Theorem lgsvalmod 15692

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, (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

Step	Hyp	Ref	Expression
1		eldifi 3326	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	adantl 277	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. Prime )
3		prmz 12628	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
4	2, 3	syl 14	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. ZZ )
5		lgscl 15687	. . . . . 6 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( A /L P ) e. ZZ )
6	4, 5	syldan 282	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. ZZ )
7	6	peano2zd 9568	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. ZZ )
8		zq 9817	. . . 4 |- ( ( ( A /L P ) + 1 ) e. ZZ -> ( ( A /L P ) + 1 ) e. QQ )
9	7, 8	syl 14	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. QQ )
10		oddprm 12777	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
11	10	adantl 277	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN )
12	11	nnnn0d 9418	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN0 )
13		zexpcl 10771	. . . . . 6 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
14	12, 13	syldan 282	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
15	14	peano2zd 9568	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ )
16		zq 9817	. . . 4 |- ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. QQ )
17	15, 16	syl 14	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. QQ )
18		neg1z 9474	. . . 4 |- -u 1 e. ZZ
19		zq 9817	. . . 4 |- ( -u 1 e. ZZ -> -u 1 e. QQ )
20	18, 19	mp1i 10	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> -u 1 e. QQ )
21		prmnn 12627	. . . . 5 |- ( P e. Prime -> P e. NN )
22	2, 21	syl 14	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. NN )
23		nnq 9824	. . . 4 |- ( P e. NN -> P e. QQ )
24	22, 23	syl 14	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. QQ )
25	22	nngt0d 9150	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 0 < P )
26		lgsval3 15691	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
27	26	eqcomd 2235	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( A /L P ) )
28	15, 22	zmodcld 10562	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 )
29	28	nn0cnd 9420	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. CC )
30		1cnd 8158	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 1 e. CC )
31	6	zred 9565	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. RR )
32	31	recnd 8171	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. CC )
33	29, 30, 32	subadd2d 8472	. . . . . 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 ) ) )
34	27, 33	mpbid 147	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
35	34	oveq1d 6015	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) mod P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) mod P ) )
36		modqabs2 10575	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) mod P ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
37	17, 24, 25, 36	syl3anc 1271	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) mod P ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
38	35, 37	eqtrd 2262	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) mod P ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
39	9, 17, 20, 24, 25, 38	modqadd1 10578	. 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 ) )
40		peano2re 8278	. . . . . . 7 |- ( ( A /L P ) e. RR -> ( ( A /L P ) + 1 ) e. RR )
41	31, 40	syl 14	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. RR )
42	41	recnd 8171	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. CC )
43		ax-1cn 8088	. . . . 5 |- 1 e. CC
44		negsub 8390	. . . . 5 |- ( ( ( ( A /L P ) + 1 ) e. CC /\ 1 e. CC ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( ( ( A /L P ) + 1 ) - 1 ) )
45	42, 43, 44	sylancl 413	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( ( ( A /L P ) + 1 ) - 1 ) )
46		pncan 8348	. . . . 5 |- ( ( ( A /L P ) e. CC /\ 1 e. CC ) -> ( ( ( A /L P ) + 1 ) - 1 ) = ( A /L P ) )
47	32, 43, 46	sylancl 413	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) - 1 ) = ( A /L P ) )
48	45, 47	eqtrd 2262	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( A /L P ) )
49	48	oveq1d 6015	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A /L P ) + 1 ) + -u 1 ) mod P ) = ( ( A /L P ) mod P ) )
50	14	zred 9565	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
51		peano2re 8278	. . . . . . 7 |- ( ( A ^ ( ( P - 1 ) / 2 ) ) e. RR -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. RR )
52	50, 51	syl 14	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. RR )
53	52	recnd 8171	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. CC )
54		negsub 8390	. . . . 5 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. CC /\ 1 e. CC ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) )
55	53, 43, 54	sylancl 413	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) )
56	50	recnd 8171	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
57		pncan 8348	. . . . 5 |- ( ( ( A ^ ( ( P - 1 ) / 2 ) ) e. CC /\ 1 e. CC ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
58	56, 43, 57	sylancl 413	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
59	55, 58	eqtrd 2262	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
60	59	oveq1d 6015	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )
61	39, 49, 60	3eqtr3d 2270	1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   \ cdif 3194   {csn 3666   class class class wbr 4082  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   1c1 7996   + caddc 7998   < clt 8177   - cmin 8313   -ucneg 8314   / cdiv 8815   NNcn 9106   2c2 9157   NN0cn0 9365   ZZcz 9442   QQcq 9810   mod cmo 10539   ^cexp 10755   Primecprime 12624   /Lclgs 15670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-2o 6561  df-oadd 6564  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-ihash 10993  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-proddc 12057  df-dvds 12294  df-gcd 12470  df-prm 12625  df-phi 12728  df-pc 12803  df-lgs 15671

This theorem is referenced by:  lgsdirprm  15707  lgsne0  15711  gausslemma2d  15742