Theorem 2lgsoddprm 15800

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 3326	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2		prmz 12641	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
3	1, 2	syl 14	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
4		8nn 9286	. . . . . . . . 9 |- 8 e. NN
5	4	a1i 9	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> 8 e. NN )
6	3, 5	zmodcld 10575	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. NN0 )
7	6	nn0zd 9575	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. ZZ )
8		1zzd 9481	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 1 e. ZZ )
9		zdceq 9530	. . . . . 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 9285	. . . . . . . 8 |- 7 e. NN
12	11	nnzi 9475	. . . . . . 7 |- 7 e. ZZ
13	12	a1i 9	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 7 e. ZZ )
14		zdceq 9530	. . . . . 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 941	. . . . 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 3686	. . . . . 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 843	. . . 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 15791	. . . 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 2231	. . . . . . . . . 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 12791	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
28		nnz 9473	. . . . . . . . . . . . 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 12405	. . . . . . . . . . 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 12420	. . . . . . . . . 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 15799	. . . . . . . . . 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 2262	. . . . 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 9482	. . . . . . . 8 |- 2 e. ZZ
43		lgscl1 15710	. . . . . . . 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 3711	. . . . . . . 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 671	. . . . . . . . . . . . . 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 12419	. . . . . . . . . . . . 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 597	. . . . . . . . . . . 12 |- ( ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) -> ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = -u 1 )
53	52	eqcomd 2235	. . . . . . . . . . 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 2262	. . . . . . . . 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 3795	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
59		simpr 110	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ P =/= 2 ) -> P =/= 2 )
60	59	necomd 2486	. . . . . . . . . . 11 |- ( ( P e. Prime /\ P =/= 2 ) -> 2 =/= P )
61	58, 60	sylbi 121	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> 2 =/= P )
62		2prm 12657	. . . . . . . . . . 11 |- 2 e. Prime
63		prmrp 12675	. . . . . . . . . . 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 15725	. . . . . . . . . 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 2410	. . . . . . . 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		pm2.24 624	. . . . . . . 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 1337	. . . . . 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 887	. . 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 713  DECID wdc 839   \/ w3o 1001   = wceq 1395   e. wcel 2200   =/= wne 2400   \ cdif 3194   {csn 3666   {cpr 3667   {ctp 3668   class class class wbr 4083  (class class class)co 6007   0cc0 8007   1c1 8008   - cmin 8325   -ucneg 8326   / cdiv 8827   NNcn 9118   2c2 9169   7c7 9174   8c8 9175   NN0cn0 9377   ZZcz 9454   mod cmo 10552   ^cexp 10768   || cdvds 12306   gcd cgcd 12482   Primecprime 12637   /Lclgs 15684

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127

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-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-2o 6569  df-oadd 6572  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-ioo 10096  df-ico 10098  df-fz 10213  df-fzo 10347  df-fl 10498  df-mod 10553  df-seqfrec 10678  df-exp 10769  df-fac 10956  df-ihash 11006  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-clim 11798  df-proddc 12070  df-dvds 12307  df-gcd 12483  df-prm 12638  df-phi 12741  df-pc 12816  df-lgs 15685

This theorem is referenced by: (None)