Theorem 2lgs 15625

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 15538) 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 12492	. 2 |- ( P e. Prime -> ( P = 2 \/ -. 2 || P ) )
2		2lgslem4 15624	. . . . . 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 5959	. . . . . 6 |- ( P = 2 -> ( 2 /L P ) = ( 2 /L 2 ) )
5	4	eqeq1d 2215	. . . . 5 |- ( P = 2 -> ( ( 2 /L P ) = 1 <-> ( 2 /L 2 ) = 1 ) )
6		oveq1 5958	. . . . . 6 |- ( P = 2 -> ( P mod 8 ) = ( 2 mod 8 ) )
7	6	eleq1d 2275	. . . . 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 12493	. . . . . . . . . 10 |- 2 e. Prime
11		prmnn 12476	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
12		dvdsprime 12488	. . . . . . . . . 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 12269	. . . . . . . . . . . . 13 |- 2 || 2
15		breq2 4051	. . . . . . . . . . . . 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 2269	. . . . . . . . . . . 12 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
19		1nprm 12480	. . . . . . . . . . . . 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 718	. . . . . . . . . 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 9407	. . . . . . 7 |- 2 e. ZZ
27	25, 26	jctil 312	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( 2 e. ZZ /\ -. P || 2 ) )
28		2lgslem1 15612	. . . . . . 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 2212	. . . . . 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 12629	. . . . . . . . . 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 1021	. . . . . . . 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 2206	. . . . . . . 8 |- ( ( P - 1 ) / 2 ) = ( ( P - 1 ) / 2 )
34		eqid 2206	. . . . . . . 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 2206	. . . . . . . 8 |- ( |_ ` ( P / 4 ) ) = ( |_ ` ( P / 4 ) )
36		eqid 2206	. . . . . . . 8 |- ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
37	32, 33, 34, 35, 36	gausslemma2d 15590	. . . . . . 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 2215	. . . . . 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 1352	. . . . 5 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( 2 /L P ) = 1 <-> ( -u 1 ^ ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) ) = 1 ) )
40	36	2lgslem2 15613	. . . . . 6 |- ( ( P e. Prime /\ -. 2 || P ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) e. ZZ )
41		m1exp1 12256	. . . . . 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 9205	. . . . . . 7 |- 2 e. NN
44		dvdsval3 12146	. . . . . . 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 15622	. . . . . . . 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 2215	. . . . . 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 12477	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ZZ )
50		8nn 9211	. . . . . . . . . . . . . . . 16 |- 8 e. NN
51	50	a1i 9	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> 8 e. NN )
52	49, 51	zmodcld 10497	. . . . . . . . . . . . . 14 |- ( P e. Prime -> ( P mod 8 ) e. NN0 )
53	52	nn0zd 9500	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( P mod 8 ) e. ZZ )
54		1z 9405	. . . . . . . . . . . . 13 |- 1 e. ZZ
55		zdceq 9455	. . . . . . . . . . . . 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 9210	. . . . . . . . . . . . . 14 |- 7 e. NN
58	57	nnzi 9400	. . . . . . . . . . . . 13 |- 7 e. ZZ
59		zdceq 9455	. . . . . . . . . . . . 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 938	. . . . . . . . . . . 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 3654	. . . . . . . . . . . . 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 840	. . . . . . . . . . 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 838	. . . . . . . . . 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 3580	. . . . . . . . . . . 12 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 1 )
70	69	eqeq1d 2215	. . . . . . . . . . 11 |- ( -. ( P mod 8 ) e. { 1 , 7 } -> ( if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 <-> 1 = 0 ) )
71		1ne0 9111	. . . . . . . . . . . 12 |- 1 =/= 0
72		eqneqall 2387	. . . . . . . . . . . 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 798	. . . . . . . . 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 3577	. . . . . . . 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 718	. 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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   E.wrex 2486   {crab 2489   \ cdif 3164   ifcif 3572   {csn 3634   {cpr 3635   class class class wbr 4047   |-> cmpt 4109   ` cfv 5276  (class class class)co 5951   0cc0 7932   1c1 7933   x. cmul 7937   < clt 8114   - cmin 8250   -ucneg 8251   / cdiv 8752   NNcn 9043   2c2 9094   4c4 9096   7c7 9099   8c8 9100   NN0cn0 9302   ZZcz 9379   ...cfz 10137   |_cfl 10418   mod cmo 10474   ^cexp 10690  ♯chash 10927   || cdvds 12142   Primecprime 12473   /Lclgs 15518

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-tp 3642  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-2o 6510  df-oadd 6513  df-er 6627  df-en 6835  df-dom 6836  df-fin 6837  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-ioo 10021  df-ico 10023  df-fz 10138  df-fzo 10272  df-fl 10420  df-mod 10475  df-seqfrec 10600  df-exp 10691  df-fac 10878  df-ihash 10928  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-proddc 11906  df-dvds 12143  df-gcd 12319  df-prm 12474  df-phi 12577  df-pc 12652  df-lgs 15519

This theorem is referenced by:  2lgsoddprm  15634