Theorem 2lgsoddprm 16215

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 3351	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2		prmz 12872	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
3	1, 2	syl 14	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
4		8nn 9455	. . . . . . . . 9 |- 8 e. NN
5	4	a1i 9	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> 8 e. NN )
6	3, 5	zmodcld 10765	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. NN0 )
7	6	nn0zd 9749	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. ZZ )
8		1zzd 9654	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 1 e. ZZ )
9		zdceq 9703	. . . . . 6 |- ( ( ( P mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( P mod 8 ) = 1 )
10	7, 8, 9	syl2anc 415	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> DECID ( P mod 8 ) = 1 )
11		7nn 9454	. . . . . . . 8 |- 7 e. NN
12	11	nnzi 9648	. . . . . . 7 |- 7 e. ZZ
13	12	a1i 9	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 7 e. ZZ )
14		zdceq 9703	. . . . . 6 |- ( ( ( P mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( P mod 8 ) = 7 )
15	7, 13, 14	syl2anc 415	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> DECID ( P mod 8 ) = 7 )
16		dcor 948	. . . . 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 3728	. . . . . 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 850	. . . 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 16206	. . . 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 2240	. . . . . . . . . 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 13022	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
28		nnz 9646	. . . . . . . . . . . . 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 12636	. . . . . . . . . . 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 12651	. . . . . . . . . 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 16214	. . . . . . . . . 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 2271	. . . . 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 9655	. . . . . . . 8 |- 2 e. ZZ
43		lgscl1 16125	. . . . . . . 8 |- ( ( 2 e. ZZ /\ P e. ZZ ) -> ( 2 /L P ) e. { -u 1 , 0 , 1 } )
44	42, 3, 43	sylancr 418	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) e. { -u 1 , 0 , 1 } )
45		eltpg 3753	. . . . . . . 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 677	. . . . . . . . . . . . . 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 12650	. . . . . . . . . . . . 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 603	. . . . . . . . . . . 12 |- ( ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) -> ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = -u 1 )
53	52	eqcomd 2244	. . . . . . . . . . 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 2271	. . . . . . . . 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 3839	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
59		simpr 110	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ P =/= 2 ) -> P =/= 2 )
60	59	necomd 2506	. . . . . . . . . . 11 |- ( ( P e. Prime /\ P =/= 2 ) -> 2 =/= P )
61	58, 60	sylbi 121	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> 2 =/= P )
62		2prm 12888	. . . . . . . . . . 11 |- 2 e. Prime
63		prmrp 12906	. . . . . . . . . . 11 |- ( ( 2 e. Prime /\ P e. Prime ) -> ( ( 2 gcd P ) = 1 <-> 2 =/= P ) )
64	62, 1, 63	sylancr 418	. . . . . . . . . 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 16140	. . . . . . . . . 10 |- ( ( 2 e. ZZ /\ P e. ZZ ) -> ( ( 2 /L P ) =/= 0 <-> ( 2 gcd P ) = 1 ) )
67	42, 3, 66	sylancr 418	. . . . . . . . 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 2430	. . . . . . . 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		pm2.24 630	. . . . . . . 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 1344	. . . . . 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 894	. . 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 720  DECID wdc 846   \/ w3o 1008   = wceq 1402   e. wcel 2209   =/= wne 2420   \ cdif 3217   {csn 3708   {cpr 3709   {ctp 3710   class class class wbr 4128  (class class class)co 6079   0cc0 8173   1c1 8174   - cmin 8491   -ucneg 8492   / cdiv 8996   NNcn 9287   2c2 9338   7c7 9343   8c8 9344   NN0cn0 9546   ZZcz 9627   mod cmo 10742   ^cexp 10958   || cdvds 12537   gcd cgcd 12713   Primecprime 12868   /Lclgs 16099

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-tp 3716  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-ioo 10277  df-ico 10279  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-fac 11147  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-proddc 12301  df-dvds 12538  df-gcd 12714  df-prm 12869  df-phi 12972  df-pc 13047  df-lgs 16100

This theorem is referenced by: (None)