Theorem 2lgslem3 15426

Description: Lemma 3 for 2lgs 15429. (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 9362	. . 3 |- ( P e. NN -> P e. ZZ )
2		lgsdir2lem3 15355	. . 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 3305	. . 3 |- ( ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) <-> ( ( P mod 8 ) e. { 1 , 7 } \/ ( P mod 8 ) e. { 3 , 5 } ) )
5		elpri 3646	. . . . . . . 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 15422	. . . . . . . . . . . 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 15425	. . . . . . . . . . . 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 717	. . . . . . . 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 3567	. . . . . . 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 2232	. . . . 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 3646	. . . . 5 |- ( ( P mod 8 ) e. { 3 , 5 } -> ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) )
23	6	2lgslem3b1 15423	. . . . . . . . . . 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 15424	. . . . . . . . . . 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 717	. . . . . . . . 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 8042	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
30		1lt3 9179	. . . . . . . . . . . . . . . . 17 |- 1 < 3
31	29, 30	ltneii 8140	. . . . . . . . . . . . . . . 16 |- 1 =/= 3
32	31	nesymi 2413	. . . . . . . . . . . . . . 15 |- -. 3 = 1
33		3re 9081	. . . . . . . . . . . . . . . . 17 |- 3 e. RR
34		3lt7 9195	. . . . . . . . . . . . . . . . 17 |- 3 < 7
35	33, 34	ltneii 8140	. . . . . . . . . . . . . . . 16 |- 3 =/= 7
36	35	neii 2369	. . . . . . . . . . . . . . 15 |- -. 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( -. 3 = 1 /\ -. 3 = 7 )
38		eqeq1 2203	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 1 <-> 3 = 1 ) )
39	38	notbid 668	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 1 <-> -. 3 = 1 ) )
40		eqeq1 2203	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 7 <-> 3 = 7 ) )
41	40	notbid 668	. . . . . . . . . . . . . . 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 9186	. . . . . . . . . . . . . . . . 17 |- 1 < 5
45	29, 44	ltneii 8140	. . . . . . . . . . . . . . . 16 |- 1 =/= 5
46	45	nesymi 2413	. . . . . . . . . . . . . . 15 |- -. 5 = 1
47		5re 9086	. . . . . . . . . . . . . . . . 17 |- 5 e. RR
48		5lt7 9193	. . . . . . . . . . . . . . . . 17 |- 5 < 7
49	47, 48	ltneii 8140	. . . . . . . . . . . . . . . 16 |- 5 =/= 7
50	49	neii 2369	. . . . . . . . . . . . . . 15 |- -. 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( -. 5 = 1 /\ -. 5 = 7 )
52		eqeq1 2203	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 1 <-> 5 = 1 ) )
53	52	notbid 668	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 1 <-> -. 5 = 1 ) )
54		eqeq1 2203	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 7 <-> 5 = 7 ) )
55	54	notbid 668	. . . . . . . . . . . . . . 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 717	. . . . . . . . . . . 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 753	. . . . . . . . . . 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 629	. . . . . . . . 9 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> -. ( P mod 8 ) e. { 1 , 7 } )
63	62	iffalsed 3572	. . . . . . . 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 2232	. . . . . . 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 717	. . 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 709   = wceq 1364   e. wcel 2167   u. cun 3155   ifcif 3562   {cpr 3624   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   0cc0 7896   1c1 7897   - cmin 8214   / cdiv 8716   NNcn 9007   2c2 9058   3c3 9059   4c4 9060   5c5 9061   7c7 9063   8c8 9064   ZZcz 9343   |_cfl 10375   mod cmo 10431   || cdvds 11969

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-ico 9986  df-fz 10101  df-fl 10377  df-mod 10432  df-dvds 11970

This theorem is referenced by:  2lgs  15429