Theorem gausslemma2dlem7 15803

Description: Lemma 7 for gausslemma2d 15804. (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 15802	. 2 |- ( ph -> ( ( ! ` H ) mod P ) = ( ( ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) x. ( ! ` H ) ) mod P ) )
7	1, 2	gausslemma2dlem0b 15785	. . . . . . . . . . 11 |- ( ph -> H e. NN )
8	7	nnnn0d 9455	. . . . . . . . . 10 |- ( ph -> H e. NN0 )
9	8	faccld 10999	. . . . . . . . 9 |- ( ph -> ( ! ` H ) e. NN )
10	9	nncnd 9157	. . . . . . . 8 |- ( ph -> ( ! ` H ) e. CC )
11	10	mullidd 8197	. . . . . . 7 |- ( ph -> ( 1 x. ( ! ` H ) ) = ( ! ` H ) )
12	11	eqcomd 2237	. . . . . 6 |- ( ph -> ( ! ` H ) = ( 1 x. ( ! ` H ) ) )
13	12	oveq1d 6033	. . . . 5 |- ( ph -> ( ( ! ` H ) mod P ) = ( ( 1 x. ( ! ` H ) ) mod P ) )
14	13	eqeq1d 2240	. . . 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 9506	. . . . 5 |- ( ph -> 1 e. ZZ )
16		neg1z 9511	. . . . . . 7 |- -u 1 e. ZZ
17	1, 4, 2, 5	gausslemma2dlem0h 15791	. . . . . . 7 |- ( ph -> N e. NN0 )
18		zexpcl 10817	. . . . . . 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 9507	. . . . . . 7 |- 2 e. ZZ
21		zexpcl 10817	. . . . . . 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 9608	. . . . 5 |- ( ph -> ( ( -u 1 ^ N ) x. ( 2 ^ H ) ) e. ZZ )
24	9	nnzd 9601	. . . . 5 |- ( ph -> ( ! ` H ) e. ZZ )
25	1	gausslemma2dlem0a 15784	. . . . 5 |- ( ph -> P e. NN )
26	1, 2	gausslemma2dlem0c 15786	. . . . 5 |- ( ph -> ( ( ! ` H ) gcd P ) = 1 )
27		cncongrcoprm 12683	. . . . 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 1281	. . . 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 9867	. . . . . . . 8 |- ( P e. NN -> P e. QQ )
32	25, 31	syl 14	. . . . . . 7 |- ( ph -> P e. QQ )
33	1	eldifad 3211	. . . . . . . 8 |- ( ph -> P e. Prime )
34		prmgt1 12709	. . . . . . . 8 |- ( P e. Prime -> 1 < P )
35	33, 34	syl 14	. . . . . . 7 |- ( ph -> 1 < P )
36		q1mod 10619	. . . . . . 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 2266	. . . 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 1397   e. wcel 2202   \ cdif 3197   ifcif 3605   {csn 3669   class class class wbr 4088   |-> cmpt 4150   ` cfv 5326  (class class class)co 6018   1c1 8033   x. cmul 8037   < clt 8214   - cmin 8350   -ucneg 8351   / cdiv 8852   NNcn 9143   2c2 9194   4c4 9196   NN0cn0 9402   ZZcz 9479   QQcq 9853   ...cfz 10243   |_cfl 10529   mod cmo 10585   ^cexp 10801   !cfa 10988   gcd cgcd 12529   Primecprime 12684

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-ioo 10127  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-fac 10989  df-ihash 11039  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-clim 11844  df-proddc 12117  df-dvds 12354  df-gcd 12530  df-prm 12685

This theorem is referenced by:  gausslemma2d  15804