Theorem gausslemma2dlem5 15931

Description: Lemma 5 for gausslemma2d 15934. (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 15930	. 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 15914	. . . . . . . . . . 11 |- ( ph -> P e. NN )
7	6	nnzd 9698	. . . . . . . . . 10 |- ( ph -> P e. ZZ )
8		4nn 9400	. . . . . . . . . 10 |- 4 e. NN
9		znq 9955	. . . . . . . . . 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 10636	. . . . . . . 8 |- ( ph -> ( |_ ` ( P / 4 ) ) e. ZZ )
12	4, 11	eqeltrid 2319	. . . . . . 7 |- ( ph -> M e. ZZ )
13	12	peano2zd 9702	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
14	1, 2	gausslemma2dlem0b 15915	. . . . . . 7 |- ( ph -> H e. NN )
15	14	nnzd 9698	. . . . . 6 |- ( ph -> H e. ZZ )
16	13, 15	fzfigd 10792	. . . . 5 |- ( ph -> ( ( M + 1 ) ... H ) e. Fin )
17		neg1cn 9341	. . . . . 6 |- -u 1 e. CC
18	17	a1i 9	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> -u 1 e. CC )
19		elfzelz 10358	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> k e. ZZ )
20		2z 9604	. . . . . . . . 9 |- 2 e. ZZ
21	20	a1i 9	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> 2 e. ZZ )
22	19, 21	zmulcld 9705	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. ZZ )
23	22	zcnd 9700	. . . . . 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 12273	. . . 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 12302	. . . . . . 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 12954	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
29		nnz 9595	. . . . . . . . . . . . . 14 |- ( P e. NN -> P e. ZZ )
30		oddm1d2 12574	. . . . . . . . . . . . . 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 2319	. . . . . . . . . 10 |- ( ph -> H e. ZZ )
35	1, 4, 2	gausslemma2dlem0f 15919	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) <_ H )
36		eluz2 9858	. . . . . . . . . 10 |- ( H e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ H e. ZZ /\ ( M + 1 ) <_ H ) )
37	13, 34, 35, 36	syl3anbrc 1208	. . . . . . . . 9 |- ( ph -> H e. ( ZZ>= ` ( M + 1 ) ) )
38		hashfz 11184	. . . . . . . . 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 9700	. . . . . . . . . 10 |- ( ph -> H e. CC )
41	12	zcnd 9700	. . . . . . . . . 10 |- ( ph -> M e. CC )
42		1cnd 8289	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
43	40, 41, 42	nppcan2d 8609	. . . . . . . . 9 |- ( ph -> ( ( H - ( M + 1 ) ) + 1 ) = ( H - M ) )
44		gausslemma2d.n	. . . . . . . . 9 |- N = ( H - M )
45	43, 44	eqtr4di 2283	. . . . . . . 8 |- ( ph -> ( ( H - ( M + 1 ) ) + 1 ) = N )
46	39, 45	eqtrd 2265	. . . . . . 7 |- ( ph -> (♯ ` ( ( M + 1 ) ... H ) ) = N )
47	46	oveq2d 6065	. . . . . 6 |- ( ph -> ( -u 1 ^ (♯ ` ( ( M + 1 ) ... H ) ) ) = ( -u 1 ^ N ) )
48	27, 47	eqtrd 2265	. . . . 5 |- ( ph -> prod_ k e. ( ( M + 1 ) ... H ) -u 1 = ( -u 1 ^ N ) )
49	48	oveq1d 6064	. . . 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 2265	. . 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 6064	. 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 2265	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 1398   e. wcel 2203   \ cdif 3207   ifcif 3619   {csn 3688   class class class wbr 4108   |-> cmpt 4170   ` cfv 5351  (class class class)co 6049   Fincfn 6974   CCcc 8124   1c1 8127   + caddc 8129   x. cmul 8131   < clt 8307   <_ cle 8308   - cmin 8443   -ucneg 8444   / cdiv 8945   NNcn 9236   2c2 9287   4c4 9289   ZZcz 9576   ZZ>=cuz 9852   QQcq 9950   ...cfz 10341   |_cfl 10627   mod cmo 10683   ^cexp 10899  ♯chash 11136   prod_cprod 12232   || cdvds 12469   Primecprime 12800

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-ihash 11137  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-clim 11960  df-proddc 12233  df-dvds 12470  df-prm 12801

This theorem is referenced by:  gausslemma2dlem6  15932