Theorem 2lgs 16026

Description: The second supplement to the law of quadratic reciprocity (for the Legendre symbol extended to arbitrary primes as second argument). Two is a square modulo a prime P iff P == pm 1 (mod 8), see first case of theorem 9.5 in [ApostolNT] p. 181. This theorem justifies our definition of ( N /L 2 ) (lgs2 15939) to some degree, by demanding that reciprocity extend to the case Q = 2. (Proposed by Mario Carneiro, 19-Jun-2015.) (Contributed by AV, 16-Jul-2021.)

Assertion

Ref	Expression
2lgs	|- ( P e. Prime -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) )

Proof of Theorem 2lgs

Dummy variables i x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prm2orodd 12831	. 2 |- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
2		2lgslem4 16025	. . . . . 6 |- ( ( 2 /L 2 ) = 1 <-> ( 2 mod 8 ) e. { 1 , 7 } )
3	2	a1i 9	. . . . 5 |- ( P = 2 -> ( ( 2 /L 2 ) = 1 <-> ( 2 mod 8 ) e. { 1 , 7 } ) )
4		oveq2 6060	. . . . . 6 |- ( P = 2 -> ( 2 /L P ) = ( 2 /L 2 ) )
5	4	eqeq1d 2243	. . . . 5 |- ( P = 2 -> ( ( 2 /L P ) = 1 <-> ( 2 /L 2 ) = 1 ) )
6		oveq1 6059	. . . . . 6 |- ( P = 2 -> ( P mod 8 ) = ( 2 mod 8 ) )
7	6	eleq1d 2303	. . . . 5 |- ( P = 2 -> ( ( P mod 8 ) e. { 1 , 7 } <-> ( 2 mod 8 ) e. { 1 , 7 } ) )
8	3, 5, 7	3bitr4d 220	. . . 4 |- ( P = 2 -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) )
9	8	a1d 22	. . 3 |- ( P = 2 -> ( P e. Prime -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) ) )
10		2prm 12832	. . . . . . . . . 10 |- 2 e. Prime
11		prmnn 12815	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
12		dvdsprime 12827	. . . . . . . . . 10 |- ( ( 2 e. Prime /\ P e. NN ) -> ( P || 2 <-> ( P = 2 \/ P = 1 ) ) )
13	10, 11, 12	sylancr 414	. . . . . . . . 9 |- ( P e. Prime -> ( P || 2 <-> ( P = 2 \/ P = 1 ) ) )
14		z2even 12608	. . . . . . . . . . . . 13 |- 2 || 2
15		breq2 4115	. . . . . . . . . . . . 13 |- ( P = 2 -> ( 2 || P <-> 2 || 2 ) )
16	14, 15	mpbiri 168	. . . . . . . . . . . 12 |- ( P = 2 -> 2 || P )
17	16	a1d 22	. . . . . . . . . . 11 |- ( P = 2 -> ( P e. Prime -> 2 || P ) )
18		eleq1 2297	. . . . . . . . . . . 12 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
19		1nprm 12819	. . . . . . . . . . . . 13 |- -. 1 e. Prime
20	19	pm2.21i 651	. . . . . . . . . . . 12 |- ( 1 e. Prime -> 2 || P )
21	18, 20	biimtrdi 163	. . . . . . . . . . 11 |- ( P = 1 -> ( P e. Prime -> 2 || P ) )
22	17, 21	jaoi 724	. . . . . . . . . 10 |- ( ( P = 2 \/ P = 1 ) -> ( P e. Prime -> 2 || P ) )
23	22	com12 30	. . . . . . . . 9 |- ( P e. Prime -> ( ( P = 2 \/ P = 1 ) -> 2 || P ) )
24	13, 23	sylbid 150	. . . . . . . 8 |- ( P e. Prime -> ( P || 2 -> 2 || P ) )
25	24	con3dimp 640	. . . . . . 7 |- ( ( P e. Prime /\ -. 2 || P ) -> -. P || 2 )
26		2z 9610	. . . . . . 7 |- 2 e. ZZ
27	25, 26	jctil 312	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( 2 e. ZZ /\ -. P || 2 ) )
28		2lgslem1 16013	. . . . . . 7 |- ( ( P e. Prime /\ -. 2 || P ) -> (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) )
29	28	eqcomd 2240	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) )
30		nnoddn2prmb 12968	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ -. 2 || P ) )
31	30	biimpri 133	. . . . . . . . 9 |- ( ( P e. Prime /\ -. 2 || P ) -> P e. ( Prime \ { 2 } ) )
32	31	3ad2ant1 1045	. . . . . . . 8 |- ( ( ( P e. Prime /\ -. 2 || P ) /\ ( 2 e. ZZ /\ -. P || 2 ) /\ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) ) -> P e. ( Prime \ { 2 } ) )
33		eqid 2234	. . . . . . . 8 |- ( ( P - 1 ) / 2 ) = ( ( P - 1 ) / 2 )
