Theorem gausslemma2d 15320

Description: Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer 2: Let p be an odd prime. Let S = {2, 4, 6, ..., p - 1}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)

Hypotheses

Ref	Expression
gausslemma2d.p	|- ( ph -> P e. ( Prime \ { 2 } ) )
gausslemma2d.h	|- H = ( ( P - 1 ) / 2 )
gausslemma2d.r	|- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )
gausslemma2d.m	|- M = ( |_ ` ( P / 4 ) )
gausslemma2d.n	|- N = ( H - M )

Assertion

Ref	Expression
gausslemma2d	|- ( ph -> ( 2 /L P ) = ( -u 1 ^ N ) )

Distinct variable groups:   x, H   x, P   ph, x   x, M

Allowed substitution hints:   R( x)   N( x)

Proof of Theorem gausslemma2d

Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . 3 |- ( ph -> P e. ( Prime \ { 2 } ) )
2		gausslemma2d.h	. . 3 |- H = ( ( P - 1 ) / 2 )
3		gausslemma2d.r	. . 3 |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )
4		gausslemma2d.m	. . 3 |- M = ( |_ ` ( P / 4 ) )
5		gausslemma2d.n	. . 3 |- N = ( H - M )
6	1, 2, 3, 4, 5	gausslemma2dlem7 15319	. 2 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 )
7	1	gausslemma2dlem0a 15300	. . . . . . 7 |- ( ph -> P e. NN )
8		nnq 9709	. . . . . . 7 |- ( P e. NN -> P e. QQ )
9	7, 8	syl 14	. . . . . 6 |- ( ph -> P e. QQ )
10		eldifi 3286	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
11		prmgt1 12310	. . . . . . 7 |- ( P e. Prime -> 1 < P )
12	1, 10, 11	3syl 17	. . . . . 6 |- ( ph -> 1 < P )
13		q1mod 10450	. . . . . 6 |- ( ( P e. QQ /\ 1 < P ) -> ( 1 mod P ) = 1 )
14	9, 12, 13	syl2anc 411	. . . . 5 |- ( ph -> ( 1 mod P ) = 1 )
15	14	eqcomd 2202	. . . 4 |- ( ph -> 1 = ( 1 mod P ) )
16	15	eqeq2d 2208	. . 3 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 <-> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) )
17		neg1z 9360	. . . . . . . . . 10 |- -u 1 e. ZZ
18	1, 4, 2, 5	gausslemma2dlem0h 15307	. . . . . . . . . 10 |- ( ph -> N e. NN0 )
19		zexpcl 10648	. . . . . . . . . 10 |- ( ( -u 1 e. ZZ /\ N e. NN0 ) -> ( -u 1 ^ N ) e. ZZ )
20	17, 18, 19	sylancr 414	. . . . . . . . 9 |- ( ph -> ( -u 1 ^ N ) e. ZZ )
21		2nn 9154	. . . . . . . . . . . 12 |- 2 e. NN
22	21	a1i 9	. . . . . . . . . . 11 |- ( ph -> 2 e. NN )
23	1, 2	gausslemma2dlem0b 15301	. . . . . . . . . . . 12 |- ( ph -> H e. NN )
24	23	nnnn0d 9304	. . . . . . . . . . 11 |- ( ph -> H e. NN0 )
25	22, 24	nnexpcld 10789	. . . . . . . . . 10 |- ( ph -> ( 2 ^ H ) e. NN )
26	25	nnzd 9449	. . . . . . . . 9 |- ( ph -> ( 2 ^ H ) e. ZZ )
27	20, 26	zmulcld 9456	. . . . . . . 8 |- ( ph -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. ZZ )
28		zq 9702	. . . . . . . 8 |- ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. ZZ -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. QQ )
29	27, 28	syl 14	. . . . . . 7 |- ( ph -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. QQ )
30	29	adantr 276	. . . . . 6 |- ( ( ph /\ ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. QQ )
31		1z 9354	. . . . . . 7 |- 1 e. ZZ
32		zq 9702	. . . . . . 7 |- ( 1 e. ZZ -> 1 e. QQ )
33	31, 32	mp1i 10	. . . . . 6 |- ( ( ph /\ ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) -> 1 e. QQ )
34	20	adantr 276	. . . . . 6 |- ( ( ph /\ ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) -> ( -u 1 ^ N ) e. ZZ )
35	9	adantr 276	. . . . . 6 |- ( ( ph /\ ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) -> P e. QQ )
36	7	nngt0d 9036	. . . . . . 7 |- ( ph -> 0 < P )
37	36	adantr 276	. . . . . 6 |- ( ( ph /\ ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) -> 0 < P )
38		simpr 110	. . . . . 6 |- ( ( ph /\ ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) )
39	30, 33, 34, 35, 37, 38	modqmul1 10471	. . . . 5 |- ( ( ph /\ ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) mod P ) = ( ( 1 x. ( -u 1 ^ N ) ) mod P ) )
40	39	ex 115	. . . 4 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) mod P ) = ( ( 1 x. ( -u 1 ^ N ) ) mod P ) ) )
41	20	zcnd 9451	. . . . . . . . 9 |- ( ph -> ( -u 1 ^ N ) e. CC )
42	25	nncnd 9006	. . . . . . . . 9 |- ( ph -> ( 2 ^ H ) e. CC )
43	41, 42, 41	mul32d 8181	. . . . . . . 8 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) = ( ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) x. ( 2 ^ H ) ) )
44	18	nn0cnd 9306	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
45	44	2timesd 9236	. . . . . . . . . . . 12 |- ( ph -> ( 2 x. N ) = ( N + N ) )
46	45	eqcomd 2202	. . . . . . . . . . 11 |- ( ph -> ( N + N ) = ( 2 x. N ) )
