Theorem 2lgslem3 15823

Description: Lemma 3 for 2lgs 15826. (Contributed by AV, 16-Jul-2021.)

Hypothesis

Ref	Expression
2lgslem2.n	|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )

Assertion

Ref	Expression
2lgslem3	|- ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )

Proof of Theorem 2lgslem3

Step	Hyp	Ref	Expression
1		nnz 9491	. . 3 |- ( P e. NN -> P e. ZZ )
2		lgsdir2lem3 15752	. . 3 |- ( ( P e. ZZ /\ -. 2 || P ) -> ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) )
3	1, 2	sylan 283	. 2 |- ( ( P e. NN /\ -. 2 || P ) -> ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) )
4		elun 3346	. . 3 |- ( ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) <-> ( ( P mod 8 ) e. { 1 , 7 } \/ ( P mod 8 ) e. { 3 , 5 } ) )
5		elpri 3690	. . . . . . . 8 |- ( ( P mod 8 ) e. { 1 , 7 } -> ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) )
6		2lgslem2.n	. . . . . . . . . . . . 13 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
7	6	2lgslem3a1 15819	. . . . . . . . . . . 12 |- ( ( P e. NN /\ ( P mod 8 ) = 1 ) -> ( N mod 2 ) = 0 )
8	7	a1d 22	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( P mod 8 ) = 1 ) -> ( -. 2 || P -> ( N mod 2 ) = 0 ) )
9	8	expcom 116	. . . . . . . . . 10 |- ( ( P mod 8 ) = 1 -> ( P e. NN -> ( -. 2 || P -> ( N mod 2 ) = 0 ) ) )
10	9	impd 254	. . . . . . . . 9 |- ( ( P mod 8 ) = 1 -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = 0 ) )
11	6	2lgslem3d1 15822	. . . . . . . . . . . 12 |- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( N mod 2 ) = 0 )
12	11	a1d 22	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( -. 2 || P -> ( N mod 2 ) = 0 ) )
13	12	expcom 116	. . . . . . . . . 10 |- ( ( P mod 8 ) = 7 -> ( P e. NN -> ( -. 2 || P -> ( N mod 2 ) = 0 ) ) )
14	13	impd 254	. . . . . . . . 9 |- ( ( P mod 8 ) = 7 -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = 0 ) )
15	10, 14	jaoi 721	. . . . . . . 8 |- ( ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = 0 ) )
16	5, 15	syl 14	. . . . . . 7 |- ( ( P mod 8 ) e. { 1 , 7 } -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = 0 ) )
17	16	imp 124	. . . . . 6 |- ( ( ( P mod 8 ) e. { 1 , 7 } /\ ( P e. NN /\ -. 2 || P ) ) -> ( N mod 2 ) = 0 )
18		iftrue 3608	. . . . . . 7 |- ( ( P mod 8 ) e. { 1 , 7 } -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 )
19	18	adantr 276	. . . . . 6 |- ( ( ( P mod 8 ) e. { 1 , 7 } /\ ( P e. NN /\ -. 2 || P ) ) -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 )
20	17, 19	eqtr4d 2265	. . . . 5 |- ( ( ( P mod 8 ) e. { 1 , 7 } /\ ( P e. NN /\ -. 2 || P ) ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )
21	20	ex 115	. . . 4 |- ( ( P mod 8 ) e. { 1 , 7 } -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) ) )
22		elpri 3690	. . . . 5 |- ( ( P mod 8 ) e. { 3 , 5 } -> ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) )
23	6	2lgslem3b1 15820	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( P mod 8 ) = 3 ) -> ( N mod 2 ) = 1 )
24	23	expcom 116	. . . . . . . . . 10 |- ( ( P mod 8 ) = 3 -> ( P e. NN -> ( N mod 2 ) = 1 ) )
25	6	2lgslem3c1 15821	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( P mod 8 ) = 5 ) -> ( N mod 2 ) = 1 )
26	25	expcom 116	. . . . . . . . . 10 |- ( ( P mod 8 ) = 5 -> ( P e. NN -> ( N mod 2 ) = 1 ) )
27	24, 26	jaoi 721	. . . . . . . . 9 |- ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) -> ( P e. NN -> ( N mod 2 ) = 1 ) )
28	27	imp 124	. . . . . . . 8 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> ( N mod 2 ) = 1 )
29		1re 8171	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
30		1lt3 9308	. . . . . . . . . . . . . . . . 17 |- 1 < 3
31	29, 30	ltneii 8269	. . . . . . . . . . . . . . . 16 |- 1 =/= 3
32	31	nesymi 2446	. . . . . . . . . . . . . . 15 |- -. 3 = 1
33		3re 9210	. . . . . . . . . . . . . . . . 17 |- 3 e. RR
34		3lt7 9324	. . . . . . . . . . . . . . . . 17 |- 3 < 7
