Theorem gausslemma2dlem5 15730

Description: Lemma 5 for gausslemma2d 15733. (Contributed by AV, 9-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
gausslemma2dlem5	|- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) mod P ) = ( ( ( -u 1 ^ N ) x. prod_ k e. ( ( M + 1 ) ... H ) ( k x. 2 ) ) mod P ) )

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

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

Proof of Theorem gausslemma2dlem5

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	1, 2, 3, 4	gausslemma2dlem5a 15729	. 2 |- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) mod P ) = ( prod_ k e. ( ( M + 1 ) ... H ) ( -u 1 x. ( k x. 2 ) ) mod P ) )
6	1	gausslemma2dlem0a 15713	. . . . . . . . . . 11 |- ( ph -> P e. NN )
7	6	nnzd 9556	. . . . . . . . . 10 |- ( ph -> P e. ZZ )
8		4nn 9262	. . . . . . . . . 10 |- 4 e. NN
9		znq 9807	. . . . . . . . . 10 |- ( ( P e. ZZ /\ 4 e. NN ) -> ( P / 4 ) e. QQ )
10	7, 8, 9	sylancl 413	. . . . . . . . 9 |- ( ph -> ( P / 4 ) e. QQ )
11	10	flqcld 10484	. . . . . . . 8 |- ( ph -> ( |_ ` ( P / 4 ) ) e. ZZ )
12	4, 11	eqeltrid 2316	. . . . . . 7 |- ( ph -> M e. ZZ )
13	12	peano2zd 9560	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
14	1, 2	gausslemma2dlem0b 15714	. . . . . . 7 |- ( ph -> H e. NN )
15	14	nnzd 9556	. . . . . 6 |- ( ph -> H e. ZZ )
16	13, 15	fzfigd 10640	. . . . 5 |- ( ph -> ( ( M + 1 ) ... H ) e. Fin )
17		neg1cn 9203	. . . . . 6 |- -u 1 e. CC
18	17	a1i 9	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> -u 1 e. CC )
19		elfzelz 10209	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> k e. ZZ )
20		2z 9462	. . . . . . . . 9 |- 2 e. ZZ
21	20	a1i 9	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> 2 e. ZZ )
22	19, 21	zmulcld 9563	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. ZZ )
23	22	zcnd 9558	. . . . . 6 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. CC )
24	23	adantl 277	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> ( k x. 2 ) e. CC )
25	16, 18, 24	fprodmul 12088	. . . 4 |- ( ph -> prod_ k e. ( ( M + 1 ) ... H ) ( -u 1 x. ( k x. 2 ) ) = ( prod_ k e. ( ( M + 1 ) ... H ) -u 1 x. prod_ k e. ( ( M + 1 ) ... H ) ( k x. 2 ) ) )
26		fprodconst 12117	. . . . . . 7 |- ( ( ( ( M + 1 ) ... H ) e. Fin /\ -u 1 e. CC ) -> prod_ k e. ( ( M + 1 ) ... H ) -u 1 = ( -u 1 ^ (♯ ` ( ( M + 1 ) ... H ) ) ) )
27	16, 17, 26	sylancl 413	. . . . . 6 |- ( ph -> prod_ k e. ( ( M + 1 ) ... H ) -u 1 = ( -u 1 ^ (♯ ` ( ( M + 1 ) ... H ) ) ) )
28		nnoddn2prm 12769	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
29		nnz 9453	. . . . . . . . . . . . . 14 |- ( P e. NN -> P e. ZZ )
30		oddm1d2 12389	. . . . . . . . . . . . . 14 |- ( P e. ZZ -> ( -. 2 || P <-> ( ( P - 1 ) / 2 ) e. ZZ ) )
31	29, 30	syl 14	. . . . . . . . . . . . 13 |- ( P e. NN -> ( -. 2 || P <-> ( ( P - 1 ) / 2 ) e. ZZ ) )
32	31	biimpa 296	. . . . . . . . . . . 12 |- ( ( P e. NN /\ -. 2 || P ) -> ( ( P - 1 ) / 2 ) e. ZZ )
