Theorem 2lgs 15791

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 15704) 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 12656	. 2 |- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
2		2lgslem4 15790	. . . . . 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 6015	. . . . . 6 |- ( P = 2 -> ( 2 /L P ) = ( 2 /L 2 ) )
5	4	eqeq1d 2238	. . . . 5 |- ( P = 2 -> ( ( 2 /L P ) = 1 <-> ( 2 /L 2 ) = 1 ) )
6		oveq1 6014	. . . . . 6 |- ( P = 2 -> ( P mod 8 ) = ( 2 mod 8 ) )
7	6	eleq1d 2298	. . . . 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 12657	. . . . . . . . . 10 |- 2 e. Prime
11		prmnn 12640	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
12		dvdsprime 12652	. . . . . . . . . 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 12433	. . . . . . . . . . . . 13 |- 2 || 2
15		breq2 4087	. . . . . . . . . . . . 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 2292	. . . . . . . . . . . 12 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
19		1nprm 12644	. . . . . . . . . . . . 13 |- -. 1 e. Prime
20	19	pm2.21i 649	. . . . . . . . . . . 12 |- ( 1 e. Prime -> 2 || P )
21	18, 20	biimtrdi 163	. . . . . . . . . . 11 |- ( P = 1 -> ( P e. Prime -> 2 || P ) )
22	17, 21	jaoi 721	. . . . . . . . . 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 638	. . . . . . 7 |- ( ( P e. Prime /\ -. 2 || P ) -> -. P || 2 )
26		2z 9482	. . . . . . 7 |- 2 e. ZZ
27	25, 26	jctil 312	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( 2 e. ZZ /\ -. P || 2 ) )
28		2lgslem1 15778	. . . . . . 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 2235	. . . . . 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 12793	. . . . . . . . . 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 1042	. . . . . . . 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 2229	. . . . . . . 8 |- ( ( P - 1 ) / 2 ) = ( ( P - 1 ) / 2 )
34		eqid 2229	. . . . . . . 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 2229	. . . . . . . 8 |- ( |_ ` ( P / 4 ) ) = ( |_ ` ( P / 4 ) )
36		eqid 2229	. . . . . . . 8 |- ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
37	32, 33, 34, 35, 36	gausslemma2d 15756	. . . . . . 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 2238	. . . . . 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 1373	. . . . 5 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( 2 /L P ) = 1 <-> ( -u 1 ^ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) = 1 ) )
40	36	2lgslem2 15779	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) e. ZZ )
41		m1exp1 12420	. . . . . 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 9280	. . . . . . 7 |- 2 e. NN
44		dvdsval3 12310	. . . . . . 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 15788	. . . . . . . 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 2238	. . . . . 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 12641	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ZZ )
50		8nn 9286	. . . . . . . . . . . . . . . 16 |- 8 e. NN
51	50	a1i 9	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> 8 e. NN )
52	49, 51	zmodcld 10575	. . . . . . . . . . . . . 14 |- ( P e. Prime -> ( P mod 8 ) e. NN0 )
53	52	nn0zd 9575	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P mod 8 ) e. ZZ )
54		1z 9480	. . . . . . . . . . . . 13 |- 1 e. ZZ
55		zdceq 9530	. . . . . . . . . . . . 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 9285	. . . . . . . . . . . . . 14 |- 7 e. NN
58	57	nnzi 9475	. . . . . . . . . . . . 13 |- 7 e. ZZ
59		zdceq 9530	. . . . . . . . . . . . 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 941	. . . . . . . . . . . 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 3686	. . . . . . . . . . . . 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 843	. . . . . . . . . . 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 841	. . . . . . . . . 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 3610	. . . . . . . . . . . 12 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 1 )
70	69	eqeq1d 2238	. . . . . . . . . . 11 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 <-> 1 = 0 ) )
71		1ne0 9186	. . . . . . . . . . . 12 |- 1 =/= 0
72		eqneqall 2410	. . . . . . . . . . . 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 801	. . . . . . . . 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 3607	. . . . . . . 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 721	. 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 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509   {crab 2512   \ cdif 3194   ifcif 3602   {csn 3666   {cpr 3667   class class class wbr 4083   |-> cmpt 4145   ` cfv 5318  (class class class)co 6007   0cc0 8007   1c1 8008   x. cmul 8012   < clt 8189   - cmin 8325   -ucneg 8326   / cdiv 8827   NNcn 9118   2c2 9169   4c4 9171   7c7 9174   8c8 9175   NN0cn0 9377   ZZcz 9454   ...cfz 10212   |_cfl 10496   mod cmo 10552   ^cexp 10768  ♯chash 11005   || cdvds 12306   Primecprime 12637   /Lclgs 15684

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-2o 6569  df-oadd 6572  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-n0 9378  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-ioo 10096  df-ico 10098  df-fz 10213  df-fzo 10347  df-fl 10498  df-mod 10553  df-seqfrec 10678  df-exp 10769  df-fac 10956  df-ihash 11006  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-clim 11798  df-proddc 12070  df-dvds 12307  df-gcd 12483  df-prm 12638  df-phi 12741  df-pc 12816  df-lgs 15685

This theorem is referenced by:  2lgsoddprm  15800