Theorem 2lgsoddprm 15270

Description: The second supplement to the law of quadratic reciprocity for odd primes (common representation, see theorem 9.5 in [ApostolNT] p. 181): The Legendre symbol for 2 at an odd prime is minus one to the power of the square of the odd prime minus one divided by eight ( ( 2 /L P ) = -1^(((P^2)-1)/8) ). (Contributed by AV, 20-Jul-2021.)

Assertion

Ref	Expression
2lgsoddprm	|- ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )

Proof of Theorem 2lgsoddprm

Step	Hyp	Ref	Expression
1		eldifi 3282	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2		prmz 12252	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
3	1, 2	syl 14	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
4		8nn 9152	. . . . . . . . 9 |- 8 e. NN
5	4	a1i 9	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> 8 e. NN )
6	3, 5	zmodcld 10419	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. NN0 )
7	6	nn0zd 9440	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. ZZ )
8		1zzd 9347	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 1 e. ZZ )
9		zdceq 9395	. . . . . 6 |- ( ( ( P mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( P mod 8 ) = 1 )
10	7, 8, 9	syl2anc 411	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> DECID ( P mod 8 ) = 1 )
11		7nn 9151	. . . . . . . 8 |- 7 e. NN
12	11	nnzi 9341	. . . . . . 7 |- 7 e. ZZ
13	12	a1i 9	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 7 e. ZZ )
14		zdceq 9395	. . . . . 6 |- ( ( ( P mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( P mod 8 ) = 7 )
15	7, 13, 14	syl2anc 411	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> DECID ( P mod 8 ) = 7 )
16		dcor 937	. . . . 5 |- (DECID ( P mod 8 ) = 1 -> (DECID ( P mod 8 ) = 7 -> DECID ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) ) )
17	10, 15, 16	sylc 62	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> DECID ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) )
18		elprg 3639	. . . . . 6 |- ( ( P mod 8 ) e. NN0 -> ( ( P mod 8 ) e. { 1 , 7 } <-> ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) ) )
19	6, 18	syl 14	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> ( ( P mod 8 ) e. { 1 , 7 } <-> ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) ) )
20	19	dcbid 839	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> (DECID ( P mod 8 ) e. { 1 , 7 } <-> DECID ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) ) )
21	17, 20	mpbird 167	. . 3 |- ( P e. ( Prime \ { 2 } ) -> DECID ( P mod 8 ) e. { 1 , 7 } )
22		2lgs 15261	. . . 4 |- ( P e. Prime -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) )
23	1, 22	syl 14	. . 3 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) )
24		simpl 109	. . . . . 6 |- ( ( ( 2 /L P ) = 1 /\ ( ( P mod 8 ) e. { 1 , 7 } /\ P e. ( Prime \ { 2 } ) ) ) -> ( 2 /L P ) = 1 )
25		eqcom 2195	. . . . . . . . . 10 |- ( 1 = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) <-> ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = 1 )
26	25	a1i 9	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( 1 = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) <-> ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = 1 ) )
27		nnoddn2prm 12401	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
28		nnz 9339	. . . . . . . . . . . . 13 |- ( P e. NN -> P e. ZZ )
29	28	anim1i 340	. . . . . . . . . . . 12 |- ( ( P e. NN /\ -. 2 || P ) -> ( P e. ZZ /\ -. 2 || P ) )
30	27, 29	syl 14	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> ( P e. ZZ /\ -. 2 || P ) )
31		sqoddm1div8z 12030	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ -. 2 || P ) -> ( ( ( P ^ 2 ) - 1 ) / 8 ) e. ZZ )
