Theorem gausslemma2dlem5a 15592

Description: Lemma for gausslemma2dlem5 15593. (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 15590	. . 3 |- ( ph -> A. k e. ( ( M + 1 ) ... H ) ( R ` k ) = ( P - ( k x. 2 ) ) )
6		prodeq2 11918	. . . 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 5969	. . 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 3179	. . . . . . . . 9 |- ( ph -> P e. Prime )
10		prmz 12483	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
11	9, 10	syl 14	. . . . . . . 8 |- ( ph -> P e. ZZ )
12		4nn 9213	. . . . . . . 8 |- 4 e. NN
13		znq 9758	. . . . . . . 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 10433	. . . . . 6 |- ( ph -> ( |_ ` ( P / 4 ) ) e. ZZ )
16	4, 15	eqeltrid 2293	. . . . 5 |- ( ph -> M e. ZZ )
17	16	peano2zd 9511	. . . 4 |- ( ph -> ( M + 1 ) e. ZZ )
18	1, 2	gausslemma2dlem0b 15577	. . . . 5 |- ( ph -> H e. NN )
19	18	nnzd 9507	. . . 4 |- ( ph -> H e. ZZ )
20	17, 19	fzfigd 10589	. . 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 10160	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> k e. ZZ )
23		2z 9413	. . . . . . . 8 |- 2 e. ZZ
24	23	a1i 9	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> 2 e. ZZ )
25	22, 24	zmulcld 9514	. . . . . 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 9513	. . . 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 9417	. . . . . 6 |- -u 1 e. ZZ
30	29	a1i 9	. . . . 5 |- ( k e. ( ( M + 1 ) ... H ) -> -u 1 e. ZZ )
31	30, 25	zmulcld 9514	. . . 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 12482	. . . 4 |- ( P e. Prime -> P e. NN )
34	9, 33	syl 14	. . 3 |- ( ph -> P e. NN )
35	25	zcnd 9509	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. CC )
36	35	mulm1d 8495	. . . . . . 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 5969	. . . . 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 9760	. . . . . . 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 9760	. . . . . . 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 9093	. . . . . . 7 |- ( P e. Prime -> 0 < P )
44	43	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> 0 < P )
45		qnegmod 10527	. . . . . 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 1250	. . . . 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 2240	. . . 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 12002	. 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 2239	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 1373   e. wcel 2177   A.wral 2485   \ cdif 3165   ifcif 3573   {csn 3635   class class class wbr 4048   |-> cmpt 4110   ` cfv 5277  (class class class)co 5954   0cc0 7938   1c1 7939   + caddc 7941   x. cmul 7943   < clt 8120   - cmin 8256   -ucneg 8257   / cdiv 8758   NNcn 9049   2c2 9100   4c4 9102   ZZcz 9385   QQcq 9753   ...cfz 10143   |_cfl 10424   mod cmo 10480   prod_cprod 11911   Primecprime 12479

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-mulrcl 8037  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-0lt1 8044  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-precex 8048  ax-cnre 8049  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-pre-apti 8053  ax-pre-ltadd 8054  ax-pre-mulgt0 8055  ax-pre-mulext 8056  ax-arch 8057  ax-caucvg 8058

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-ilim 4421  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-isom 5286  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-irdg 6466  df-frec 6487  df-1o 6512  df-2o 6513  df-oadd 6516  df-er 6630  df-en 6838  df-dom 6839  df-fin 6840  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-sub 8258  df-neg 8259  df-reap 8661  df-ap 8668  df-div 8759  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-n0 9309  df-z 9386  df-uz 9662  df-q 9754  df-rp 9789  df-fz 10144  df-fzo 10278  df-fl 10426  df-mod 10481  df-seqfrec 10606  df-exp 10697  df-ihash 10934  df-cj 11203  df-re 11204  df-im 11205  df-rsqrt 11359  df-abs 11360  df-clim 11640  df-proddc 11912  df-dvds 12149  df-prm 12480

This theorem is referenced by:  gausslemma2dlem5  15593