Theorem 2lgs 15964

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 15877) 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 12816	. 2 |- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
2		2lgslem4 15963	. . . . . 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 6057	. . . . . 6 |- ( P = 2 -> ( 2 /L P ) = ( 2 /L 2 ) )
5	4	eqeq1d 2241	. . . . 5 |- ( P = 2 -> ( ( 2 /L P ) = 1 <-> ( 2 /L 2 ) = 1 ) )
6		oveq1 6056	. . . . . 6 |- ( P = 2 -> ( P mod 8 ) = ( 2 mod 8 ) )
7	6	eleq1d 2301	. . . . 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 12817	. . . . . . . . . 10 |- 2 e. Prime
11		prmnn 12800	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
12		dvdsprime 12812	. . . . . . . . . 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 12593	. . . . . . . . . . . . 13 |- 2 || 2
15		breq2 4112	. . . . . . . . . . . . 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 2295	. . . . . . . . . . . 12 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
19		1nprm 12804	. . . . . . . . . . . . 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 9601	. . . . . . 7 |- 2 e. ZZ
27	25, 26	jctil 312	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( 2 e. ZZ /\ -. P || 2 ) )
28		2lgslem1 15951	. . . . . . 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 2238	. . . . . 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 12953	. . . . . . . . . 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 2232	. . . . . . . 8 |- ( ( P - 1 ) / 2 ) = ( ( P - 1 ) / 2 )
34		eqid 2232	. . . . . . . 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 2232	. . . . . . . 8 |- ( |_ ` ( P / 4 ) ) = ( |_ ` ( P / 4 ) )
36		eqid 2232	. . . . . . . 8 |- ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
37	32, 33, 34, 35, 36	gausslemma2d 15929	. . . . . . 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 2241	. . . . . 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 15952	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) e. ZZ )
41		m1exp1 12580	. . . . . 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 9395	. . . . . . 7 |- 2 e. NN
44		dvdsval3 12470	. . . . . . 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 15961	. . . . . . . 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 2241	. . . . . 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 12801	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ZZ )
50		8nn 9401	. . . . . . . . . . . . . . . 16 |- 8 e. NN
51	50	a1i 9	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> 8 e. NN )
52	49, 51	zmodcld 10703	. . . . . . . . . . . . . 14 |- ( P e. Prime -> ( P mod 8 ) e. NN0 )
53	52	nn0zd 9694	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P mod 8 ) e. ZZ )
54		1z 9599	. . . . . . . . . . . . 13 |- 1 e. ZZ
55		zdceq 9649	. . . . . . . . . . . . 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 9400	. . . . . . . . . . . . . 14 |- 7 e. NN
58	57	nnzi 9594	. . . . . . . . . . . . 13 |- 7 e. ZZ
59		zdceq 9649	. . . . . . . . . . . . 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 3708	. . . . . . . . . . . . 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 3629	. . . . . . . . . . . 12 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 1 )
70	69	eqeq1d 2241	. . . . . . . . . . 11 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 <-> 1 = 0 ) )
71		1ne0 9301	. . . . . . . . . . . 12 |- 1 =/= 0
72		eqneqall 2422	. . . . . . . . . . . 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 3626	. . . . . . . 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 2203   =/= wne 2412   E.wrex 2521   {crab 2524   \ cdif 3207   ifcif 3619   {csn 3688   {cpr 3689   class class class wbr 4108   |-> cmpt 4170   ` cfv 5351  (class class class)co 6049   0cc0 8123   1c1 8124   x. cmul 8128   < clt 8304   - cmin 8440   -ucneg 8441   / cdiv 8942   NNcn 9233   2c2 9284   4c4 9286   7c7 9289   8c8 9290   NN0cn0 9492   ZZcz 9573   ...cfz 10338   |_cfl 10624   mod cmo 10680   ^cexp 10896  ♯chash 11133   || cdvds 12466   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-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:  2lgsoddprm  15973