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Theorem lgsvalmod 25041
Description: The Legendre symbol is equivalent to a ^ ( ( p - 1 ) / 2 ), mod p. This theorem is also called "Euler's criterion", see theorem 9.2 in [ApostolNT] p. 180, or a representation of Euler's criterion using the Legendre symbol, see also lgsqr 25076. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
lgsvalmod |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )
Proof of Theorem lgsvalmod
StepHypRef Expression
1 eldifi 3732 . . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
21adantl 482 . . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. Prime )
3 prmz 15389 . . . . . . 7 |- ( P e. Prime -> P e. ZZ )
42, 3syl 17 . . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. ZZ )
5 lgscl 25036 . . . . . 6 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( A /L P ) e. ZZ )
64, 5syldan 487 . . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. ZZ )
76zred 11482 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. RR )
8 peano2re 10209 . . . 4 |- ( ( A /L P ) e. RR -> ( ( A /L P ) + 1 ) e. RR )
97, 8syl 17 . . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. RR )
10 oddprm 15515 . . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
1110adantl 482 . . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN )
1211nnnn0d 11351 . . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN0 )
13 zexpcl 12875 . . . . . 6 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
1412, 13syldan 487 . . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
1514zred 11482 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
16 peano2re 10209 . . . 4 |- ( ( A ^ ( ( P - 1 ) / 2 ) ) e. RR -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. RR )
1715, 16syl 17 . . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. RR )
18 neg1rr 11125 . . . 4 |- -u 1 e. RR
1918a1i 11 . . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> -u 1 e. RR )
20 prmnn 15388 . . . . 5 |- ( P e. Prime -> P e. NN )
212, 20syl 17 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. NN )
2221nnrpd 11870 . . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. RR+ )
23 lgsval3 25040 . . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
2423eqcomd 2628 . . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( A /L P ) )
2517, 22modcld 12674 . . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. RR )
2625recnd 10068 . . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. CC )
27 ax-1cn 9994 . . . . . . . 8 |- 1 e. CC
2827a1i 11 . . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 1 e. CC )
297recnd 10068 . . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. CC )
3026, 28, 29subadd2d 10411 . . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( A /L P ) <-> ( ( A /L P ) + 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) ) )
3124, 30mpbid 222 . . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
3231oveq1d 6665 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) mod P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) mod P ) )
33 modabs2 12704 . . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. RR /\ P e. RR+ ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) mod P ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
3417, 22, 33syl2anc 693 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) mod P ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
3532, 34eqtrd 2656 . . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) mod P ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
36 modadd1 12707 . . 3 |- ( ( ( ( ( A /L P ) + 1 ) e. RR /\ ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. RR ) /\ ( -u 1 e. RR /\ P e. RR+ ) /\ ( ( ( A /L P ) + 1 ) mod P ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) ) -> ( ( ( ( A /L P ) + 1 ) + -u 1 ) mod P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) mod P ) )
379, 17, 19, 22, 35, 36syl221anc 1337 . 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A /L P ) + 1 ) + -u 1 ) mod P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) mod P ) )
389recnd 10068 . . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. CC )
39 negsub 10329 . . . . 5 |- ( ( ( ( A /L P ) + 1 ) e. CC /\ 1 e. CC ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( ( ( A /L P ) + 1 ) - 1 ) )
4038, 27, 39sylancl 694 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( ( ( A /L P ) + 1 ) - 1 ) )
41 pncan 10287 . . . . 5 |- ( ( ( A /L P ) e. CC /\ 1 e. CC ) -> ( ( ( A /L P ) + 1 ) - 1 ) = ( A /L P ) )
4229, 27, 41sylancl 694 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) - 1 ) = ( A /L P ) )
4340, 42eqtrd 2656 . . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( A /L P ) )
4443oveq1d 6665 . 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A /L P ) + 1 ) + -u 1 ) mod P ) = ( ( A /L P ) mod P ) )
4517recnd 10068 . . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. CC )
46 negsub 10329 . . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. CC /\ 1 e. CC ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) )
4745, 27, 46sylancl 694 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) )
4815recnd 10068 . . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
49 pncan 10287 . . . . 5 |- ( ( ( A ^ ( ( P - 1 ) / 2 ) ) e. CC /\ 1 e. CC ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
5048, 27, 49sylancl 694 . . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
5147, 50eqtrd 2656 . . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
5251oveq1d 6665 . 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )
5337, 44, 523eqtr3d 2664 1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   {csn 4177  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   RR+crp 11832   mod cmo 12668   ^cexp 12860   Primecprime 15385   /Lclgs 25019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542  df-lgs 25020
This theorem is referenced by:  lgsdirprm  25056  lgsne0  25060  lgsqrlem3  25073  gausslemma2d  25099  fmtnoprmfac2lem1  41478
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