Theorem 2lgslem3 25129

Description: Lemma 3 for 2lgs 25132. (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 11399	. . 3 |- ( P e. NN -> P e. ZZ )
2		lgsdir2lem3 25052	. . 3 |- ( ( P e. ZZ /\ -. 2 || P ) -> ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) )
3	1, 2	sylan 488	. 2 |- ( ( P e. NN /\ -. 2 || P ) -> ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) )
4		elun 3753	. . 3 |- ( ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) <-> ( ( P mod 8 ) e. { 1 , 7 } \/ ( P mod 8 ) e. { 3 , 5 } ) )
5		ovex 6678	. . . . . . . . 9 |- ( P mod 8 ) e. _V
6	5	elpr 4198	. . . . . . . 8 |- ( ( P mod 8 ) e. { 1 , 7 } <-> ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) )
7		2lgslem2.n	. . . . . . . . . . . . 13 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
8	7	2lgslem3a1 25125	. . . . . . . . . . . 12 |- ( ( P e. NN /\ ( P mod 8 ) = 1 ) -> ( N mod 2 ) = 0 )
9	8	a1d 25	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( P mod 8 ) = 1 ) -> ( -. 2 || P -> ( N mod 2 ) = 0 ) )
10	9	expcom 451	. . . . . . . . . 10 |- ( ( P mod 8 ) = 1 -> ( P e. NN -> ( -. 2 || P -> ( N mod 2 ) = 0 ) ) )
11	10	impd 447	. . . . . . . . 9 |- ( ( P mod 8 ) = 1 -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = 0 ) )
12	7	2lgslem3d1 25128	. . . . . . . . . . . 12 |- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( N mod 2 ) = 0 )
13	12	a1d 25	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( -. 2 || P -> ( N mod 2 ) = 0 ) )
14	13	expcom 451	. . . . . . . . . 10 |- ( ( P mod 8 ) = 7 -> ( P e. NN -> ( -. 2 || P -> ( N mod 2 ) = 0 ) ) )
15	14	impd 447	. . . . . . . . 9 |- ( ( P mod 8 ) = 7 -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = 0 ) )
16	11, 15	jaoi 394	. . . . . . . 8 |- ( ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = 0 ) )
17	6, 16	sylbi 207	. . . . . . 7 |- ( ( P mod 8 ) e. { 1 , 7 } -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = 0 ) )
18	17	imp 445	. . . . . 6 |- ( ( ( P mod 8 ) e. { 1 , 7 } /\ ( P e. NN /\ -. 2 || P ) ) -> ( N mod 2 ) = 0 )
19		iftrue 4092	. . . . . . 7 |- ( ( P mod 8 ) e. { 1 , 7 } -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 )
20	19	adantr 481	. . . . . 6 |- ( ( ( P mod 8 ) e. { 1 , 7 } /\ ( P e. NN /\ -. 2 || P ) ) -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 0 )
21	18, 20	eqtr4d 2659	. . . . 5 |- ( ( ( P mod 8 ) e. { 1 , 7 } /\ ( P e. NN /\ -. 2 || P ) ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )
22	21	ex 450	. . . 4 |- ( ( P mod 8 ) e. { 1 , 7 } -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) ) )
23	5	elpr 4198	. . . . 5 |- ( ( P mod 8 ) e. { 3 , 5 } <-> ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) )
24	7	2lgslem3b1 25126	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( P mod 8 ) = 3 ) -> ( N mod 2 ) = 1 )
25	24	expcom 451	. . . . . . . . . 10 |- ( ( P mod 8 ) = 3 -> ( P e. NN -> ( N mod 2 ) = 1 ) )
26	7	2lgslem3c1 25127	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( P mod 8 ) = 5 ) -> ( N mod 2 ) = 1 )
27	26	expcom 451	. . . . . . . . . 10 |- ( ( P mod 8 ) = 5 -> ( P e. NN -> ( N mod 2 ) = 1 ) )
28	25, 27	jaoi 394	. . . . . . . . 9 |- ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) -> ( P e. NN -> ( N mod 2 ) = 1 ) )
29	28	imp 445	. . . . . . . 8 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> ( N mod 2 ) = 1 )
30		1re 10039	. . . . . . . . . . . . . . . 16 |- 1 e. RR
31		1lt3 11196	. . . . . . . . . . . . . . . 16 |- 1 < 3
32	30, 31	ltneii 10150	. . . . . . . . . . . . . . 15 |- 1 =/= 3
33	32	nesymi 2851	. . . . . . . . . . . . . 14 |- -. 3 = 1
34		3re 11094	. . . . . . . . . . . . . . . 16 |- 3 e. RR
35		3lt7 11212	. . . . . . . . . . . . . . . 16 |- 3 < 7
