Theorem gausslemma2dlem5a 15123

Description: Lemma for gausslemma2dlem5 15124. (Contributed by AV, 8-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 ) )

Assertion

Ref	Expression
gausslemma2dlem5a	|- ( 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 ) )

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

Allowed substitution hint:   R( x)

Proof of Theorem gausslemma2dlem5a

Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . . 4 |- ( ph -> P e. ( Prime \ { 2 } ) )
2		gausslemma2d.h	. . . 4 |- H = ( ( P - 1 ) / 2 )
3		gausslemma2d.r	. . . 4 |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )
4		gausslemma2d.m	. . . 4 |- M = ( |_ ` ( P / 4 ) )
5	1, 2, 3, 4	gausslemma2dlem3 15121	. . 3 |- ( ph -> A. k e. ( ( M + 1 ) ... H ) ( R ` k ) = ( P - ( k x. 2 ) ) )
6		prodeq2 11687	. . . 4 |- ( A. k e. ( ( M + 1 ) ... H ) ( R ` k ) = ( P - ( k x. 2 ) ) -> prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) = prod_ k e. ( ( M + 1 ) ... H ) ( P - ( k x. 2 ) ) )
7	6	oveq1d 5925	. . 3 |- ( A. k e. ( ( M + 1 ) ... H ) ( R ` k ) = ( P - ( k x. 2 ) ) -> ( prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) mod P ) = ( prod_ k e. ( ( M + 1 ) ... H ) ( P - ( k x. 2 ) ) mod P ) )
8	5, 7	syl 14	. 2 |- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) ( R ` k ) mod P ) = ( prod_ k e. ( ( M + 1 ) ... H ) ( P - ( k x. 2 ) ) mod P ) )
9	1	eldifad 3164	. . . . . . . . 9 |- ( ph -> P e. Prime )
10		prmz 12236	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
11	9, 10	syl 14	. . . . . . . 8 |- ( ph -> P e. ZZ )
12		4nn 9135	. . . . . . . 8 |- 4 e. NN
13		znq 9679	. . . . . . . 8 |- ( ( P e. ZZ /\ 4 e. NN ) -> ( P / 4 ) e. QQ )
14	11, 12, 13	sylancl 413	. . . . . . 7 |- ( ph -> ( P / 4 ) e. QQ )
15	14	flqcld 10336	. . . . . 6 |- ( ph -> ( |_ ` ( P / 4 ) ) e. ZZ )
16	4, 15	eqeltrid 2280	. . . . 5 |- ( ph -> M e. ZZ )
17	16	peano2zd 9432	. . . 4 |- ( ph -> ( M + 1 ) e. ZZ )
18	1, 2	gausslemma2dlem0b 15108	. . . . 5 |- ( ph -> H e. NN )
19	18	nnzd 9428	. . . 4 |- ( ph -> H e. ZZ )
20	17, 19	fzfigd 10492	. . 3 |- ( ph -> ( ( M + 1 ) ... H ) e. Fin )
21	10	adantr 276	. . . . 5 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> P e. ZZ )
22		elfzelz 10081	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> k e. ZZ )
23		2z 9335	. . . . . . . 8 |- 2 e. ZZ
24	23	a1i 9	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> 2 e. ZZ )
25	22, 24	zmulcld 9435	. . . . . 6 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. ZZ )
26	25	adantl 277	. . . . 5 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> ( k x. 2 ) e. ZZ )
27	21, 26	zsubcld 9434	. . . 4 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> ( P - ( k x. 2 ) ) e. ZZ )
28	9, 27	sylan 283	. . 3 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> ( P - ( k x. 2 ) ) e. ZZ )
29		neg1z 9339	. . . . . 6 |- -u 1 e. ZZ
30	29	a1i 9	. . . . 5 |- ( k e. ( ( M + 1 ) ... H ) -> -u 1 e. ZZ )
31	30, 25	zmulcld 9435	. . . 4 |- ( k e. ( ( M + 1 ) ... H ) -> ( -u 1 x. ( k x. 2 ) ) e. ZZ )
32	31	adantl 277	. . 3 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> ( -u 1 x. ( k x. 2 ) ) e. ZZ )
33		prmnn 12235	. . . 4 |- ( P e. Prime -> P e. NN )
34	9, 33	syl 14	. . 3 |- ( ph -> P e. NN )
35	25	zcnd 9430	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. CC )
36	35	mulm1d 8419	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> ( -u 1 x. ( k x. 2 ) ) = -u ( k x. 2 ) )
37	36	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> ( -u 1 x. ( k x. 2 ) ) = -u ( k x. 2 ) )
38	37	oveq1d 5925	. . . . 5 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> ( ( -u 1 x. ( k x. 2 ) ) mod P ) = ( -u ( k x. 2 ) mod P ) )
39		zq 9681	. . . . . . 7 |- ( ( k x. 2 ) e. ZZ -> ( k x. 2 ) e. QQ )
40	26, 39	syl 14	. . . . . 6 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> ( k x. 2 ) e. QQ )
41		zq 9681	. . . . . . 7 |- ( P e. ZZ -> P e. QQ )
42	21, 41	syl 14	. . . . . 6 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> P e. QQ )
43	33	nngt0d 9016	. . . . . . 7 |- ( P e. Prime -> 0 < P )
44	43	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> 0 < P )
45		qnegmod 10430	. . . . . 6 |- ( ( ( k x. 2 ) e. QQ /\ P e. QQ /\ 0 < P ) -> ( -u ( k x. 2 ) mod P ) = ( ( P - ( k x. 2 ) ) mod P ) )
46	40, 42, 44, 45	syl3anc 1249	. . . . 5 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> ( -u ( k x. 2 ) mod P ) = ( ( P - ( k x. 2 ) ) mod P ) )
47	38, 46	eqtr2d 2227	. . . 4 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> ( ( P - ( k x. 2 ) ) mod P ) = ( ( -u 1 x. ( k x. 2 ) ) mod P ) )
48	9, 47	sylan 283	. . 3 |- ( ( ph /\ k e. ( ( M + 1 ) ... H ) ) -> ( ( P - ( k x. 2 ) ) mod P ) = ( ( -u 1 x. ( k x. 2 ) ) mod P ) )
49	20, 28, 32, 34, 48	fprodmodd 11771	. 2 |- ( ph -> ( prod_ k e. ( ( M + 1 ) ... H ) ( P - ( k x. 2 ) ) mod P ) = ( prod_ k e. ( ( M + 1 ) ... H ) ( -u 1 x. ( k x. 2 ) ) mod P ) )
50	8, 49	eqtrd 2226	1 |- ( 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   \ cdif 3150   ifcif 3557   {csn 3618   class class class wbr 4029   |-> cmpt 4090   ` cfv 5246  (class class class)co 5910   0cc0 7862   1c1 7863   + caddc 7865   x. cmul 7867   < clt 8044   - cmin 8180   -ucneg 8181   / cdiv 8681   NNcn 8972   2c2 9023   4c4 9025   ZZcz 9307   QQcq 9674   ...cfz 10064   |_cfl 10327   mod cmo 10383   prod_cprod 11680   Primecprime 12232

