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Theorem powm2modprm 14833
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 768 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  P  e.  Prime )
2 simpr 468 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  ZZ )
32adantr 472 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  A  e.  ZZ )
4 m1dvdsndvds 14828 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A  - 
1 )  ->  -.  P  ||  A ) )
54imp 436 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  -.  P  ||  A )
6 eqid 2471 . . . . . 6  |-  ( ( A ^ ( P  -  2 ) )  mod  P )  =  ( ( A ^
( P  -  2 ) )  mod  P
)
76modprminv 14829 . . . . 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 468 . . . . . 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 2477 . . . . 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 17 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  -.  P  ||  A )  ->  1  =  ( ( A  x.  ( ( A ^ ( P  - 
2 ) )  mod 
P ) )  mod 
P ) )
111, 3, 5, 10syl3anc 1292 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  1  =  ( ( A  x.  ( ( A ^
( P  -  2 ) )  mod  P
) )  mod  P
) )
12 modprm1div 14827 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A  mod  P
)  =  1  <->  P  ||  ( A  -  1 ) ) )
1312biimpar 493 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( A  mod  P )  =  1 )
1413oveq1d 6323 . . . . 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 6323 . . . 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 10965 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
1716ad2antlr 741 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  A  e.  RR )
18 prmm2nn0 14724 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  ( P  -  2 )  e. 
NN0 )
1918anim2i 579 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  P  e.  Prime )  -> 
( A  e.  ZZ  /\  ( P  -  2 )  e.  NN0 )
)
2019ancoms 460 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A  e.  ZZ  /\  ( P  -  2 )  e.  NN0 ) )
2120adantr 472 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( A  e.  ZZ  /\  ( P  -  2 )  e. 
NN0 ) )
22 zexpcl 12325 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  ZZ )
2321, 22syl 17 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( A ^ ( P  - 
2 ) )  e.  ZZ )
24 prmnn 14704 . . . . . . . . 9  |-  ( P  e.  Prime  ->  P  e.  NN )
2524adantr 472 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  P  e.  NN )
2625adantr 472 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  P  e.  NN )
2723, 26zmodcld 12150 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A ^ ( P  - 
2 ) )  mod 
P )  e.  NN0 )
2827nn0zd 11061 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A ^ ( P  - 
2 ) )  mod 
P )  e.  ZZ )
2924nnrpd 11362 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  RR+ )
3029adantr 472 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  P  e.  RR+ )
3130adantr 472 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  P  e.  RR+ )
32 modmulmod 12189 . . . . 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 1292 . . . 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 17 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ ( P  - 
2 ) )  e.  ZZ )
3534, 25zmodcld 12150 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  e.  NN0 )
3635nn0cnd 10951 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A ^ ( P  -  2 ) )  mod  P )  e.  CC )
3736mulid2d 9679 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
1  x.  ( ( A ^ ( P  -  2 ) )  mod  P ) )  =  ( ( A ^ ( P  - 
2 ) )  mod 
P ) )
3837oveq1d 6323 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( 1  x.  (
( A ^ ( P  -  2 ) )  mod  P ) )  mod  P )  =  ( ( ( A ^ ( P  -  2 ) )  mod  P )  mod 
P ) )
3938adantr 472 . . . . 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 12327 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( P  -  2
)  e.  NN0 )  ->  ( A ^ ( P  -  2 ) )  e.  RR )
4116, 18, 40syl2anr 486 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ ( P  - 
2 ) )  e.  RR )
4241, 30jca 541 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  (
( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ ) )
4342adantr 472 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A ^ ( P  - 
2 ) )  e.  RR  /\  P  e.  RR+ ) )
44 modabs2 12164 . . . . . 6  |-  ( ( ( A ^ ( P  -  2 ) )  e.  RR  /\  P  e.  RR+ )  -> 
( ( ( A ^ ( P  - 
2 ) )  mod 
P )  mod  P
)  =  ( ( A ^ ( P  -  2 ) )  mod  P ) )
4543, 44syl 17 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( (
( A ^ ( P  -  2 ) )  mod  P )  mod  P )  =  ( ( A ^
( P  -  2 ) )  mod  P
) )
4639, 45eqtrd 2505 . . . 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 2513 . . 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 2506 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  P  ||  ( A  -  1 ) )  ->  ( ( A ^ ( P  - 
2 ) )  mod 
P )  =  1 )
4948ex 441 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   class class class wbr 4395  (class class class)co 6308   RRcr 9556   1c1 9558    x. cmul 9562    - cmin 9880   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   RR+crp 11325   ...cfz 11810    mod cmo 12129   ^cexp 12310    || cdvds 14382   Primecprime 14701
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-rep 4508  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-int 4227  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-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  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-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793
This theorem is referenced by:  numclwwlk5  25919
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