Theorem 2lgslem3 15961

Description: Lemma 3 for 2lgs 15964. (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 9592	. . 3 |- ( P e. NN -> P e. ZZ )
2		lgsdir2lem3 15890	. . 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 3359	. . 3 |- ( ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) <-> ( ( P mod 8 ) e. { 1 , 7 } \/ ( P mod 8 ) e. { 3 , 5 } ) )
5		elpri 3711	. . . . . . . 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 15957	. . . . . . . . . . . 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 15960	. . . . . . . . . . . 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 724	. . . . . . . 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 3626	. . . . . . 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 2268	. . . . 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 3711	. . . . 5 |- ( ( P mod 8 ) e. { 3 , 5 } -> ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) )
23	6	2lgslem3b1 15958	. . . . . . . . . . 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 15959	. . . . . . . . . . 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 724	. . . . . . . . 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 8269	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
30		1lt3 9405	. . . . . . . . . . . . . . . . 17 |- 1 < 3
31	29, 30	ltneii 8366	. . . . . . . . . . . . . . . 16 |- 1 =/= 3
32	31	nesymi 2458	. . . . . . . . . . . . . . 15 |- -. 3 = 1
33		3re 9307	. . . . . . . . . . . . . . . . 17 |- 3 e. RR
34		3lt7 9421	. . . . . . . . . . . . . . . . 17 |- 3 < 7
35	33, 34	ltneii 8366	. . . . . . . . . . . . . . . 16 |- 3 =/= 7
36	35	neii 2414	. . . . . . . . . . . . . . 15 |- -. 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( -. 3 = 1 /\ -. 3 = 7 )
38		eqeq1 2239	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 1 <-> 3 = 1 ) )
39	38	notbid 673	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 1 <-> -. 3 = 1 ) )
40		eqeq1 2239	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 7 <-> 3 = 7 ) )
41	40	notbid 673	. . . . . . . . . . . . . . 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 9412	. . . . . . . . . . . . . . . . 17 |- 1 < 5
45	29, 44	ltneii 8366	. . . . . . . . . . . . . . . 16 |- 1 =/= 5
46	45	nesymi 2458	. . . . . . . . . . . . . . 15 |- -. 5 = 1
47		5re 9312	. . . . . . . . . . . . . . . . 17 |- 5 e. RR
48		5lt7 9419	. . . . . . . . . . . . . . . . 17 |- 5 < 7
49	47, 48	ltneii 8366	. . . . . . . . . . . . . . . 16 |- 5 =/= 7
50	49	neii 2414	. . . . . . . . . . . . . . 15 |- -. 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( -. 5 = 1 /\ -. 5 = 7 )
52		eqeq1 2239	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 1 <-> 5 = 1 ) )
53	52	notbid 673	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 1 <-> -. 5 = 1 ) )
54		eqeq1 2239	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 7 <-> 5 = 7 ) )
55	54	notbid 673	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . 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 760	. . . . . . . . . . 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 633	. . . . . . . . 9 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> -. ( P mod 8 ) e. { 1 , 7 } )
63	62	iffalsed 3631	. . . . . . . 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 2268	. . . . . . 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 724	. . 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 716   = wceq 1398   e. wcel 2203   u. cun 3208   ifcif 3619   {cpr 3689   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   0cc0 8123   1c1 8124   - cmin 8440   / cdiv 8942   NNcn 9233   2c2 9284   3c3 9285   4c4 9286   5c5 9287   7c7 9289   8c8 9290   ZZcz 9573   |_cfl 10624   mod cmo 10680   || cdvds 12466

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-n0 9493  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-ico 10223  df-fz 10339  df-fl 10626  df-mod 10681  df-dvds 12467

This theorem is referenced by:  2lgs  15964