Theorem 2lgslem3 16203

Description: Lemma 3 for 2lgs 16206. (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 9646	. . 3 |- ( P e. NN -> P e. ZZ )
2		lgsdir2lem3 16132	. . 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 3370	. . 3 |- ( ( P mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) <-> ( ( P mod 8 ) e. { 1 , 7 } \/ ( P mod 8 ) e. { 3 , 5 } ) )
5		elpri 3731	. . . . . . . 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 16199	. . . . . . . . . . . 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 16202	. . . . . . . . . . . 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 728	. . . . . . . 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 3645	. . . . . . 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 2274	. . . . 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 3731	. . . . 5 |- ( ( P mod 8 ) e. { 3 , 5 } -> ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) )
23	6	2lgslem3b1 16200	. . . . . . . . . . 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 16201	. . . . . . . . . . 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 728	. . . . . . . . 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 8319	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
30		1lt3 9459	. . . . . . . . . . . . . . . . 17 |- 1 < 3
31	29, 30	ltneii 8416	. . . . . . . . . . . . . . . 16 |- 1 =/= 3
32	31	nesymi 2466	. . . . . . . . . . . . . . 15 |- -. 3 = 1
33		3re 9361	. . . . . . . . . . . . . . . . 17 |- 3 e. RR
34		3lt7 9475	. . . . . . . . . . . . . . . . 17 |- 3 < 7
35	33, 34	ltneii 8416	. . . . . . . . . . . . . . . 16 |- 3 =/= 7
36	35	neii 2422	. . . . . . . . . . . . . . 15 |- -. 3 = 7
37	32, 36	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( -. 3 = 1 /\ -. 3 = 7 )
38		eqeq1 2245	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 1 <-> 3 = 1 ) )
39	38	notbid 677	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 1 <-> -. 3 = 1 ) )
40		eqeq1 2245	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 3 -> ( ( P mod 8 ) = 7 <-> 3 = 7 ) )
41	40	notbid 677	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 3 -> ( -. ( P mod 8 ) = 7 <-> -. 3 = 7 ) )
42	39, 41	anbi12d 477	. . . . . . . . . . . . . 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 9466	. . . . . . . . . . . . . . . . 17 |- 1 < 5
45	29, 44	ltneii 8416	. . . . . . . . . . . . . . . 16 |- 1 =/= 5
46	45	nesymi 2466	. . . . . . . . . . . . . . 15 |- -. 5 = 1
47		5re 9366	. . . . . . . . . . . . . . . . 17 |- 5 e. RR
48		5lt7 9473	. . . . . . . . . . . . . . . . 17 |- 5 < 7
49	47, 48	ltneii 8416	. . . . . . . . . . . . . . . 16 |- 5 =/= 7
50	49	neii 2422	. . . . . . . . . . . . . . 15 |- -. 5 = 7
51	46, 50	pm3.2i 272	. . . . . . . . . . . . . 14 |- ( -. 5 = 1 /\ -. 5 = 7 )
52		eqeq1 2245	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 1 <-> 5 = 1 ) )
53	52	notbid 677	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 1 <-> -. 5 = 1 ) )
54		eqeq1 2245	. . . . . . . . . . . . . . . 16 |- ( ( P mod 8 ) = 5 -> ( ( P mod 8 ) = 7 <-> 5 = 7 ) )
55	54	notbid 677	. . . . . . . . . . . . . . 15 |- ( ( P mod 8 ) = 5 -> ( -. ( P mod 8 ) = 7 <-> -. 5 = 7 ) )
56	53, 55	anbi12d 477	. . . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 764	. . . . . . . . . . 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 637	. . . . . . . . 9 |- ( ( ( ( P mod 8 ) = 3 \/ ( P mod 8 ) = 5 ) /\ P e. NN ) -> -. ( P mod 8 ) e. { 1 , 7 } )
63	62	iffalsed 3650	. . . . . . . 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 2274	. . . . . . 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 728	. . 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 720   = wceq 1402   e. wcel 2209   u. cun 3218   ifcif 3638   {cpr 3709   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   0cc0 8173   1c1 8174   - cmin 8491   / cdiv 8996   NNcn 9287   2c2 9338   3c3 9339   4c4 9340   5c5 9341   7c7 9343   8c8 9344   ZZcz 9627   |_cfl 10686   mod cmo 10742   || cdvds 12537

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-ico 10279  df-fz 10395  df-fl 10688  df-mod 10743  df-dvds 12538

This theorem is referenced by:  2lgs  16206