Theorem gausslemma2dlem5 15317

Description: Lemma 5 for gausslemma2d 15320. (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 15316	. 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 15300	. . . . . . . . . . 11 |- ( ph -> P e. NN )
7	6	nnzd 9449	. . . . . . . . . 10 |- ( ph -> P e. ZZ )
8		4nn 9156	. . . . . . . . . 10 |- 4 e. NN
9		znq 9700	. . . . . . . . . 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 10369	. . . . . . . 8 |- ( ph -> ( |_ ` ( P / 4 ) ) e. ZZ )
12	4, 11	eqeltrid 2283	. . . . . . 7 |- ( ph -> M e. ZZ )
13	12	peano2zd 9453	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
14	1, 2	gausslemma2dlem0b 15301	. . . . . . 7 |- ( ph -> H e. NN )
15	14	nnzd 9449	. . . . . 6 |- ( ph -> H e. ZZ )
16	13, 15	fzfigd 10525	. . . . 5 |- ( ph -> ( ( M + 1 ) ... H ) e. Fin )
17		neg1cn 9097	. . . . . 6 |- -u 1 e. CC
18	17	a1i 9	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> -u 1 e. CC )
19		elfzelz 10102	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> k e. ZZ )
20		2z 9356	. . . . . . . . 9 |- 2 e. ZZ
21	20	a1i 9	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> 2 e. ZZ )
22	19, 21	zmulcld 9456	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. ZZ )
23	22	zcnd 9451	. . . . . 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 11758	. . . 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 11787	. . . . . . 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 12439	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
29		nnz 9347	. . . . . . . . . . . . . 14 |- ( P e. NN -> P e. ZZ )
30		oddm1d2 12059	. . . . . . . . . . . . . 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 2283	. . . . . . . . . 10 |- ( ph -> H e. ZZ )
35	1, 4, 2	gausslemma2dlem0f 15305	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) <_ H )
36		eluz2 9609	. . . . . . . . . 10 |- ( H e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ H e. ZZ /\ ( M + 1 ) <_ H ) )
37	13, 34, 35, 36	syl3anbrc 1183	. . . . . . . . 9 |- ( ph -> H e. ( ZZ>= ` ( M + 1 ) ) )
38		hashfz 10915	. . . . . . . . 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 9451	. . . . . . . . . 10 |- ( ph -> H e. CC )
41	12	zcnd 9451	. . . . . . . . . 10 |- ( ph -> M e. CC )
42		1cnd 8044	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
43	40, 41, 42	nppcan2d 8365	. . . . . . . . 9 |- ( ph -> ( ( H - ( M + 1 ) ) + 1 ) = ( H - M ) )
44		gausslemma2d.n	. . . . . . . . 9 |- N = ( H - M )
45	43, 44	eqtr4di 2247	. . . . . . . 8 |- ( ph -> ( ( H - ( M + 1 ) ) + 1 ) = N )
46	39, 45	eqtrd 2229	. . . . . . 7 |- ( ph -> (♯ ` ( ( M + 1 ) ... H ) ) = N )
47	46	oveq2d 5939	. . . . . 6 |- ( ph -> ( -u 1 ^ (♯ ` ( ( M + 1 ) ... H ) ) ) = ( -u 1 ^ N ) )
48	27, 47	eqtrd 2229	. . . . 5 |- ( ph -> prod_ k e. ( ( M + 1 ) ... H ) -u 1 = ( -u 1 ^ N ) )
49	48	oveq1d 5938	. . . 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 2229	. . 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 5938	. 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 2229	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 1364   e. wcel 2167   \ cdif 3154   ifcif 3562   {csn 3623   class class class wbr 4034   |-> cmpt 4095   ` cfv 5259  (class class class)co 5923   Fincfn 6800   CCcc 7879   1c1 7882   + caddc 7884   x. cmul 7886   < clt 8063   <_ cle 8064   - cmin 8199   -ucneg 8200   / cdiv 8701   NNcn 8992   2c2 9043   4c4 9045   ZZcz 9328   ZZ>=cuz 9603   QQcq 9695   ...cfz 10085   |_cfl 10360   mod cmo 10416   ^cexp 10632  ♯chash 10869   prod_cprod 11717   || cdvds 11954   Primecprime 12285

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-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-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-n0 9252  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-fz 10086  df-fzo 10220  df-fl 10362  df-mod 10417  df-seqfrec 10542  df-exp 10633  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-prm 12286

This theorem is referenced by:  gausslemma2dlem6  15318