Theorem gausslemma2dlem5a 16167

Description: Lemma for gausslemma2dlem5 16168. (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 16165	. . 3 |- ( ph -> A. k e. ( ( M + 1 ) ... H ) ( R ` k ) = ( P - ( k x. 2 ) ) )
6		prodeq2 12307	. . . 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 6094	. . 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 3231	. . . . . . . . 9 |- ( ph -> P e. Prime )
10		prmz 12872	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
11	9, 10	syl 14	. . . . . . . 8 |- ( ph -> P e. ZZ )
12		4nn 9451	. . . . . . . 8 |- 4 e. NN
13		znq 10007	. . . . . . . 8 |- ( ( P e. ZZ /\ 4 e. NN ) -> ( P / 4 ) e. QQ )
14	11, 12, 13	sylancl 417	. . . . . . 7 |- ( ph -> ( P / 4 ) e. QQ )
15	14	flqcld 10695	. . . . . 6 |- ( ph -> ( |_ ` ( P / 4 ) ) e. ZZ )
16	4, 15	eqeltrid 2325	. . . . 5 |- ( ph -> M e. ZZ )
17	16	peano2zd 9754	. . . 4 |- ( ph -> ( M + 1 ) e. ZZ )
18	1, 2	gausslemma2dlem0b 16152	. . . . 5 |- ( ph -> H e. NN )
19	18	nnzd 9750	. . . 4 |- ( ph -> H e. ZZ )
20	17, 19	fzfigd 10851	. . 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 10411	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> k e. ZZ )
23		2z 9655	. . . . . . . 8 |- 2 e. ZZ
24	23	a1i 9	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> 2 e. ZZ )
25	22, 24	zmulcld 9757	. . . . . 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 9756	. . . 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 9659	. . . . . 6 |- -u 1 e. ZZ
30	29	a1i 9	. . . . 5 |- ( k e. ( ( M + 1 ) ... H ) -> -u 1 e. ZZ )
31	30, 25	zmulcld 9757	. . . 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 12871	. . . 4 |- ( P e. Prime -> P e. NN )
34	9, 33	syl 14	. . 3 |- ( ph -> P e. NN )
35	25	zcnd 9752	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. CC )
36	35	mulm1d 8731	. . . . . . 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 6094	. . . . 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 10009	. . . . . . 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 10009	. . . . . . 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 9331	. . . . . . 7 |- ( P e. Prime -> 0 < P )
44	43	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> 0 < P )
45		qnegmod 10789	. . . . . 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 1278	. . . . 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 2272	. . . 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 12391	. 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 2271	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 1402   e. wcel 2209   A.wral 2528   \ cdif 3217   ifcif 3638   {csn 3708   class class class wbr 4128   |-> cmpt 4190   ` cfv 5375  (class class class)co 6079   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   < clt 8354   - cmin 8491   -ucneg 8492   / cdiv 8996   NNcn 9287   2c2 9338   4c4 9340   ZZcz 9627   QQcq 10002   ...cfz 10394   |_cfl 10686   mod cmo 10742   prod_cprod 12300   Primecprime 12868

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-proddc 12301  df-dvds 12538  df-prm 12869

This theorem is referenced by:  gausslemma2dlem5  16168