Theorem gausslemma2dlem5 25096

Description: Lemma 5 for gausslemma2d 25099. (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 25095	. 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		fzfi 12771	. . . . . 6 |- ( ( M + 1 ) ... H ) e. Fin
7	6	a1i 11	. . . . 5 |- ( ph -> ( ( M + 1 ) ... H ) e. Fin )
8		neg1cn 11124	. . . . . 6 |- -u 1 e. CC
9	8	a1i 11	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> -u 1 e. CC )
10		elfzelz 12342	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> k e. ZZ )
11		2z 11409	. . . . . . . . 9 |- 2 e. ZZ
12	11	a1i 11	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> 2 e. ZZ )
13	10, 12	zmulcld 11488	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. ZZ )
14	13	zcnd 11483	. . . . . 6 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. CC )
15	14	adantl 482	. . . . 5 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> ( k x. 2 ) e. CC )
16	7, 9, 15	fprodmul 14690	. . . 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 ) ) )
17	6, 8	pm3.2i 471	. . . . . . 7 |- ( ( ( M + 1 ) ... H ) e. Fin /\ -u 1 e. CC )
18		fprodconst 14708	. . . . . . 7 |- ( ( ( ( M + 1 ) ... H ) e. Fin /\ -u 1 e. CC ) -> prod_ k e. ( ( M + 1 ) ... H ) -u 1 = ( -u 1 ^ ( # ` ( ( M + 1 ) ... H ) ) ) )
19	17, 18	mp1i 13	. . . . . 6 |- ( ph -> prod_ k e. ( ( M + 1 ) ... H ) -u 1 = ( -u 1 ^ ( # ` ( ( M + 1 ) ... H ) ) ) )
20		nnoddn2prm 15516	. . . . . . . . . . . . . . 15 |- ( P e. ( Prime \ { 2 } ) -> ( P e. NN /\ -. 2 || P ) )
21		nnre 11027	. . . . . . . . . . . . . . . 16 |- ( P e. NN -> P e. RR )
22	21	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( P e. NN /\ -. 2 || P ) -> P e. RR )
23	1, 20, 22	3syl 18	. . . . . . . . . . . . . 14 |- ( ph -> P e. RR )
24		4re 11097	. . . . . . . . . . . . . . 15 |- 4 e. RR
25	24	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> 4 e. RR )
26		4ne0 11117	. . . . . . . . . . . . . . 15 |- 4 =/= 0
27	26	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> 4 =/= 0 )
28	23, 25, 27	redivcld 10853	. . . . . . . . . . . . 13 |- ( ph -> ( P / 4 ) e. RR )
29	28	flcld 12599	. . . . . . . . . . . 12 |- ( ph -> ( |_ ` ( P / 4 ) ) e. ZZ )
30	4, 29	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> M e. ZZ )
31	30	peano2zd 11485	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) e. ZZ )
32		nnz 11399	. . . . . . . . . . . . . 14 |- ( P e. NN -> P e. ZZ )
33		oddm1d2 15084	. . . . . . . . . . . . . 14 |- ( P e. ZZ -> ( -. 2 || P <-> ( ( P - 1 ) / 2 ) e. ZZ ) )
34	32, 33	syl 17	. . . . . . . . . . . . 13 |- ( P e. NN -> ( -. 2 || P <-> ( ( P - 1 ) / 2 ) e. ZZ ) )
35	34	biimpa 501	. . . . . . . . . . . 12 |- ( ( P e. NN /\ -. 2 || P ) -> ( ( P - 1 ) / 2 ) e. ZZ )
36	1, 20, 35	3syl 18	. . . . . . . . . . 11 |- ( ph -> ( ( P - 1 ) / 2 ) e. ZZ )
37	2, 36	syl5eqel 2705	. . . . . . . . . 10 |- ( ph -> H e. ZZ )
38	1, 4, 2	gausslemma2dlem0f 25086	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) <_ H )
39		eluz2 11693	. . . . . . . . . 10 |- ( H e. ( ZZ>= ` ( M + 1 ) ) <-> ( ( M + 1 ) e. ZZ /\ H e. ZZ /\ ( M + 1 ) <_ H ) )
40	31, 37, 38, 39	syl3anbrc 1246	. . . . . . . . 9 |- ( ph -> H e. ( ZZ>= ` ( M + 1 ) ) )
41		hashfz 13214	. . . . . . . . 9 |- ( H e. ( ZZ>= ` ( M + 1 ) ) -> ( # ` ( ( M + 1 ) ... H ) ) = ( ( H - ( M + 1 ) ) + 1 ) )
42	40, 41	syl 17	. . . . . . . 8 |- ( ph -> ( # ` ( ( M + 1 ) ... H ) ) = ( ( H - ( M + 1 ) ) + 1 ) )
43	37	zcnd 11483	. . . . . . . . . 10 |- ( ph -> H e. CC )
44	30	zcnd 11483	. . . . . . . . . 10 |- ( ph -> M e. CC )
45		1cnd 10056	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
46	43, 44, 45	nppcan2d 10418	. . . . . . . . 9 |- ( ph -> ( ( H - ( M + 1 ) ) + 1 ) = ( H - M ) )
47		gausslemma2d.n	. . . . . . . . 9 |- N = ( H - M )
48	46, 47	syl6eqr 2674	. . . . . . . 8 |- ( ph -> ( ( H - ( M + 1 ) ) + 1 ) = N )
49	42, 48	eqtrd 2656	. . . . . . 7 |- ( ph -> ( # ` ( ( M + 1 ) ... H ) ) = N )
50	49	oveq2d 6666	. . . . . 6 |- ( ph -> ( -u 1 ^ ( # ` ( ( M + 1 ) ... H ) ) ) = ( -u 1 ^ N ) )
51	19, 50	eqtrd 2656	. . . . 5 |- ( ph -> prod_ k e. ( ( M + 1 ) ... H ) -u 1 = ( -u 1 ^ N ) )
52	51	oveq1d 6665	. . . 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 ) ) )
53	16, 52	eqtrd 2656	. . 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 ) ) )
54	53	oveq1d 6665	. 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 ) )
55	5, 54	eqtrd 2656	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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   4c4 11072   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   |_cfl 12591   mod cmo 12668   ^cexp 12860   #chash 13117   prod_cprod 14635   || cdvds 14983   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-dvds 14984  df-prm 15386

This theorem is referenced by:  gausslemma2dlem6  25097