Theorem 2lgsoddprm 16035

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 3343	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2		prmz 12816	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
3	1, 2	syl 14	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
4		8nn 9410	. . . . . . . . 9 |- 8 e. NN
5	4	a1i 9	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> 8 e. NN )
6	3, 5	zmodcld 10714	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. NN0 )
7	6	nn0zd 9704	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( P mod 8 ) e. ZZ )
8		1zzd 9609	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 1 e. ZZ )
9		zdceq 9658	. . . . . 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 9409	. . . . . . . 8 |- 7 e. NN
12	11	nnzi 9603	. . . . . . 7 |- 7 e. ZZ
13	12	a1i 9	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> 7 e. ZZ )
14		zdceq 9658	. . . . . 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 3711	. . . . . 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 16026	. . . 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 2236	. . . . . . . . . 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 12966	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
28		nnz 9601	. . . . . . . . . . . . 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 12580	. . . . . . . . . . 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 12595	. . . . . . . . . 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 16034	. . . . . . . . . 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 2267	. . . . 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 9610	. . . . . . . 8 |- 2 e. ZZ
43		lgscl1 15945	. . . . . . . 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 3736	. . . . . . . 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 12594	. . . . . . . . . . . . 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 2240	. . . . . . . . . . 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 2267	. . . . . . . . 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 3822	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
59		simpr 110	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ P =/= 2 ) -> P =/= 2 )
60	59	necomd 2500	. . . . . . . . . . 11 |- ( ( P e. Prime /\ P =/= 2 ) -> 2 =/= P )
61	58, 60	sylbi 121	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> 2 =/= P )
62		2prm 12832	. . . . . . . . . . 11 |- 2 e. Prime
63		prmrp 12850	. . . . . . . . . . 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 15960	. . . . . . . . . 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 2424	. . . . . . . 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 2205   =/= wne 2414   \ cdif 3210   {csn 3691   {cpr 3692   {ctp 3693   class class class wbr 4111  (class class class)co 6052   0cc0 8132   1c1 8133   - cmin 8449   -ucneg 8450   / cdiv 8951   NNcn 9242   2c2 9293   7c7 9298   8c8 9299   NN0cn0 9501   ZZcz 9582   mod cmo 10691   ^cexp 10907   || cdvds 12481   gcd cgcd 12657   Primecprime 12812   /Lclgs 15919

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-ioo 10231  df-ico 10233  df-fz 10349  df-fzo 10484  df-fl 10637  df-mod 10692  df-seqfrec 10817  df-exp 10908  df-fac 11096  df-ihash 11147  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-clim 11972  df-proddc 12245  df-dvds 12482  df-gcd 12658  df-prm 12813  df-phi 12916  df-pc 12991  df-lgs 15920

This theorem is referenced by: (None)