Theorem gausslemma2dlem5a 16306

Description: Lemma for gausslemma2dlem5 16307. (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 16304	. . 3 |- ( ph -> A. k e. ( ( M + 1 ) ... H ) ( R ` k ) = ( P - ( k x. 2 ) ) )
6		prodeq2 12342	. . . 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 6100	. . 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 12907	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
11	9, 10	syl 14	. . . . . . . 8 |- ( ph -> P e. ZZ )
12		4nn 9473	. . . . . . . 8 |- 4 e. NN
13		znq 10034	. . . . . . . 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 10725	. . . . . 6 |- ( ph -> ( |_ ` ( P / 4 ) ) e. ZZ )
16	4, 15	eqeltrid 2325	. . . . 5 |- ( ph -> M e. ZZ )
17	16	peano2zd 9776	. . . 4 |- ( ph -> ( M + 1 ) e. ZZ )
18	1, 2	gausslemma2dlem0b 16291	. . . . 5 |- ( ph -> H e. NN )
19	18	nnzd 9772	. . . 4 |- ( ph -> H e. ZZ )
20	17, 19	fzfigd 10882	. . 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 10439	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> k e. ZZ )
23		2z 9677	. . . . . . . 8 |- 2 e. ZZ
24	23	a1i 9	. . . . . . 7 |- ( k e. ( ( M + 1 ) ... H ) -> 2 e. ZZ )
25	22, 24	zmulcld 9779	. . . . . 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 9778	. . . 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 9681	. . . . . 6 |- -u 1 e. ZZ
30	29	a1i 9	. . . . 5 |- ( k e. ( ( M + 1 ) ... H ) -> -u 1 e. ZZ )
31	30, 25	zmulcld 9779	. . . 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 12906	. . . 4 |- ( P e. Prime -> P e. NN )
34	9, 33	syl 14	. . 3 |- ( ph -> P e. NN )
35	25	zcnd 9774	. . . . . . . 8 |- ( k e. ( ( M + 1 ) ... H ) -> ( k x. 2 ) e. CC )
36	35	mulm1d 8739	. . . . . . 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 6100	. . . . 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 10036	. . . . . . 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 10036	. . . . . . 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 9351	. . . . . . 7 |- ( P e. Prime -> 0 < P )
44	43	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ k e. ( ( M + 1 ) ... H ) ) -> 0 < P )
45		qnegmod 10820	. . . . . 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 12426	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   \ cdif 3217   ifcif 3638   {csn 3709   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   < clt 8361   - cmin 8499   -ucneg 8500   / cdiv 9005   NNcn 9307   2c2 9358   4c4 9360   ZZcz 9649   QQcq 10029   ...cfz 10422   |_cfl 10714   mod cmo 10773   prod_cprod 12335   Primecprime 12903

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-fz 10423  df-fzo 10561  df-fl 10716  df-mod 10774  df-seqfrec 10899  df-exp 10990  df-ihash 11230  df-cj 11622  df-re 11623  df-im 11624  df-rsqrt 11779  df-abs 11780  df-clim 12063  df-proddc 12336  df-dvds 12573  df-prm 12904

This theorem is used by:  gausslemma2dlem5  16307