Theorem gausslemma2d 15596

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 15595	. 2 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 )
7	1	gausslemma2dlem0a 15576	. . . . . . 7 |- ( ph -> P e. NN )
8		nnq 9767	. . . . . . 7 |- ( P e. NN -> P e. QQ )
9	7, 8	syl 14	. . . . . 6 |- ( ph -> P e. QQ )
10		eldifi 3297	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
11		prmgt1 12504	. . . . . . 7 |- ( P e. Prime -> 1 < P )
12	1, 10, 11	3syl 17	. . . . . 6 |- ( ph -> 1 < P )
13		q1mod 10514	. . . . . 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 2212	. . . 4 |- ( ph -> 1 = ( 1 mod P ) )
16	15	eqeq2d 2218	. . 3 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 <-> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = ( 1 mod P ) ) )
17		neg1z 9417	. . . . . . . . . 10 |- -u 1 e. ZZ
18	1, 4, 2, 5	gausslemma2dlem0h 15583	. . . . . . . . . 10 |- ( ph -> N e. NN0 )
19		zexpcl 10712	. . . . . . . . . 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 9211	. . . . . . . . . . . 12 |- 2 e. NN
22	21	a1i 9	. . . . . . . . . . 11 |- ( ph -> 2 e. NN )
23	1, 2	gausslemma2dlem0b 15577	. . . . . . . . . . . 12 |- ( ph -> H e. NN )
24	23	nnnn0d 9361	. . . . . . . . . . 11 |- ( ph -> H e. NN0 )
25	22, 24	nnexpcld 10853	. . . . . . . . . 10 |- ( ph -> ( 2 ^ H ) e. NN )
26	25	nnzd 9507	. . . . . . . . 9 |- ( ph -> ( 2 ^ H ) e. ZZ )
27	20, 26	zmulcld 9514	. . . . . . . 8 |- ( ph -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. ZZ )
28		zq 9760	. . . . . . . 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 9411	. . . . . . 7 |- 1 e. ZZ
32		zq 9760	. . . . . . 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 9093	. . . . . . 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 10535	. . . . 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 9509	. . . . . . . . 9 |- ( ph -> ( -u 1 ^ N ) e. CC )
42	25	nncnd 9063	. . . . . . . . 9 |- ( ph -> ( 2 ^ H ) e. CC )
43	41, 42, 41	mul32d 8238	. . . . . . . 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 9363	. . . . . . . . . . . . 13 |- ( ph -> N e. CC )
45	44	2timesd 9293	. . . . . . . . . . . 12 |- ( ph -> ( 2 x. N ) = ( N + N ) )
46	45	eqcomd 2212	. . . . . . . . . . 11 |- ( ph -> ( N + N ) = ( 2 x. N ) )
47	46	oveq2d 5970	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( N + N ) ) = ( -u 1 ^ ( 2 x. N ) ) )
48		neg1cn 9154	. . . . . . . . . . . 12 |- -u 1 e. CC
49	48	a1i 9	. . . . . . . . . . 11 |- ( ph -> -u 1 e. CC )
50	49, 18, 18	expaddd 10833	. . . . . . . . . 10 |- ( ph -> ( -u 1 ^ ( N + N ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) )
51	18	nn0zd 9506	. . . . . . . . . . 11 |- ( ph -> N e. ZZ )
52		m1expeven 10744	. . . . . . . . . . 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 2247	. . . . . . . . 9 |- ( ph -> ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) = 1 )
55	54	oveq1d 5969	. . . . . . . 8 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( -u 1 ^ N ) ) x. ( 2 ^ H ) ) = ( 1 x. ( 2 ^ H ) ) )
56	42	mullidd 8103	. . . . . . . 8 |- ( ph -> ( 1 x. ( 2 ^ H ) ) = ( 2 ^ H ) )
57	43, 55, 56	3eqtrd 2243	. . . . . . 7 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) = ( 2 ^ H ) )
58	57	oveq1d 5969	. . . . . 6 |- ( ph -> ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( -u 1 ^ N ) ) mod P ) = ( ( 2 ^ H ) mod P ) )
59	41	mullidd 8103	. . . . . . 7 |- ( ph -> ( 1 x. ( -u 1 ^ N ) ) = ( -u 1 ^ N ) )
60	59	oveq1d 5969	. . . . . 6 |- ( ph -> ( ( 1 x. ( -u 1 ^ N ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) )
61	58, 60	eqeq12d 2221	. . . . 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 5965	. . . . . . . 8 |- ( 2 ^ H ) = ( 2 ^ ( ( P - 1 ) / 2 ) )
63	62	oveq1i 5964	. . . . . . 7 |- ( ( 2 ^ H ) mod P ) = ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P )
64	63	eqeq1i 2214	. . . . . 6 |- ( ( ( 2 ^ H ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( -u 1 ^ N ) mod P ) )
65		2z 9413	. . . . . . . . . 10 |- 2 e. ZZ
66		lgsvalmod 15546	. . . . . . . . . 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 2212	. . . . . . . 8 |- ( ph -> ( ( 2 ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( ( 2 /L P ) mod P ) )
69	68	eqeq1d 2215	. . . . . . 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 15584	. . . . . . 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 1373   e. wcel 2177   \ cdif 3165   ifcif 3573   {csn 3635   class class class wbr 4048   |-> cmpt 4110   ` cfv 5277  (class class class)co 5954   CCcc 7936   0cc0 7938   1c1 7939   + caddc 7941   x. cmul 7943   < clt 8120   - cmin 8256   -ucneg 8257   / cdiv 8758   NNcn 9049   2c2 9100   4c4 9102   NN0cn0 9308   ZZcz 9385   QQcq 9753   ...cfz 10143   |_cfl 10424   mod cmo 10480   ^cexp 10696   Primecprime 12479   /Lclgs 15524

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 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-mulrcl 8037  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-0lt1 8044  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-precex 8048  ax-cnre 8049  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-pre-apti 8053  ax-pre-ltadd 8054  ax-pre-mulgt0 8055  ax-pre-mulext 8056  ax-arch 8057  ax-caucvg 8058

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 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-tp 3643  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-ilim 4421  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-isom 5286  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-irdg 6466  df-frec 6487  df-1o 6512  df-2o 6513  df-oadd 6516  df-er 6630  df-en 6838  df-dom 6839  df-fin 6840  df-sup 7098  df-inf 7099  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-sub 8258  df-neg 8259  df-reap 8661  df-ap 8668  df-div 8759  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-n0 9309  df-z 9386  df-uz 9662  df-q 9754  df-rp 9789  df-ioo 10027  df-fz 10144  df-fzo 10278  df-fl 10426  df-mod 10481  df-seqfrec 10606  df-exp 10697  df-fac 10884  df-ihash 10934  df-cj 11203  df-re 11204  df-im 11205  df-rsqrt 11359  df-abs 11360  df-clim 11640  df-proddc 11912  df-dvds 12149  df-gcd 12325  df-prm 12480  df-phi 12583  df-pc 12658  df-lgs 15525

This theorem is referenced by:  2lgs  15631