35	33, 34	ltneii 8269	. . . . . . . . . . . . . . . 16 |- 3 =/= 7
36	35	neii 2402	. . . . . . . . . . . . . . 15 |- -. 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( -. 3 = 1 /\ -. 3 = 7 )
38		eqeq1 2236	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 1 <-> 3 = 1 ) )
39	38	notbid 671	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 1 <-> -. 3 = 1 ) )
40		eqeq1 2236	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 7 <-> 3 = 7 ) )
41	40	notbid 671	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 7 <-> -. 3 = 7 ) )
42	39, 41	anbi12d 473	. . . . . . . . . . . . . 14 |- ( ( P mod 8 ) = 3 -> ( ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) <-> ( -. 3 = 1 /\ -. 3 = 7 ) ) )
43	37, 42	mpbiri 168	. . . . . . . . . . . . 13 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
44		1lt5 9315	. . . . . . . . . . . . . . . . 17 |- 1 < 5
45	29, 44	ltneii 8269	. . . . . . . . . . . . . . . 16 |- 1 =/= 5
46	45	nesymi 2446	. . . . . . . . . . . . . . 15 |- -. 5 = 1
47		5re 9215	. . . . . . . . . . . . . . . . 17 |- 5 e. RR
48		5lt7 9322	. . . . . . . . . . . . . . . . 17 |- 5 < 7
49	47, 48	ltneii 8269	. . . . . . . . . . . . . . . 16 |- 5 =/= 7
50	49	neii 2402	. . . . . . . . . . . . . . 15 |- -. 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( -. 5 = 1 /\ -. 5 = 7 )
52		eqeq1 2236	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 1 <-> 5 = 1 ) )
53	52	notbid 671	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 1 <-> -. 5 = 1 ) )
54		eqeq1 2236	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 7 <-> 5 = 7 ) )
55	54	notbid 671	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 7 <-> -. 5 = 7 ) )
56	53, 55	anbi12d 473	. . . . . . . . . . . . . 14 |- ( ( P mod 8 ) = 5 -> ( ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) <-> ( -. 5 = 1 /\ -. 5 = 7 ) ) )
57	51, 56	mpbiri 168	. . . . . . . . . . . . 13 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
58	43, 57	jaoi 721	. . . . . . . . . . . 12 |- ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) -> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
59	58	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
60		ioran 757	. . . . . . . . . . 11 |- ( -. ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) <-> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
61	59, 60	sylibr 134	. . . . . . . . . 10 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> -. ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) )
62	61, 5	nsyl 631	. . . . . . . . 9 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> -. ( P mod 8 ) e. { 1 , 7 } )
63	62	iffalsed 3613	. . . . . . . 8 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 1 )
64	28, 63	eqtr4d 2265	. . . . . . 7 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )
65	64	a1d 22	. . . . . 6 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> ( -. 2 || P -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) ) )
66	65	expimpd 363	. . . . 5 |- ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) ) )
67	22, 66	syl 14	. . . 4 |- ( ( P mod 8 ) e. { 3 , 5 } -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) ) )
68	21, 67	jaoi 721	. . 3 |- ( ( ( P mod 8 ) e. { 1 , 7 } \/ ( P mod 8 ) e. { 3 , 5 } ) -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) ) )
69	4, 68	sylbi 121	. 2 |- ( ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) ) )
70	3, 69	mpcom 36	1 |- ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   u. cun 3196   ifcif 3603   {cpr 3668   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   0cc0 8025   1c1 8026   - cmin 8343   / cdiv 8845   NNcn 9136   2c2 9187   3c3 9188   4c4 9189   5c5 9190   7c7 9192   8c8 9193   ZZcz 9472   |_cfl 10521   mod cmo 10577   || cdvds 12341

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-ico 10122  df-fz 10237  df-fl 10523  df-mod 10578  df-dvds 12342

This theorem is referenced by:  2lgs  15826