47	46	oveq2d 5939	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( N + N ) ) = ( -u 1 ^ ( 2 x. N ) ) )
48		neg1cn 9097	. . . . . . . . . . . 12 |- -u 1 e. CC
49	48	a1i 9	. . . . . . . . . . 11 |- ( ph -> -u 1 e. CC )
50	49, 18, 18	expaddd 10769	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
51	18	nn0zd 9448	. . . . . . . . . . 11 |- ( ph -> N e. ZZ )
52		m1expeven 10680	. . . . . . . . . . 11 |- ( N e. ZZ -> ( -u 1 ^ ( 2 x. N ) ) = 1 )
53	51, 52	syl 14	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( 2 x. N ) ) = 1 )
54	47, 50, 53	3eqtr3d 2237	. . . . . . . . 9 |- ( ph -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
55	54	oveq1d 5938	. . . . . . . 8 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) x. ( 2 ^ H ) ) = ( 1 x. ( 2 ^ H ) ) )
56	42	mullidd 8046	. . . . . . . 8 |- ( ph -> ( 1 x. ( 2 ^ H ) ) = ( 2 ^ H ) )
57	43, 55, 56	3eqtrd 2233	. . . . . . 7 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) = ( 2 ^ H ) )
58	57	oveq1d 5938	. . . . . 6 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) mod P ) = ( ( 2 ^ H ) mod P ) )
59	41	mullidd 8046	. . . . . . 7 |- ( ph -> ( 1 x. ( -u 1 ^ N ) ) = ( -u 1 ^ N ) )
60	59	oveq1d 5938	. . . . . 6 |- ( ph -> ( ( 1 x. ( -u 1 ^ N ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) )
61	58, 60	eqeq12d 2211	. . . . 5 |- ( ph -> ( ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) mod P ) = ( ( 1 x. ( -u 1 ^ N ) ) mod P ) <-> ( ( 2 ^ H ) mod P ) = ( ( -u 1 ^ N ) mod P ) ) )
62	2	oveq2i 5934	. . . . . . . 8 |- ( 2 ^ H ) = ( 2 ^ ( ( P - 1 ) / 2 ) )
63	62	oveq1i 5933	. . . . . . 7 |- ( ( 2 ^ H ) mod P ) = ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P )
64	63	eqeq1i 2204	. . . . . 6 |- ( ( ( 2 ^ H ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) )
65		2z 9356	. . . . . . . . . 10 |- 2 e. ZZ
66		lgsvalmod 15270	. . . . . . . . . 10 |- ( ( 2 e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( 2 /L P ) mod P ) = ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) )
67	65, 1, 66	sylancr 414	. . . . . . . . 9 |- ( ph -> ( ( 2 /L P ) mod P ) = ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) )
68	67	eqcomd 2202	. . . . . . . 8 |- ( ph -> ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( 2 /L P ) mod P ) )
69	68	eqeq1d 2205	. . . . . . 7 |- ( ph -> ( ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) ) )
70	1, 4, 2, 5	gausslemma2dlem0i 15308	. . . . . . 7 |- ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )
71	69, 70	sylbid 150	. . . . . 6 |- ( ph -> ( ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )
72	64, 71	biimtrid 152	. . . . 5 |- ( ph -> ( ( ( 2 ^ H ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )
73	61, 72	sylbid 150	. . . 4 |- ( ph -> ( ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) mod P ) = ( ( 1 x. ( -u 1 ^ N ) ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )
74	40, 73	syld 45	. . 3 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) )
75	16, 74	sylbid 150	. 2 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 -> ( 2 /L P ) = ( -u 1 ^ N ) ) )
76	6, 75	mpd 13	1 |- ( ph -> ( 2 /L P ) = ( -u 1 ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   \ cdif 3154   ifcif 3562   {csn 3623   class class class wbr 4034   |-> cmpt 4095   ` cfv 5259  (class class class)co 5923   CCcc 7879   0cc0 7881   1c1 7882   + caddc 7884   x. cmul 7886   < clt 8063   - cmin 8199   -ucneg 8200   / cdiv 8701   NNcn 8992   2c2 9043   4c4 9045   NN0cn0 9251   ZZcz 9328   QQcq 9695   ...cfz 10085   |_cfl 10360   mod cmo 10416   ^cexp 10632   Primecprime 12285   /Lclgs 15248

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000  ax-caucvg 8001

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-irdg 6429  df-frec 6450  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6593  df-en 6801  df-dom 6802  df-fin 6803  df-sup 7051  df-inf 7052  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-5 9054  df-6 9055  df-7 9056  df-8 9057  df-n0 9252  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-ioo 9969  df-fz 10086  df-fzo 10220  df-fl 10362  df-mod 10417  df-seqfrec 10542  df-exp 10633  df-fac 10820  df-ihash 10870  df-cj 11009  df-re 11010  df-im 11011  df-rsqrt 11165  df-abs 11166  df-clim 11446  df-proddc 11718  df-dvds 11955  df-gcd 12131  df-prm 12286  df-phi 12389  df-pc 12464  df-lgs 15249

This theorem is referenced by:  2lgs  15355