Theorem 2lgsoddprm 15973

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 3340	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2		prmz 12801	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
3	1, 2	syl 14	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
4		8nn 9401	. . . . . . . . 9 |- 8 e. NN
5	4	a1i 9	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> 8 e. NN )
6	3, 5	zmodcld 10703	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. NN0 )
7	6	nn0zd 9694	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. ZZ )
8		1zzd 9600	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 1 e. ZZ )
9		zdceq 9649	. . . . . 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 9400	. . . . . . . 8 |- 7 e. NN
12	11	nnzi 9594	. . . . . . 7 |- 7 e. ZZ
13	12	a1i 9	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 7 e. ZZ )
14		zdceq 9649	. . . . . 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 944	. . . . 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 3708	. . . . . 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 846	. . . 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 15964	. . . 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 2234	. . . . . . . . . 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 12951	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
28		nnz 9592	. . . . . . . . . . . . 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 12565	. . . . . . . . . . 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 12580	. . . . . . . . . 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 15972	. . . . . . . . . 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 2265	. . . . 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 9601	. . . . . . . 8 |- 2 e. ZZ
43		lgscl1 15883	. . . . . . . 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 3733	. . . . . . . 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 673	. . . . . . . . . . . . . 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 12579	. . . . . . . . . . . . 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 599	. . . . . . . . . . . 12 |- ( ( P e. ( Prime \ { 2 } ) /\ -. ( P mod 8 ) e. { 1 , 7 } ) -> ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) = -u 1 )
53	52	eqcomd 2238	. . . . . . . . . . 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 2265	. . . . . . . . 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 3819	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
59		simpr 110	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ P =/= 2 ) -> P =/= 2 )
60	59	necomd 2498	. . . . . . . . . . 11 |- ( ( P e. Prime /\ P =/= 2 ) -> 2 =/= P )
61	58, 60	sylbi 121	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> 2 =/= P )
62		2prm 12817	. . . . . . . . . . 11 |- 2 e. Prime
63		prmrp 12835	. . . . . . . . . . 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 15898	. . . . . . . . . 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 2422	. . . . . . . 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		pm2.24 626	. . . . . . . 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 1340	. . . . . 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 890	. . 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 716  DECID wdc 842   \/ w3o 1004   = wceq 1398   e. wcel 2203   =/= wne 2412   \ cdif 3207   {csn 3688   {cpr 3689   {ctp 3690   class class class wbr 4108  (class class class)co 6049   0cc0 8123   1c1 8124   - cmin 8440   -ucneg 8441   / cdiv 8942   NNcn 9233   2c2 9284   7c7 9289   8c8 9290   NN0cn0 9492   ZZcz 9573   mod cmo 10680   ^cexp 10896   || cdvds 12466   gcd cgcd 12642   Primecprime 12797   /Lclgs 15857

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242  ax-caucvg 8243

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-sup 7274  df-inf 7275  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-ioo 10221  df-ico 10223  df-fz 10339  df-fzo 10473  df-fl 10626  df-mod 10681  df-seqfrec 10806  df-exp 10897  df-fac 11084  df-ihash 11134  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677  df-clim 11957  df-proddc 12230  df-dvds 12467  df-gcd 12643  df-prm 12798  df-phi 12901  df-pc 12976  df-lgs 15858

This theorem is referenced by: (None)