Theorem lgsvalmod 13714

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 3249	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	adantl 275	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> P e. Prime )
3		prmz 12065	. . . . . . 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 13709	. . . . . 6 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( A /L P ) e. ZZ )
6	4, 5	syldan 280	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. ZZ )
7	6	peano2zd 9337	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. ZZ )
8		zq 9585	. . . 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 12213	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
11	10	adantl 275	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN )
12	11	nnnn0d 9188	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN0 )
13		zexpcl 10491	. . . . . 6 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
14	12, 13	syldan 280	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
15	14	peano2zd 9337	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ )
16		zq 9585	. . . 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 9244	. . . 4 |- -u 1 e. ZZ
19		zq 9585	. . . 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 12064	. . . . 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 9592	. . . 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 8922	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 0 < P )
26		lgsval3 13713	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
27	26	eqcomd 2176	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( A /L P ) )
28	15, 22	zmodcld 10301	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 )
29	28	nn0cnd 9190	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. CC )
30		1cnd 7936	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 1 e. CC )
31	6	zred 9334	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. RR )
32	31	recnd 7948	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. CC )
33	29, 30, 32	subadd2d 8249	. . . . . 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 146	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
35	34	oveq1d 5868	. . . 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 10314	. . . . 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 1233	. . . 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 2203	. . 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 10317	. 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 8055	. . . . . . 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 7948	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. CC )
43		ax-1cn 7867	. . . . 5 |- 1 e. CC
44		negsub 8167	. . . . 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 411	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( ( ( A /L P ) + 1 ) - 1 ) )
46		pncan 8125	. . . . 5 |- ( ( ( A /L P ) e. CC /\ 1 e. CC ) -> ( ( ( A /L P ) + 1 ) - 1 ) = ( A /L P ) )
47	32, 43, 46	sylancl 411	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) - 1 ) = ( A /L P ) )
48	45, 47	eqtrd 2203	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( A /L P ) )
49	48	oveq1d 5868	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A /L P ) + 1 ) + -u 1 ) mod P ) = ( ( A /L P ) mod P ) )
50	14	zred 9334	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
51		peano2re 8055	. . . . . . 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 7948	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. CC )
54		negsub 8167	. . . . 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 411	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) )
56	50	recnd 7948	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
57		pncan 8125	. . . . 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 411	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
59	55, 58	eqtrd 2203	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
60	59	oveq1d 5868	. 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 2211	1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) mod P ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   \ cdif 3118   {csn 3583   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   < clt 7954   - cmin 8090   -ucneg 8091   / cdiv 8589   NNcn 8878   2c2 8929   NN0cn0 9135   ZZcz 9212   QQcq 9578   mod cmo 10278   ^cexp 10475   Primecprime 12061   /Lclgs 13692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-2o 6396  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-proddc 11514  df-dvds 11750  df-gcd 11898  df-prm 12062  df-phi 12165  df-pc 12239  df-lgs 13693

This theorem is referenced by:  lgsdirprm  13729  lgsne0  13733