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-iinf 4616  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-mulrcl 7961  ax-addcom 7962  ax-mulcom 7963  ax-addass 7964  ax-mulass 7965  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-1rid 7969  ax-0id 7970  ax-rnegex 7971  ax-precex 7972  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-apti 7977  ax-pre-ltadd 7978  ax-pre-mulgt0 7979  ax-pre-mulext 7980  ax-arch 7981  ax-caucvg 7982

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4322  df-po 4325  df-iso 4326  df-iord 4395  df-on 4397  df-ilim 4398  df-suc 4400  df-iom 4619  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-isom 5255  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-recs 6349  df-irdg 6414  df-frec 6435  df-1o 6460  df-2o 6461  df-oadd 6464  df-er 6578  df-en 6786  df-dom 6787  df-fin 6788  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-reap 8584  df-ap 8591  df-div 8682  df-inn 8973  df-2 9031  df-3 9032  df-4 9033  df-n0 9231  df-z 9308  df-uz 9583  df-q 9675  df-rp 9710  df-fz 10065  df-fzo 10199  df-fl 10329  df-mod 10384  df-seqfrec 10509  df-exp 10597  df-ihash 10834  df-cj 10973  df-re 10974  df-im 10975  df-rsqrt 11129  df-abs 11130  df-clim 11409  df-proddc 11681  df-dvds 11918  df-prm 12233

This theorem is referenced by:  gausslemma2dlem5  15124