Theorem gausslemma2d 16171

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 16170	. 2 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 )
7	1	gausslemma2dlem0a 16151	. . . . . . 7 |- ( ph -> P e. NN )
8		nnq 10016	. . . . . . 7 |- ( P e. NN -> P e. QQ )
9	7, 8	syl 14	. . . . . 6 |- ( ph -> P e. QQ )
10		eldifi 3351	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
11		prmgt1 12893	. . . . . . 7 |- ( P e. Prime -> 1 < P )
12	1, 10, 11	3syl 17	. . . . . 6 |- ( ph -> 1 < P )
13		q1mod 10776	. . . . . 6 |- ( ( P e. QQ /\ 1 < P ) -> ( 1 mod P ) = 1 )
14	9, 12, 13	syl2anc 415	. . . . 5 |- ( ph -> ( 1 mod P ) = 1 )
15	14	eqcomd 2244	. . . 4 |- ( ph -> 1 = ( 1 mod P ) )
16	15	eqeq2d 2250	. . 3 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 <-> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) )
17		neg1z 9659	. . . . . . . . . 10 |- -u 1 e. ZZ
18	1, 4, 2, 5	gausslemma2dlem0h 16158	. . . . . . . . . 10 |- ( ph -> N e. NN0 )
19		zexpcl 10974	. . . . . . . . . 10 |- ( ( -u 1 e. ZZ /\ N e. NN0 ) -> ( -u 1 ^ N ) e. ZZ )
20	17, 18, 19	sylancr 418	. . . . . . . . 9 |- ( ph -> ( -u 1 ^ N ) e. ZZ )
21		2nn 9449	. . . . . . . . . . . 12 |- 2 e. NN
22	21	a1i 9	. . . . . . . . . . 11 |- ( ph -> 2 e. NN )
23	1, 2	gausslemma2dlem0b 16152	. . . . . . . . . . . 12 |- ( ph -> H e. NN )
24	23	nnnn0d 9603	. . . . . . . . . . 11 |- ( ph -> H e. NN0 )
25	22, 24	nnexpcld 11116	. . . . . . . . . 10 |- ( ph -> ( 2 ^ H ) e. NN )
26	25	nnzd 9750	. . . . . . . . 9 |- ( ph -> ( 2 ^ H ) e. ZZ )
27	20, 26	zmulcld 9757	. . . . . . . 8 |- ( ph -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. ZZ )
28		zq 10009	. . . . . . . 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 9653	. . . . . . 7 |- 1 e. ZZ
32		zq 10009	. . . . . . 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 9331	. . . . . . 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 10797	. . . . 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 9752	. . . . . . . . 9 |- ( ph -> ( -u 1 ^ N ) e. CC )
42	25	nncnd 9301	. . . . . . . . 9 |- ( ph -> ( 2 ^ H ) e. CC )
43	41, 42, 41	mul32d 8473	. . . . . . . 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 9605	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
45	44	2timesd 9531	. . . . . . . . . . . 12 |- ( ph -> ( 2 x. N ) = ( N + N ) )
46	45	eqcomd 2244	. . . . . . . . . . 11 |- ( ph -> ( N + N ) = ( 2 x. N ) )
47	46	oveq2d 6095	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( N + N ) ) = ( -u 1 ^ ( 2 x. N ) ) )
48		neg1cn 9392	. . . . . . . . . . . 12 |- -u 1 e. CC
49	48	a1i 9	. . . . . . . . . . 11 |- ( ph -> -u 1 e. CC )
50	49, 18, 18	expaddd 11096	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
51	18	nn0zd 9749	. . . . . . . . . . 11 |- ( ph -> N e. ZZ )
52		m1expeven 11006	. . . . . . . . . . 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 2279	. . . . . . . . 9 |- ( ph -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
55	54	oveq1d 6094	. . . . . . . 8 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) x. ( 2 ^ H ) ) = ( 1 x. ( 2 ^ H ) ) )
56	42	mullidd 8338	. . . . . . . 8 |- ( ph -> ( 1 x. ( 2 ^ H ) ) = ( 2 ^ H ) )
57	43, 55, 56	3eqtrd 2275	. . . . . . 7 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) = ( 2 ^ H ) )
58	57	oveq1d 6094	. . . . . 6 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) mod P ) = ( ( 2 ^ H ) mod P ) )
59	41	mullidd 8338	. . . . . . 7 |- ( ph -> ( 1 x. ( -u 1 ^ N ) ) = ( -u 1 ^ N ) )
60	59	oveq1d 6094	. . . . . 6 |- ( ph -> ( ( 1 x. ( -u 1 ^ N ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) )
61	58, 60	eqeq12d 2253	. . . . 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 6090	. . . . . . . 8 |- ( 2 ^ H ) = ( 2 ^ ( ( P - 1 ) / 2 ) )
63	62	oveq1i 6089	. . . . . . 7 |- ( ( 2 ^ H ) mod P ) = ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P )
64	63	eqeq1i 2246	. . . . . 6 |- ( ( ( 2 ^ H ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) )
65		2z 9655	. . . . . . . . . 10 |- 2 e. ZZ
66		lgsvalmod 16121	. . . . . . . . . 10 |- ( ( 2 e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( 2 /L P ) mod P ) = ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) )
67	65, 1, 66	sylancr 418	. . . . . . . . 9 |- ( ph -> ( ( 2 /L P ) mod P ) = ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) )
68	67	eqcomd 2244	. . . . . . . 8 |- ( ph -> ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( 2 /L P ) mod P ) )
69	68	eqeq1d 2247	. . . . . . 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 16159	. . . . . . 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 1402   e. wcel 2209   \ cdif 3217   ifcif 3638   {csn 3708   class class class wbr 4128   |-> cmpt 4190   ` cfv 5375  (class class class)co 6079   CCcc 8171   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   < clt 8354   - cmin 8491   -ucneg 8492   / cdiv 8996   NNcn 9287   2c2 9338   4c4 9340   NN0cn0 9546   ZZcz 9627   QQcq 10002   ...cfz 10394   |_cfl 10686   mod cmo 10742   ^cexp 10958   Primecprime 12868   /Lclgs 16099

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-tp 3716  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-ioo 10277  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-fac 11147  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-proddc 12301  df-dvds 12538  df-gcd 12714  df-prm 12869  df-phi 12972  df-pc 13047  df-lgs 16100

This theorem is referenced by:  2lgs  16206