Theorem gausslemma2d 15185

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 15184	. 2 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 )
7	1	gausslemma2dlem0a 15165	. . . . . . 7 |- ( ph -> P e. NN )
8		nnq 9698	. . . . . . 7 |- ( P e. NN -> P e. QQ )
9	7, 8	syl 14	. . . . . 6 |- ( ph -> P e. QQ )
10		eldifi 3281	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
11		prmgt1 12270	. . . . . . 7 |- ( P e. Prime -> 1 < P )
12	1, 10, 11	3syl 17	. . . . . 6 |- ( ph -> 1 < P )
13		q1mod 10427	. . . . . 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 2199	. . . 4 |- ( ph -> 1 = ( 1 mod P ) )
16	15	eqeq2d 2205	. . 3 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 <-> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) )
17		neg1z 9349	. . . . . . . . . 10 |- -u 1 e. ZZ
18	1, 4, 2, 5	gausslemma2dlem0h 15172	. . . . . . . . . 10 |- ( ph -> N e. NN0 )
19		zexpcl 10625	. . . . . . . . . 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 9143	. . . . . . . . . . . 12 |- 2 e. NN
22	21	a1i 9	. . . . . . . . . . 11 |- ( ph -> 2 e. NN )
23	1, 2	gausslemma2dlem0b 15166	. . . . . . . . . . . 12 |- ( ph -> H e. NN )
24	23	nnnn0d 9293	. . . . . . . . . . 11 |- ( ph -> H e. NN0 )
25	22, 24	nnexpcld 10766	. . . . . . . . . 10 |- ( ph -> ( 2 ^ H ) e. NN )
26	25	nnzd 9438	. . . . . . . . 9 |- ( ph -> ( 2 ^ H ) e. ZZ )
27	20, 26	zmulcld 9445	. . . . . . . 8 |- ( ph -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. ZZ )
28		zq 9691	. . . . . . . 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 9343	. . . . . . 7 |- 1 e. ZZ
32		zq 9691	. . . . . . 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 9026	. . . . . . 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 10448	. . . . 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 9440	. . . . . . . . 9 |- ( ph -> ( -u 1 ^ N ) e. CC )
42	25	nncnd 8996	. . . . . . . . 9 |- ( ph -> ( 2 ^ H ) e. CC )
43	41, 42, 41	mul32d 8172	. . . . . . . 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 9295	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
45	44	2timesd 9225	. . . . . . . . . . . 12 |- ( ph -> ( 2 x. N ) = ( N + N ) )
46	45	eqcomd 2199	. . . . . . . . . . 11 |- ( ph -> ( N + N ) = ( 2 x. N ) )
47	46	oveq2d 5934	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( N + N ) ) = ( -u 1 ^ ( 2 x. N ) ) )
48		neg1cn 9087	. . . . . . . . . . . 12 |- -u 1 e. CC
49	48	a1i 9	. . . . . . . . . . 11 |- ( ph -> -u 1 e. CC )
50	49, 18, 18	expaddd 10746	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
51	18	nn0zd 9437	. . . . . . . . . . 11 |- ( ph -> N e. ZZ )
52		m1expeven 10657	. . . . . . . . . . 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 2234	. . . . . . . . 9 |- ( ph -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
55	54	oveq1d 5933	. . . . . . . 8 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) x. ( 2 ^ H ) ) = ( 1 x. ( 2 ^ H ) ) )
56	42	mullidd 8037	. . . . . . . 8 |- ( ph -> ( 1 x. ( 2 ^ H ) ) = ( 2 ^ H ) )
57	43, 55, 56	3eqtrd 2230	. . . . . . 7 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) = ( 2 ^ H ) )
58	57	oveq1d 5933	. . . . . 6 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) mod P ) = ( ( 2 ^ H ) mod P ) )
59	41	mullidd 8037	. . . . . . 7 |- ( ph -> ( 1 x. ( -u 1 ^ N ) ) = ( -u 1 ^ N ) )
60	59	oveq1d 5933	. . . . . 6 |- ( ph -> ( ( 1 x. ( -u 1 ^ N ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) )
61	58, 60	eqeq12d 2208	. . . . 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 5929	. . . . . . . 8 |- ( 2 ^ H ) = ( 2 ^ ( ( P - 1 ) / 2 ) )
63	62	oveq1i 5928	. . . . . . 7 |- ( ( 2 ^ H ) mod P ) = ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P )
64	63	eqeq1i 2201	. . . . . 6 |- ( ( ( 2 ^ H ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) )
65		2z 9345	. . . . . . . . . 10 |- 2 e. ZZ
66		lgsvalmod 15135	. . . . . . . . . 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 2199	. . . . . . . 8 |- ( ph -> ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( 2 /L P ) mod P ) )
69	68	eqeq1d 2202	. . . . . . 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 15173	. . . . . . 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 2164   \ cdif 3150   ifcif 3557   {csn 3618   class class class wbr 4029   |-> cmpt 4090   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   - cmin 8190   -ucneg 8191   / cdiv 8691   NNcn 8982   2c2 9033   4c4 9035   NN0cn0 9240   ZZcz 9317   QQcq 9684   ...cfz 10074   |_cfl 10337   mod cmo 10393   ^cexp 10609   Primecprime 12245   /Lclgs 15113

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 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-2o 6470  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-ioo 9958  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-proddc 11694  df-dvds 11931  df-gcd 12080  df-prm 12246  df-phi 12349  df-pc 12423  df-lgs 15114

This theorem is referenced by: (None)