )  -> 
( ( A  mod  N )  -  ( ( A  -  1 )  mod  N ) )  <  ( N  - 
1 ) ) )
8355, 76, 82mp2and 685 . 2  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) )  <  ( N  -  1 ) )
8435, 83pm2.61ian 799 1  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  mod  N
)  -  ( ( A  -  1 )  mod  N ) )  <  ( N  - 
1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   class class class wbr 4402  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609   2c2 10659   ZZcz 10937   RR+crp 11302   ...cfz 11784    mod cmo 12096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fl 12028  df-mod 12097
This theorem is referenced by: (None)
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