32	30, 31	syl 14	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> ( ( ( P ^ 2 ) - 1 ) / 8 ) e. ZZ )
33		m1exp1 12045	. . . . . . . . . 10 |- ( ( ( ( P ^ 2 ) - 1 ) / 8 ) e. ZZ -> ( ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = 1 <-> 2 || ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
34	32, 33	syl 14	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = 1 <-> 2 || ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
35		2lgsoddprmlem4 15269	. . . . . . . . . 10 |- ( ( P e. ZZ /\ -. 2 || P ) -> ( 2 || ( ( ( P ^ 2 ) - 1 ) / 8 ) <-> ( P mod 8 ) e. { 1 , 7 } ) )
36	30, 35	syl 14	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( 2 || ( ( ( P ^ 2 ) - 1 ) / 8 ) <-> ( P mod 8 ) e. { 1 , 7 } ) )
37	26, 34, 36	3bitrd 214	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( 1 = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) <-> ( P mod 8 ) e. { 1 , 7 } ) )
38	37	biimparc 299	. . . . . . 7 |- ( ( ( P mod 8 ) e. { 1 , 7 } /\ P e. ( Prime \ { 2 } ) ) -> 1 = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
39	38	adantl 277	. . . . . 6 |- ( ( ( 2 /L P ) = 1 /\ ( ( P mod 8 ) e. { 1 , 7 } /\ P e. ( Prime \ { 2 } ) ) ) -> 1 = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
40	24, 39	eqtrd 2226	. . . . 5 |- ( ( ( 2 /L P ) = 1 /\ ( ( P mod 8 ) e. { 1 , 7 } /\ P e. ( Prime \ { 2 } ) ) ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
41	40	exp32 365	. . . 4 |- ( ( 2 /L P ) = 1 -> ( ( P mod 8 ) e. { 1 , 7 } -> ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) )
42		2z 9348	. . . . . . . 8 |- 2 e. ZZ
43		lgscl1 15180	. . . . . . . 8 |- ( ( 2 e. ZZ /\ P e. ZZ ) -> ( 2 /L P ) e. { -u 1 , 0 , 1 } )
44	42, 3, 43	sylancr 414	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) e. { -u 1 , 0 , 1 } )
45		eltpg 3664	. . . . . . . 8 |- ( ( 2 /L P ) e. { -u 1 , 0 , 1 } -> ( ( 2 /L P ) e. { -u 1 , 0 , 1 } <-> ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) ) )
46	44, 45	syl 14	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 /L P ) e. { -u 1 , 0 , 1 } <-> ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) ) )
47	44, 46	mpbid 147	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) )
48		simpl 109	. . . . . . . . . 10 |- ( ( ( 2 /L P ) = -u 1 /\ ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) ) -> ( 2 /L P ) = -u 1 )
49	36	notbid 668	. . . . . . . . . . . . . 14 |- ( P e. ( Prime \ { 2 } ) -> ( -. 2 || ( ( ( P ^ 2 ) - 1 ) / 8 ) <-> -. ( P mod 8 ) e. { 1 , 7 } ) )
50	49	biimpar 297	. . . . . . . . . . . . 13 |- ( ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) -> -. 2 || ( ( ( P ^ 2 ) - 1 ) / 8 ) )
51		m1expo 12044	. . . . . . . . . . . . 13 |- ( ( ( ( ( P ^ 2 ) - 1 ) / 8 ) e. ZZ /\ -. 2 || ( ( ( P ^ 2 ) - 1 ) / 8 ) ) -> ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = -u 1 )
52	32, 50, 51	syl2an2r 595	. . . . . . . . . . . 12 |- ( ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) -> ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = -u 1 )
53	52	eqcomd 2199	. . . . . . . . . . 11 |- ( ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) -> -u 1 = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
54	53	adantl 277	. . . . . . . . . 10 |- ( ( ( 2 /L P ) = -u 1 /\ ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) ) -> -u 1 = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
55	48, 54	eqtrd 2226	. . . . . . . . 9 |- ( ( ( 2 /L P ) = -u 1 /\ ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
56	55	a1d 22	. . . . . . . 8 |- ( ( ( 2 /L P ) = -u 1 /\ ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) ) -> ( -. ( 2 /L P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) )
