Theorem gausslemma2dlem7 16053

Description: Lemma 7 for gausslemma2d 16054. (Contributed by AV, 13-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
gausslemma2dlem7	|- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 )

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

Allowed substitution hints:   R( x)   N( x)

Proof of Theorem gausslemma2dlem7

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		gausslemma2d.n	. . 3 |- N = ( H - M )
6	1, 2, 3, 4, 5	gausslemma2dlem6 16052	. 2 |- ( ph -> ( ( ! ` H ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) )
7	1, 2	gausslemma2dlem0b 16035	. . . . . . . . . . 11 |- ( ph -> H e. NN )
8	7	nnnn0d 9570	. . . . . . . . . 10 |- ( ph -> H e. NN0 )
9	8	faccld 11123	. . . . . . . . 9 |- ( ph -> ( ! ` H ) e. NN )
10	9	nncnd 9268	. . . . . . . 8 |- ( ph -> ( ! ` H ) e. CC )
11	10	mullidd 8308	. . . . . . 7 |- ( ph -> ( 1 x. ( ! ` H ) ) = ( ! ` H ) )
12	11	eqcomd 2240	. . . . . 6 |- ( ph -> ( ! ` H ) = ( 1 x. ( ! ` H ) ) )
13	12	oveq1d 6073	. . . . 5 |- ( ph -> ( ( ! ` H ) mod P ) = ( ( 1 x. ( ! ` H ) ) mod P ) )
14	13	eqeq1d 2243	. . . 4 |- ( ph -> ( ( ( ! ` H ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) <-> ( ( 1 x. ( ! ` H ) ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) ) )
15		1zzd 9621	. . . . 5 |- ( ph -> 1 e. ZZ )
16		neg1z 9626	. . . . . . 7 |- -u 1 e. ZZ
17	1, 4, 2, 5	gausslemma2dlem0h 16041	. . . . . . 7 |- ( ph -> N e. NN0 )
18		zexpcl 10940	. . . . . . 7 |- ( ( -u 1 e. ZZ /\ N e. NN0 ) -> ( -u 1 ^ N ) e. ZZ )
19	16, 17, 18	sylancr 414	. . . . . 6 |- ( ph -> ( -u 1 ^ N ) e. ZZ )
20		2z 9622	. . . . . . 7 |- 2 e. ZZ
21		zexpcl 10940	. . . . . . 7 |- ( ( 2 e. ZZ /\ H e. NN0 ) -> ( 2 ^ H ) e. ZZ )
22	20, 8, 21	sylancr 414	. . . . . 6 |- ( ph -> ( 2 ^ H ) e. ZZ )
23	19, 22	zmulcld 9724	. . . . 5 |- ( ph -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. ZZ )
24	9	nnzd 9717	. . . . 5 |- ( ph -> ( ! ` H ) e. ZZ )
25	1	gausslemma2dlem0a 16034	. . . . 5 |- ( ph -> P e. NN )
26	1, 2	gausslemma2dlem0c 16036	. . . . 5 |- ( ph -> ( ( ! ` H ) gcd P ) = 1 )
27		cncongrcoprm 12828	. . . . 5 |- ( ( ( 1 e. ZZ /\ ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. ZZ /\ ( ! ` H ) e. ZZ ) /\ ( P e. NN /\ ( ( ! ` H ) gcd P ) = 1 ) ) -> ( ( ( 1 x. ( ! ` H ) ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) <-> ( 1 mod P ) = ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) ) )
28	15, 23, 24, 25, 26, 27	syl32anc 1282	. . . 4 |- ( ph -> ( ( ( 1 x. ( ! ` H ) ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) <-> ( 1 mod P ) = ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) ) )
29	14, 28	bitrd 188	. . 3 |- ( ph -> ( ( ( ! ` H ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) <-> ( 1 mod P ) = ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) ) )
30		simpr 110	. . . . 5 |- ( ( ph /\ ( 1 mod P ) = ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) ) -> ( 1 mod P ) = ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) )
31		nnq 9983	. . . . . . . 8 |- ( P e. NN -> P e. QQ )
32	25, 31	syl 14	. . . . . . 7 |- ( ph -> P e. QQ )
33	1	eldifad 3225	. . . . . . . 8 |- ( ph -> P e. Prime )
34		prmgt1 12854	. . . . . . . 8 |- ( P e. Prime -> 1 < P )
35	33, 34	syl 14	. . . . . . 7 |- ( ph -> 1 < P )
36		q1mod 10742	. . . . . . 7 |- ( ( P e. QQ /\ 1 < P ) -> ( 1 mod P ) = 1 )
37	32, 35, 36	syl2anc 411	. . . . . 6 |- ( ph -> ( 1 mod P ) = 1 )
38	37	adantr 276	. . . . 5 |- ( ( ph /\ ( 1 mod P ) = ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) ) -> ( 1 mod P ) = 1 )
39	30, 38	eqtr3d 2269	. . . 4 |- ( ( ph /\ ( 1 mod P ) = ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) ) -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 )
40	39	ex 115	. . 3 |- ( ph -> ( ( 1 mod P ) = ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 ) )
41	29, 40	sylbid 150	. 2 |- ( ph -> ( ( ( ! ` H ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 ) )
42	6, 41	mpd 13	1 |- ( ph -> ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) mod P ) = 1 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   \ cdif 3211   ifcif 3624   {csn 3694   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   1c1 8144   x. cmul 8148   < clt 8324   - cmin 8460   -ucneg 8461   / cdiv 8963   NNcn 9254   2c2 9305   4c4 9307   NN0cn0 9513   ZZcz 9594   QQcq 9969   ...cfz 10361   |_cfl 10652   mod cmo 10708   ^cexp 10924   !cfa 11112   gcd cgcd 12674   Primecprime 12829

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-ioo 10244  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-proddc 12262  df-dvds 12499  df-gcd 12675  df-prm 12830

This theorem is referenced by:  gausslemma2d  16054