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Theorem powm2modprm 14204
Description: If an integer minus 1 is divisible by a prime number, then the integer to the power of the prime number minus 2 is 1 modulo the prime number. (Contributed by Alexander van der Vekens, 30-Aug-2018.)
Assertion
Ref Expression
powm2modprm  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A  - 
1 )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  =  1 ) )

Proof of Theorem powm2modprm
StepHypRef Expression
1 simpll 753 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  P  e.  Prime )
2 simpr 461 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  ZZ )
32adantr 465 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  A  e.  ZZ )
4 m1dvdsndvds 14201 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A  - 
1 )  ->  -.  P  ||  A ) )
54imp 429 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  -.  P  ||  A )
6 eqid 2467 . . . . . 6  |-  ( ( A ^ ( P  -  2 ) )  mod  P )  =  ( ( A ^
( P  -  2 ) )  mod  P
)
76modprminv 14202 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  (
( ( A ^
( P  -  2 ) )  mod  P
)  e.  ( 1 ... ( P  - 
1 ) )  /\  ( ( A  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  mod  P
)  =  1 ) )
8 simpr 461 . . . . . 6  |-  ( ( ( ( A ^
( P  -  2 ) )  mod  P
)  e.  ( 1 ... ( P  - 
1 ) )  /\  ( ( A  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  mod  P
)  =  1 )  ->  ( ( A  x.  ( ( A ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P )  =  1 )
98eqcomd 2475 . . . . 5  |-  ( ( ( ( A ^
( P  -  2 ) )  mod  P
)  e.  ( 1 ... ( P  - 
1 ) )  /\  ( ( A  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  mod  P
)  =  1 )  ->  1  =  ( ( A  x.  (
( A ^ ( P  -  2 ) )  mod  P ) )  mod  P ) )
107, 9syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  1  =  ( ( A  x.  ( ( A ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P ) )
111, 3, 5, 10syl3anc 1228 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  1  =  ( ( A  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  mod  P
) )
12 modprm1div 14200 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A  mod  P
)  =  1  <->  P  ||  ( A  -  1 ) ) )
1312biimpar 485 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( A  mod  P )  =  1 )
1413oveq1d 6310 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A  mod  P )  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  =  ( 1  x.  ( ( A ^ ( P  -  2 ) )  mod  P ) ) )
1514oveq1d 6310 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( (
( A  mod  P
)  x.  ( ( A ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  ( ( 1  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  mod  P
) )
16 zre 10880 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
1716ad2antlr 726 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  A  e.  RR )
18 prmm2nn0 14114 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  ( P  -  2 )  e. 
NN0 )
1918anim2i 569 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  Prime )  -> 
( A  e.  ZZ  /\  ( P  -  2 )  e.  NN0 )
)
2019ancoms 453 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A  e.  ZZ  /\  ( P  -  2 )  e.  NN0 ) )
2120adantr 465 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( A  e.  ZZ  /\  ( P  -  2 )  e. 
NN0 ) )
22 zexpcl 12161 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  ZZ )
2321, 22syl 16 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( A ^ ( P  - 
2 ) )  e.  ZZ )
24 prmnn 14096 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
2524adantr 465 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  P  e.  NN )
2625adantr 465 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  P  e.  NN )
2723, 26zmodcld 11996 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A ^ ( P  - 
2 ) )  mod 
P )  e.  NN0 )
2827nn0zd 10976 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A ^ ( P  - 
2 ) )  mod 
P )  e.  ZZ )
2924nnrpd 11267 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  RR+ )
3029adantr 465 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  P  e.  RR+ )
3130adantr 465 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  P  e.  RR+ )
32 modmulmod 12032 . . . . 5  |-  ( ( A  e.  RR  /\  ( ( A ^
( P  -  2 ) )  mod  P
)  e.  ZZ  /\  P  e.  RR+ )  -> 
( ( ( A  mod  P )  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  mod  P
)  =  ( ( A  x.  ( ( A ^ ( P  -  2 ) )  mod  P ) )  mod  P ) )
3317, 28, 31, 32syl3anc 1228 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( (
( A  mod  P
)  x.  ( ( A ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  ( ( A  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  mod  P
) )
3420, 22syl 16 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ ( P  - 
2 ) )  e.  ZZ )
3534, 25zmodcld 11996 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  e.  NN0 )
3635nn0cnd 10866 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  e.  CC )
3736mulid2d 9626 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
1  x.  ( ( A ^ ( P  -  2 ) )  mod  P ) )  =  ( ( A ^ ( P  - 
2 ) )  mod 
P ) )
3837oveq1d 6310 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( 1  x.  (
( A ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  ( ( ( A ^ ( P  -  2 ) )  mod  P )  mod 
P ) )
3938adantr 465 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( (
1  x.  ( ( A ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  ( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
) )
40 reexpcl 12163 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  RR )
4116, 18, 40syl2anr 478 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ ( P  - 
2 ) )  e.  RR )
4241, 30jca 532 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ ) )
4342adantr 465 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A ^ ( P  - 
2 ) )  e.  RR  /\  P  e.  RR+ ) )
44 modabs2 12010 . . . . . 6  |-  ( ( ( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ )  -> 
( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)  =  ( ( A ^ ( P  -  2 ) )  mod  P ) )
4543, 44syl 16 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( (
( A ^ ( P  -  2 ) )  mod  P )  mod  P )  =  ( ( A ^
( P  -  2 ) )  mod  P
) )
4639, 45eqtrd 2508 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( (
1  x.  ( ( A ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  ( ( A ^
( P  -  2 ) )  mod  P
) )
4715, 33, 463eqtr3d 2516 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A  x.  ( ( A ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P )  =  ( ( A ^ ( P  -  2 ) )  mod  P ) )
4811, 47eqtr2d 2509 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A ^ ( P  - 
2 ) )  mod 
P )  =  1 )
4948ex 434 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A  - 
1 )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453  (class class class)co 6295   RRcr 9503   1c1 9505    x. cmul 9509    - cmin 9817   NNcn 10548   2c2 10597   NN0cn0 10807   ZZcz 10876   RR+crp 11232   ...cfz 11684    mod cmo 11976   ^cexp 12146    || cdivides 13864   Primecprime 14093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-prm 14094  df-phi 14172
This theorem is referenced by:  numclwwlk5  24936
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