57	56	exp32 365	. . . . . . 7 |- ( ( 2 /L P ) = -u 1 -> ( P e. ( Prime \ { 2 } ) -> ( -. ( P mod 8 ) e. { 1 , 7 } -> ( -. ( 2 /L P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) ) )
58		eldifsn 3746	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
59		simpr 110	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ P =/= 2 ) -> P =/= 2 )
60	59	necomd 2450	. . . . . . . . . . 11 |- ( ( P e. Prime /\ P =/= 2 ) -> 2 =/= P )
61	58, 60	sylbi 121	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> 2 =/= P )
62		2prm 12268	. . . . . . . . . . 11 |- 2 e. Prime
63		prmrp 12286	. . . . . . . . . . 11 |- ( ( 2 e. Prime /\ P e. Prime ) -> ( ( 2 gcd P ) = 1 <-> 2 =/= P ) )
64	62, 1, 63	sylancr 414	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 gcd P ) = 1 <-> 2 =/= P ) )
65	61, 64	mpbird 167	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( 2 gcd P ) = 1 )
66		lgsne0 15195	. . . . . . . . . 10 |- ( ( 2 e. ZZ /\ P e. ZZ ) -> ( ( 2 /L P ) =/= 0 <-> ( 2 gcd P ) = 1 ) )
67	42, 3, 66	sylancr 414	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 /L P ) =/= 0 <-> ( 2 gcd P ) = 1 ) )
68	65, 67	mpbird 167	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) =/= 0 )
69		eqneqall 2374	. . . . . . . 8 |- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) =/= 0 -> ( -. ( P mod 8 ) e. { 1 , 7 } -> ( -. ( 2 /L P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) ) )
70	68, 69	syl5 32	. . . . . . 7 |- ( ( 2 /L P ) = 0 -> ( P e. ( Prime \ { 2 } ) -> ( -. ( P mod 8 ) e. { 1 , 7 } -> ( -. ( 2 /L P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) ) )
71		pm2.24 622	. . . . . . . 8 |- ( ( 2 /L P ) = 1 -> ( -. ( 2 /L P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) )
72	71	2a1d 23	. . . . . . 7 |- ( ( 2 /L P ) = 1 -> ( P e. ( Prime \ { 2 } ) -> ( -. ( P mod 8 ) e. { 1 , 7 } -> ( -. ( 2 /L P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) ) )
73	57, 70, 72	3jaoi 1314	. . . . . 6 |- ( ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) -> ( P e. ( Prime \ { 2 } ) -> ( -. ( P mod 8 ) e. { 1 , 7 } -> ( -. ( 2 /L P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) ) )
74	47, 73	mpcom 36	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> ( -. ( P mod 8 ) e. { 1 , 7 } -> ( -. ( 2 /L P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) )
75	74	com13 80	. . . 4 |- ( -. ( 2 /L P ) = 1 -> ( -. ( P mod 8 ) e. { 1 , 7 } -> ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) )
76	41, 75	bijadc 883	. . 3 |- (DECID ( P mod 8 ) e. { 1 , 7 } -> ( ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) -> ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) ) )
77	21, 23, 76	sylc 62	. 2 |- ( P e. ( Prime \ { 2 } ) -> ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) ) )
78	77	pm2.43i 49	1 |- ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   \/ w3o 979   = wceq 1364   e. wcel 2164   =/= wne 2364   \ cdif 3151   {csn 3619   {cpr 3620   {ctp 3621   class class class wbr 4030  (class class class)co 5919   0cc0 7874   1c1 7875   - cmin 8192   -ucneg 8193   / cdiv 8693   NNcn 8984   2c2 9035   7c7 9040   8c8 9041   NN0cn0 9243   ZZcz 9320   mod cmo 10396   ^cexp 10612   || cdvds 11933   gcd cgcd 12082   Primecprime 12248   /Lclgs 15154

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-2o 6472  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-ioo 9961  df-ico 9963  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-proddc 11697  df-dvds 11934  df-gcd 12083  df-prm 12249  df-phi 12352  df-pc 12426  df-lgs 15155

This theorem is referenced by: (None)