33	1, 28, 32	3syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( P - 1 ) / 2 ) e. ZZ )
34	2, 33	eqeltrid 2316	. . . . . . . . . 10 |- ( ph -> H e. ZZ )
35	1, 4, 2	gausslemma2dlem0f 15718	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) <_ H )
36		eluz2 9716	. . . . . . . . . 10 |- ( H e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ H e. ZZ /\ ( M + 1 ) <_ H ) )
37	13, 34, 35, 36	syl3anbrc 1205	. . . . . . . . 9 |- ( ph -> H e. ( ZZ>= ` ( M + 1 ) ) )
38		hashfz 11030	. . . . . . . . 9 |- ( H e. ( ZZ>= ` ( M + 1 ) ) -> (♯ ` ( ( M + 1 ) ... H ) ) = ( ( H - ( M + 1 ) ) + 1 ) )
39	37, 38	syl 14	. . . . . . . 8 |- ( ph -> (♯ ` ( ( M + 1 ) ... H ) ) = ( ( H - ( M + 1 ) ) + 1 ) )
40	34	zcnd 9558	. . . . . . . . . 10 |- ( ph -> H e. CC )
41	12	zcnd 9558	. . . . . . . . . 10 |- ( ph -> M e. CC )
42		1cnd 8150	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
43	40, 41, 42	nppcan2d 8471	. . . . . . . . 9 |- ( ph -> ( ( H - ( M + 1 ) ) + 1 ) = ( H - M ) )
44		gausslemma2d.n	. . . . . . . . 9 |- N = ( H - M )
45	43, 44	eqtr4di 2280	. . . . . . . 8 |- ( ph -> ( ( H - ( M + 1 ) ) + 1 ) = N )
46	39, 45	eqtrd 2262	. . . . . . 7 |- ( ph -> (♯ ` ( ( M + 1 ) ... H ) ) = N )
47	46	oveq2d 6010	. . . . . 6 |- ( ph -> ( -u 1 ^ (♯ ` ( ( M + 1 ) ... H ) ) ) = ( -u 1 ^ N ) )
48	27, 47	eqtrd 2262	. . . . 5 |- ( ph -> prod_ k e. ( ( M + 1 ) ... H ) -u 1 = ( -u 1 ^ N ) )
49	48	oveq1d 6009	. . . 4 |- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) -u 1 x. prod_ k e. ( ( M + 1 ) ... H ) ( k x. 2 ) ) = ( ( -u 1 ^ N ) x. prod_ k e. ( ( M + 1 ) ... H ) ( k x. 2 ) ) )
50	25, 49	eqtrd 2262	. . 3 |- ( ph -> prod_ k e. ( ( M + 1 ) ... H ) ( -u 1 x. ( k x. 2 ) ) = ( ( -u 1 ^ N ) x. prod_ k e. ( ( M + 1 ) ... H ) ( k x. 2 ) ) )
51	50	oveq1d 6009	. 2 |- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) ( -u 1 x. ( k x. 2 ) ) mod P ) = ( ( ( -u 1 ^ N ) x. prod_ k e. ( ( M + 1 ) ... H ) ( k x. 2 ) ) mod P ) )
52	5, 51	eqtrd 2262	1 |- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) mod P ) = ( ( ( -u 1 ^ N ) x. prod_ k e. ( ( M + 1 ) ... H ) ( k x. 2 ) ) mod P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   \ cdif 3194   ifcif 3602   {csn 3666   class class class wbr 4082   |-> cmpt 4144   ` cfv 5314  (class class class)co 5994   Fincfn 6877   CCcc 7985   1c1 7988   + caddc 7990   x. cmul 7992   < clt 8169   <_ cle 8170   - cmin 8305   -ucneg 8306   / cdiv 8807   NNcn 9098   2c2 9149   4c4 9151   ZZcz 9434   ZZ>=cuz 9710   QQcq 9802   ...cfz 10192   |_cfl 10475   mod cmo 10531   ^cexp 10747  ♯chash 10984   prod_cprod 12047   || cdvds 12284   Primecprime 12615

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-frec 6527  df-1o 6552  df-2o 6553  df-oadd 6556  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-fl 10477  df-mod 10532  df-seqfrec 10657  df-exp 10748  df-ihash 10985  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-clim 11776  df-proddc 12048  df-dvds 12285  df-prm 12616

This theorem is referenced by:  gausslemma2dlem6  15731