Theorem lgsvalmod 15747

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 3329	. . . . . . . 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 12682	. . . . . . 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 15742	. . . . . 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 9604	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. ZZ )
8		zq 9859	. . . 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 12831	. . . . . . . 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 9454	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( P - 1 ) / 2 ) e. NN0 )
13		zexpcl 10815	. . . . . 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 9604	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ )
16		zq 9859	. . . 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 9510	. . . 4 |- -u 1 e. ZZ
19		zq 9859	. . . 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 12681	. . . . 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 9866	. . . 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 9186	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 0 < P )
26		lgsval3 15746	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) = ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
27	26	eqcomd 2237	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = ( A /L P ) )
28	15, 22	zmodcld 10606	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 )
29	28	nn0cnd 9456	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. CC )
30		1cnd 8194	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> 1 e. CC )
31	6	zred 9601	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. RR )
32	31	recnd 8207	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A /L P ) e. CC )
33	29, 30, 32	subadd2d 8508	. . . . . 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 6032	. . . 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 10619	. . . . 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 1273	. . . 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 2264	. . 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 10622	. 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 8314	. . . . . . 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 8207	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A /L P ) + 1 ) e. CC )
43		ax-1cn 8124	. . . . 5 |- 1 e. CC
44		negsub 8426	. . . . 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 8384	. . . . 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 2264	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A /L P ) + 1 ) + -u 1 ) = ( A /L P ) )
49	48	oveq1d 6032	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( ( A /L P ) + 1 ) + -u 1 ) mod P ) = ( ( A /L P ) mod P ) )
50	14	zred 9601	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. RR )
51		peano2re 8314	. . . . . . 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 8207	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. CC )
54		negsub 8426	. . . . 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 8207	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
57		pncan 8384	. . . . 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 2264	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) + -u 1 ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
60	59	oveq1d 6032	. 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 2272	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 1397   e. wcel 2202   \ cdif 3197   {csn 3669   class class class wbr 4088  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   < clt 8213   - cmin 8349   -ucneg 8350   / cdiv 8851   NNcn 9142   2c2 9193   NN0cn0 9401   ZZcz 9478   QQcq 9852   mod cmo 10583   ^cexp 10799   Primecprime 12678   /Lclgs 15725

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-2o 6582  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-proddc 12111  df-dvds 12348  df-gcd 12524  df-prm 12679  df-phi 12782  df-pc 12857  df-lgs 15726

This theorem is referenced by:  lgsdirprm  15762  lgsne0  15766  gausslemma2d  15797