36	34, 35	ltneii 10150	. . . . . . . . . . . . . . 15 |- 3 =/= 7
37	36	neii 2796	. . . . . . . . . . . . . 14 |- -. 3 = 7
38	33, 37	pm3.2i 471	. . . . . . . . . . . . 13 |- ( -. 3 = 1 /\ -. 3 = 7 )
39		eqeq1 2626	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 1 <-> 3 = 1 ) )
40	39	notbid 308	. . . . . . . . . . . . . 14 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 1 <-> -. 3 = 1 ) )
41		eqeq1 2626	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 7 <-> 3 = 7 ) )
42	41	notbid 308	. . . . . . . . . . . . . 14 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 7 <-> -. 3 = 7 ) )
43	40, 42	anbi12d 747	. . . . . . . . . . . . 13 |- ( ( P mod 8 ) = 3 -> ( ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) <-> ( -. 3 = 1 /\ -. 3 = 7 ) ) )
44	38, 43	mpbiri 248	. . . . . . . . . . . 12 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
45		1lt5 11203	. . . . . . . . . . . . . . . 16 |- 1 < 5
46	30, 45	ltneii 10150	. . . . . . . . . . . . . . 15 |- 1 =/= 5
47	46	nesymi 2851	. . . . . . . . . . . . . 14 |- -. 5 = 1
48		5re 11099	. . . . . . . . . . . . . . . 16 |- 5 e. RR
49		5lt7 11210	. . . . . . . . . . . . . . . 16 |- 5 < 7
50	48, 49	ltneii 10150	. . . . . . . . . . . . . . 15 |- 5 =/= 7
51	50	neii 2796	. . . . . . . . . . . . . 14 |- -. 5 = 7
52	47, 51	pm3.2i 471	. . . . . . . . . . . . 13 |- ( -. 5 = 1 /\ -. 5 = 7 )
53		eqeq1 2626	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 1 <-> 5 = 1 ) )
54	53	notbid 308	. . . . . . . . . . . . . 14 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 1 <-> -. 5 = 1 ) )
55		eqeq1 2626	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 7 <-> 5 = 7 ) )
56	55	notbid 308	. . . . . . . . . . . . . 14 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 7 <-> -. 5 = 7 ) )
57	54, 56	anbi12d 747	. . . . . . . . . . . . 13 |- ( ( P mod 8 ) = 5 -> ( ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) <-> ( -. 5 = 1 /\ -. 5 = 7 ) ) )
58	52, 57	mpbiri 248	. . . . . . . . . . . 12 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
59	44, 58	jaoi 394	. . . . . . . . . . 11 |- ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) -> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
60	59	adantr 481	. . . . . . . . . 10 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
61		ioran 511	. . . . . . . . . . 11 |- ( -. ( ( P mod 8 ) = 1 \/ ( P mod 8 ) = 7 ) <-> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
62	61, 6	xchnxbir 323	. . . . . . . . . 10 |- ( -. ( P mod 8 ) e. { 1 , 7 } <-> ( -. ( P mod 8 ) = 1 /\ -. ( P mod 8 ) = 7 ) )
63	60, 62	sylibr 224	. . . . . . . . 9 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> -. ( P mod 8 ) e. { 1 , 7 } )
64	63	iffalsed 4097	. . . . . . . 8 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) = 1 )
65	29, 64	eqtr4d 2659	. . . . . . 7 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )
66	65	a1d 25	. . . . . 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 ) ) )
67	66	expimpd 629	. . . . 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 ) ) )
68	23, 67	sylbi 207	. . . 4 |- ( ( P mod 8 ) e. { 3 , 5 } -> ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) ) )
69	22, 68	jaoi 394	. . 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 ) ) )
70	4, 69	sylbi 207	. 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 ) ) )
71	3, 70	mpcom 38	1 |- ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   ifcif 4086   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   5c5 11073   7c7 11075   8c8 11076   ZZcz 11377   |_cfl 12591   mod cmo 12668   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fl 12593  df-mod 12669  df-dvds 14984

This theorem is referenced by:  2lgs  25132