34		eqid 2234	. . . . . . . 8 |- ( y e. ( 1 ... ( ( P - 1 ) / 2 ) ) |-> if ( ( y x. 2 ) < ( P / 2 ) , ( y x. 2 ) , ( P - ( y x. 2 ) ) ) ) = ( y e. ( 1 ... ( ( P - 1 ) / 2 ) ) |-> if ( ( y x. 2 ) < ( P / 2 ) , ( y x. 2 ) , ( P - ( y x. 2 ) ) ) )
35		eqid 2234	. . . . . . . 8 |- ( |_ ` ( P / 4 ) ) = ( |_ ` ( P / 4 ) )
36		eqid 2234	. . . . . . . 8 |- ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
37	32, 33, 34, 35, 36	gausslemma2d 15991	. . . . . . 7 |- ( ( ( P e. Prime /\ -. 2 || P ) /\ ( 2 e. ZZ /\ -. P || 2 ) /\ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) )
38	37	eqeq1d 2243	. . . . . 6 |- ( ( ( P e. Prime /\ -. 2 || P ) /\ ( 2 e. ZZ /\ -. P || 2 ) /\ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) ) -> ( ( 2 /L P ) = 1 <-> ( -u 1 ^ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) = 1 ) )
39	27, 29, 38	mpd3an23 1376	. . . . 5 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( 2 /L P ) = 1 <-> ( -u 1 ^ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) = 1 ) )
40	36	2lgslem2 16014	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) e. ZZ )
41		m1exp1 12595	. . . . . 6 |- ( ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) e. ZZ -> ( ( -u 1 ^ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) = 1 <-> 2 || ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) )
42	40, 41	syl 14	. . . . 5 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( -u 1 ^ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) = 1 <-> 2 || ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) )
43		2nn 9404	. . . . . . 7 |- 2 e. NN
44		dvdsval3 12485	. . . . . . 7 |- ( ( 2 e. NN /\ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) e. ZZ ) -> ( 2 || ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) <-> ( ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) mod 2 ) = 0 ) )
45	43, 40, 44	sylancr 414	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( 2 || ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) <-> ( ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) mod 2 ) = 0 ) )
46	36	2lgslem3 16023	. . . . . . . 8 |- ( ( P e. NN /\ -. 2 || P ) -> ( ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )
47	11, 46	sylan 283	. . . . . . 7 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )
48	47	eqeq1d 2243	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) mod 2 ) = 0 <-> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 ) )
49		prmz 12816	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ZZ )
50		8nn 9410	. . . . . . . . . . . . . . . 16 |- 8 e. NN
51	50	a1i 9	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> 8 e. NN )
52	49, 51	zmodcld 10714	. . . . . . . . . . . . . 14 |- ( P e. Prime -> ( P mod 8 ) e. NN0 )
53	52	nn0zd 9704	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P mod 8 ) e. ZZ )
54		1z 9608	. . . . . . . . . . . . 13 |- 1 e. ZZ
55		zdceq 9658	. . . . . . . . . . . . 13 |- ( ( ( P mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( P mod 8 ) = 1 )
56	53, 54, 55	sylancl 413	. . . . . . . . . . . 12 |- ( P e. Prime -> DECID ( P mod 8 ) = 1 )
57		7nn 9409	. . . . . . . . . . . . . 14 |- 7 e. NN
58	57	nnzi 9603	. . . . . . . . . . . . 13 |- 7 e. ZZ
59		zdceq 9658	. . . . . . . . . . . . 13 |- ( ( ( P mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( P mod 8 ) = 7 )
60	53, 58, 59	sylancl 413	. . . . . . . . . . . 12 |- ( P e. Prime -> DECID ( P mod 8 ) = 7 )
