Theorem 2lgs 15252

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 15165) 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 12267	. 2 |- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
2		2lgslem4 15251	. . . . . 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 5927	. . . . . 6 |- ( P = 2 -> ( 2 /L P ) = ( 2 /L 2 ) )
5	4	eqeq1d 2202	. . . . 5 |- ( P = 2 -> ( ( 2 /L P ) = 1 <-> ( 2 /L 2 ) = 1 ) )
6		oveq1 5926	. . . . . 6 |- ( P = 2 -> ( P mod 8 ) = ( 2 mod 8 ) )
7	6	eleq1d 2262	. . . . 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 12268	. . . . . . . . . 10 |- 2 e. Prime
11		prmnn 12251	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
12		dvdsprime 12263	. . . . . . . . . 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 12058	. . . . . . . . . . . . 13 |- 2 || 2
15		breq2 4034	. . . . . . . . . . . . 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 2256	. . . . . . . . . . . 12 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
19		1nprm 12255	. . . . . . . . . . . . 13 |- -. 1 e. Prime
20	19	pm2.21i 647	. . . . . . . . . . . 12 |- ( 1 e. Prime -> 2 || P )
21	18, 20	biimtrdi 163	. . . . . . . . . . 11 |- ( P = 1 -> ( P e. Prime -> 2 || P ) )
22	17, 21	jaoi 717	. . . . . . . . . 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 636	. . . . . . 7 |- ( ( P e. Prime /\ -. 2 || P ) -> -. P || 2 )
26		2z 9348	. . . . . . 7 |- 2 e. ZZ
27	25, 26	jctil 312	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( 2 e. ZZ /\ -. P || 2 ) )
28		2lgslem1 15239	. . . . . . 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 2199	. . . . . 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 12403	. . . . . . . . . 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 1020	. . . . . . . 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 2193	. . . . . . . 8 |- ( ( P - 1 ) / 2 ) = ( ( P - 1 ) / 2 )
34		eqid 2193	. . . . . . . 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 2193	. . . . . . . 8 |- ( |_ ` ( P / 4 ) ) = ( |_ ` ( P / 4 ) )
36		eqid 2193	. . . . . . . 8 |- ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
37	32, 33, 34, 35, 36	gausslemma2d 15217	. . . . . . 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 2202	. . . . . 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 1350	. . . . 5 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( 2 /L P ) = 1 <-> ( -u 1 ^ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) = 1 ) )
40	36	2lgslem2 15240	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) e. ZZ )
41		m1exp1 12045	. . . . . 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 9146	. . . . . . 7 |- 2 e. NN
44		dvdsval3 11937	. . . . . . 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 15249	. . . . . . . 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 2202	. . . . . 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 12252	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ZZ )
50		8nn 9152	. . . . . . . . . . . . . . . 16 |- 8 e. NN
51	50	a1i 9	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> 8 e. NN )
52	49, 51	zmodcld 10419	. . . . . . . . . . . . . 14 |- ( P e. Prime -> ( P mod 8 ) e. NN0 )
53	52	nn0zd 9440	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P mod 8 ) e. ZZ )
54		1z 9346	. . . . . . . . . . . . 13 |- 1 e. ZZ
55		zdceq 9395	. . . . . . . . . . . . 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 9151	. . . . . . . . . . . . . 14 |- 7 e. NN
58	57	nnzi 9341	. . . . . . . . . . . . 13 |- 7 e. ZZ
59		zdceq 9395	. . . . . . . . . . . . 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 937	. . . . . . . . . . . 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 3639	. . . . . . . . . . . . 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 839	. . . . . . . . . . 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 837	. . . . . . . . . 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 3566	. . . . . . . . . . . 12 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 1 )
70	69	eqeq1d 2202	. . . . . . . . . . 11 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 <-> 1 = 0 ) )
71		1ne0 9052	. . . . . . . . . . . 12 |- 1 =/= 0
72		eqneqall 2374	. . . . . . . . . . . 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 797	. . . . . . . . 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 3563	. . . . . . . 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 717	. 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 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   =/= wne 2364   E.wrex 2473   {crab 2476   \ cdif 3151   ifcif 3558   {csn 3619   {cpr 3620   class class class wbr 4030   |-> cmpt 4091   ` cfv 5255  (class class class)co 5919   0cc0 7874   1c1 7875   x. cmul 7879   < clt 8056   - cmin 8192   -ucneg 8193   / cdiv 8693   NNcn 8984   2c2 9035   4c4 9037   7c7 9040   8c8 9041   NN0cn0 9243   ZZcz 9320   ...cfz 10077   |_cfl 10340   mod cmo 10396   ^cexp 10612  ♯chash 10849   || cdvds 11933   Primecprime 12248   /Lclgs 15145

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-2o 6472  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-ioo 9961  df-ico 9963  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-proddc 11697  df-dvds 11934  df-gcd 12083  df-prm 12249  df-phi 12352  df-pc 12426  df-lgs 15146

This theorem is referenced by: (None)