Theorem lgsvalmod 15344

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 3286	. . . . . . . 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 12304	. . . . . . 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 15339	. . . . . 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 9468	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. ZZ )
8		zq 9717	. . . 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 12453	. . . . . . . 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 9319	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN0 )
13		zexpcl 10663	. . . . . 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 9468	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ )
16		zq 9717	. . . 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 9375	. . . 4 |- -u 1 e. ZZ
19		zq 9717	. . . 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 12303	. . . . 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 9724	. . . 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 9051	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 0 < P )
26		lgsval3 15343	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
27	26	eqcomd 2202	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( A /L P ) )
28	15, 22	zmodcld 10454	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 )
29	28	nn0cnd 9321	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. CC )
30		1cnd 8059	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 1 e. CC )
31	6	zred 9465	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. RR )
32	31	recnd 8072	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. CC )
33	29, 30, 32	subadd2d 8373	. . . . . 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 5940	. . . 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 10467	. . . . 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 1249	. . . 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 2229	. . 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 10470	. 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 8179	. . . . . . 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 8072	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. CC )
43		ax-1cn 7989	. . . . 5 |- 1 e. CC
44		negsub 8291	. . . . 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 8249	. . . . 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 2229	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( A /L P ) )
49	48	oveq1d 5940	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A /L P ) + 1 ) + -u 1 ) mod P ) = ( ( A /L P ) mod P ) )
50	14	zred 9465	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
51		peano2re 8179	. . . . . . 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 8072	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. CC )
54		negsub 8291	. . . . 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 8072	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
57		pncan 8249	. . . . 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 2229	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
60	59	oveq1d 5940	. 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 2237	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 1364   e. wcel 2167   \ cdif 3154   {csn 3623   class class class wbr 4034  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   1c1 7897   + caddc 7899   < clt 8078   - cmin 8214   -ucneg 8215   / cdiv 8716   NNcn 9007   2c2 9058   NN0cn0 9266   ZZcz 9343   QQcq 9710   mod cmo 10431   ^cexp 10647   Primecprime 12300   /Lclgs 15322

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-2o 6484  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-proddc 11733  df-dvds 11970  df-gcd 12146  df-prm 12301  df-phi 12404  df-pc 12479  df-lgs 15323

This theorem is referenced by:  lgsdirprm  15359  lgsne0  15363  gausslemma2d  15394