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Theorem difmodm1lt 40833
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 464 . . . 4  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( A  mod  N )  =  1 )
2 zre 10965 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  RR )
323ad2ant1 1051 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  A  e.  RR )
4 nnre 10638 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
543ad2ant2 1052 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  N  e.  RR )
6 1lt2 10799 . . . . . . . . . . 11  |-  1  <  2
7 1red 9676 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  1  e.  RR )
8 2re 10701 . . . . . . . . . . . . . 14  |-  2  e.  RR
98a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  2  e.  RR )
107, 9, 43jca 1210 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
1  e.  RR  /\  2  e.  RR  /\  N  e.  RR ) )
11 lttr 9728 . . . . . . . . . . . 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 690 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
2  <  N  ->  1  <  N ) )
1413a1i 11 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( N  e.  NN  ->  ( 2  <  N  -> 
1  <  N )
) )
15143imp 1224 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  1  <  N )
163, 5, 153jca 1210 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  e.  RR  /\  N  e.  RR  /\  1  < 
N ) )
1716adantl 473 . . . . . 6  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( A  e.  RR  /\  N  e.  RR  /\  1  < 
N ) )
18 m1mod0mod1 38868 . . . . . 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 240 . . . 4  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  -  1 )  mod 
N )  =  0 )
211, 20oveq12d 6326 . . 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 10690 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
2322breq1i 4402 . . . . . . . . 9  |-  ( 2  <  N  <->  ( 1  +  1 )  < 
N )
2423biimpi 199 . . . . . . . 8  |-  ( 2  <  N  ->  (
1  +  1 )  <  N )
2524adantl 473 . . . . . . 7  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
( 1  +  1 )  <  N )
26 1red 9676 . . . . . . . 8  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
1  e.  RR )
274adantr 472 . . . . . . . 8  |-  ( ( N  e.  NN  /\  2  <  N )  ->  N  e.  RR )
2826, 26, 27ltaddsub2d 10235 . . . . . . 7  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
( ( 1  +  1 )  <  N  <->  1  <  ( N  - 
1 ) ) )
2925, 28mpbid 215 . . . . . 6  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
1  <  ( N  -  1 ) )
30 1m0e1 10742 . . . . . . 7  |-  ( 1  -  0 )  =  1
3130breq1i 4402 . . . . . 6  |-  ( ( 1  -  0 )  <  ( N  - 
1 )  <->  1  <  ( N  -  1 ) )
3229, 31sylibr 217 . . . . 5  |-  ( ( N  e.  NN  /\  2  <  N )  -> 
( 1  -  0 )  <  ( N  -  1 ) )
33323adant1 1048 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
1  -  0 )  <  ( N  - 
1 ) )
3433adantl 473 . . 3  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( 1  -  0 )  <  ( N  -  1 ) )
3521, 34eqbrtrd 4416 . 2  |-  ( ( ( A  mod  N
)  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) )  <  ( N  -  1 ) )
36 zmodfz 12151 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  ( A  mod  N
)  e.  ( 0 ... ( N  - 
1 ) ) )
37363adant3 1050 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  mod  N )  e.  ( 0 ... ( N  -  1 ) ) )
38 elfzle2 11829 . . . . . 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 473 . . . 4  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( A  mod  N )  <_  ( N  -  1 ) )
41 nnrp 11334 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR+ )
42413ad2ant2 1052 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  N  e.  RR+ )
433, 42modcld 12135 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  mod  N )  e.  RR )
44 peano2rem 9961 . . . . . . . . 9  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
454, 44syl 17 . . . . . . . 8  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  RR )
46453ad2ant2 1052 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( N  -  1 )  e.  RR )
47 peano2zm 11004 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  ( A  -  1 )  e.  ZZ )
4847zred 11063 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( A  -  1 )  e.  RR )
49483ad2ant1 1051 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( A  -  1 )  e.  RR )
5049, 42modcld 12135 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  -  1 )  mod  N )  e.  RR )
5143, 46, 503jca 1210 . . . . . 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 473 . . . . 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 10129 . . . . 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 215 . . 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 541 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  -  1 )  e.  RR  /\  N  e.  RR+ ) )
5756adantl 473 . . . . . . 7  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  -  1 )  e.  RR  /\  N  e.  RR+ ) )
58 modge0 12139 . . . . . . 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 206 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  mod  N
)  =  1  <->  (
( A  -  1 )  mod  N )  =  0 ) )
6261notbid 301 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( -.  ( A  mod  N
)  =  1  <->  -.  ( ( A  - 
1 )  mod  N
)  =  0 ) )
6362biimpac 494 . . . . . . 7  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  -.  ( ( A  -  1 )  mod  N )  =  0 )
6463neqned 2650 . . . . . 6  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( A  -  1 )  mod 
N )  =/=  0
)
6559, 64jca 541 . . . . 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 9662 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  0  e.  RR )
6766, 50jca 541 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
0  e.  RR  /\  ( ( A  - 
1 )  mod  N
)  e.  RR ) )
6867adantl 473 . . . . . 6  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( 0  e.  RR  /\  ( ( A  -  1 )  mod  N )  e.  RR ) )
69 ltlen 9753 . . . . . 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 240 . . . 4  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  0  <  (
( A  -  1 )  mod  N ) )
7250, 46jca 541 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( ( A  - 
1 )  mod  N
)  e.  RR  /\  ( N  -  1
)  e.  RR ) )
7372adantl 473 . . . . 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 10127 . . . . 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 215 . . 3  |-  ( ( -.  ( A  mod  N )  =  1  /\  ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N ) )  ->  ( ( N  -  1 )  -  ( ( A  - 
1 )  mod  N
) )  <  ( N  -  1 ) )
7743, 50resubcld 10068 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( A  mod  N
)  -  ( ( A  -  1 )  mod  N ) )  e.  RR )
7846, 50resubcld 10068 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  (
( N  -  1 )  -  ( ( A  -  1 )  mod  N ) )  e.  RR )
7977, 78, 463jca 1210 . . . . 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 473 . . . 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 9742 . . . 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 693 . 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 807 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   2c2 10681   ZZcz 10961   RR+crp 11325   ...cfz 11810    mod cmo 12129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130
This theorem is referenced by: (None)
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