61		dcor 944	. . . . . . . . . . . 12 |- (DECID ( P mod 8 ) = 1 -> (DECID ( P mod 8 ) = 7 -> DECID ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) ) )
62	56, 60, 61	sylc 62	. . . . . . . . . . 11 |- ( P e. Prime -> DECID ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) )
63		elprg 3711	. . . . . . . . . . . . 13 |- ( ( P mod 8 ) e. NN0 -> ( ( P mod 8 ) e. { 1 , 7 } <-> ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) ) )
64	52, 63	syl 14	. . . . . . . . . . . 12 |- ( P e. Prime -> ( ( P mod 8 ) e. { 1 , 7 } <-> ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) ) )
65	64	dcbid 846	. . . . . . . . . . 11 |- ( P e. Prime -> (DECID ( P mod 8 ) e. { 1 , 7 } <-> DECID ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) ) )
66	62, 65	mpbird 167	. . . . . . . . . 10 |- ( P e. Prime -> DECID ( P mod 8 ) e. { 1 , 7 } )
67		exmiddc 844	. . . . . . . . . 10 |- (DECID ( P mod 8 ) e. { 1 , 7 } -> ( ( P mod 8 ) e. { 1 , 7 } \/ -. ( P mod 8 ) e. { 1 , 7 } ) )
68	66, 67	syl 14	. . . . . . . . 9 |- ( P e. Prime -> ( ( P mod 8 ) e. { 1 , 7 } \/ -. ( P mod 8 ) e. { 1 , 7 } ) )
69		iffalse 3632	. . . . . . . . . . . 12 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 1 )
70	69	eqeq1d 2243	. . . . . . . . . . 11 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 <-> 1 = 0 ) )
71		1ne0 9310	. . . . . . . . . . . 12 |- 1 =/= 0
72		eqneqall 2424	. . . . . . . . . . . 12 |- ( 1 = 0 -> ( 1 =/= 0 -> ( P mod 8 ) e. { 1 , 7 } ) )
73	71, 72	mpi 15	. . . . . . . . . . 11 |- ( 1 = 0 -> ( P mod 8 ) e. { 1 , 7 } )
74	70, 73	biimtrdi 163	. . . . . . . . . 10 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 -> ( P mod 8 ) e. { 1 , 7 } ) )
75	74	jao1i 804	. . . . . . . . 9 |- ( ( ( P mod 8 ) e. { 1 , 7 } \/ -. ( P mod 8 ) e. { 1 , 7 } ) -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 -> ( P mod 8 ) e. { 1 , 7 } ) )
76	68, 75	syl 14	. . . . . . . 8 |- ( P e. Prime -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 -> ( P mod 8 ) e. { 1 , 7 } ) )
77		iftrue 3629	. . . . . . . 8 |- ( ( P mod 8 ) e. { 1 , 7 } -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 )
78	76, 77	impbid1 142	. . . . . . 7 |- ( P e. Prime -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 <-> ( P mod 8 ) e. { 1 , 7 } ) )
79	78	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 <-> ( P mod 8 ) e. { 1 , 7 } ) )
80	45, 48, 79	3bitrd 214	. . . . 5 |- ( ( P e. Prime /\ -. 2 || P ) -> ( 2 || ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) <-> ( P mod 8 ) e. { 1 , 7 } ) )
81	39, 42, 80	3bitrd 214	. . . 4 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) )
82	81	expcom 116	. . 3 |- ( -. 2 || P -> ( P e. Prime -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) ) )
83	9, 82	jaoi 724	. 2 |- ( ( P = 2 \/ -. 2 || P ) -> ( P e. Prime -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) ) )
84	1, 83	mpcom 36	1 |- ( P e. Prime -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   E.wrex 2523   {crab 2526   \ cdif 3210   ifcif 3622   {csn 3691   {cpr 3692   class class class wbr 4111   |-> cmpt 4173   ` cfv 5354  (class class class)co 6052   0cc0 8132   1c1 8133   x. cmul 8137   < clt 8313   - cmin 8449   -ucneg 8450   / cdiv 8951   NNcn 9242   2c2 9293   4c4 9295   7c7 9298   8c8 9299   NN0cn0 9501   ZZcz 9582   ...cfz 10348   |_cfl 10635   mod cmo 10691   ^cexp 10907  ♯chash 11146   || cdvds 12481   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-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:  2lgsoddprm  16035