Theorem 2lgsoddprmlem3 25139

Description: Lemma 3 for 2lgsoddprm 25141. (Contributed by AV, 20-Jul-2021.)

Assertion

Ref	Expression
2lgsoddprmlem3	|- ( ( N e. ZZ /\ -. 2 || N /\ R = ( N mod 8 ) ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> R e. { 1 , 7 } ) )

Proof of Theorem 2lgsoddprmlem3

Step	Hyp	Ref	Expression
1		lgsdir2lem3 25052	. . 3 |- ( ( N e. ZZ /\ -. 2 || N ) -> ( N mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) )
2		eleq1 2689	. . . . 5 |- ( ( N mod 8 ) = R -> ( ( N mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) <-> R e. ( { 1 , 7 } u. { 3 , 5 } ) ) )
3	2	eqcoms 2630	. . . 4 |- ( R = ( N mod 8 ) -> ( ( N mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) <-> R e. ( { 1 , 7 } u. { 3 , 5 } ) ) )
4		elun 3753	. . . . . 6 |- ( R e. ( { 1 , 7 } u. { 3 , 5 } ) <-> ( R e. { 1 , 7 } \/ R e. { 3 , 5 } ) )
5		elpri 4197	. . . . . . . 8 |- ( R e. { 3 , 5 } -> ( R = 3 \/ R = 5 ) )
6		oveq1 6657	. . . . . . . . . . . . . 14 |- ( R = 3 -> ( R ^ 2 ) = ( 3 ^ 2 ) )
7	6	oveq1d 6665	. . . . . . . . . . . . 13 |- ( R = 3 -> ( ( R ^ 2 ) - 1 ) = ( ( 3 ^ 2 ) - 1 ) )
8	7	oveq1d 6665	. . . . . . . . . . . 12 |- ( R = 3 -> ( ( ( R ^ 2 ) - 1 ) / 8 ) = ( ( ( 3 ^ 2 ) - 1 ) / 8 ) )
9		2lgsoddprmlem3b 25136	. . . . . . . . . . . 12 |- ( ( ( 3 ^ 2 ) - 1 ) / 8 ) = 1
10	8, 9	syl6eq 2672	. . . . . . . . . . 11 |- ( R = 3 -> ( ( ( R ^ 2 ) - 1 ) / 8 ) = 1 )
11	10	breq2d 4665	. . . . . . . . . 10 |- ( R = 3 -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> 2 || 1 ) )
12		n2dvds1 15104	. . . . . . . . . . 11 |- -. 2 || 1
13	12	pm2.21i 116	. . . . . . . . . 10 |- ( 2 || 1 -> R e. { 1 , 7 } )
14	11, 13	syl6bi 243	. . . . . . . . 9 |- ( R = 3 -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) -> R e. { 1 , 7 } ) )
15		oveq1 6657	. . . . . . . . . . . . . 14 |- ( R = 5 -> ( R ^ 2 ) = ( 5 ^ 2 ) )
16	15	oveq1d 6665	. . . . . . . . . . . . 13 |- ( R = 5 -> ( ( R ^ 2 ) - 1 ) = ( ( 5 ^ 2 ) - 1 ) )
17	16	oveq1d 6665	. . . . . . . . . . . 12 |- ( R = 5 -> ( ( ( R ^ 2 ) - 1 ) / 8 ) = ( ( ( 5 ^ 2 ) - 1 ) / 8 ) )
18	17	breq2d 4665	. . . . . . . . . . 11 |- ( R = 5 -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> 2 || ( ( ( 5 ^ 2 ) - 1 ) / 8 ) ) )
19		2lgsoddprmlem3c 25137	. . . . . . . . . . . 12 |- ( ( ( 5 ^ 2 ) - 1 ) / 8 ) = 3
20	19	breq2i 4661	. . . . . . . . . . 11 |- ( 2 || ( ( ( 5 ^ 2 ) - 1 ) / 8 ) <-> 2 || 3 )
21	18, 20	syl6bb 276	. . . . . . . . . 10 |- ( R = 5 -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> 2 || 3 ) )
22		n2dvds3 15107	. . . . . . . . . . 11 |- -. 2 || 3
23	22	pm2.21i 116	. . . . . . . . . 10 |- ( 2 || 3 -> R e. { 1 , 7 } )
24	21, 23	syl6bi 243	. . . . . . . . 9 |- ( R = 5 -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) -> R e. { 1 , 7 } ) )
25	14, 24	jaoi 394	. . . . . . . 8 |- ( ( R = 3 \/ R = 5 ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) -> R e. { 1 , 7 } ) )
26	5, 25	syl 17	. . . . . . 7 |- ( R e. { 3 , 5 } -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) -> R e. { 1 , 7 } ) )
27	26	jao1i 825	. . . . . 6 |- ( ( R e. { 1 , 7 } \/ R e. { 3 , 5 } ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) -> R e. { 1 , 7 } ) )
28	4, 27	sylbi 207	. . . . 5 |- ( R e. ( { 1 , 7 } u. { 3 , 5 } ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) -> R e. { 1 , 7 } ) )
29		elpri 4197	. . . . . 6 |- ( R e. { 1 , 7 } -> ( R = 1 \/ R = 7 ) )
30		z0even 15103	. . . . . . . 8 |- 2 || 0
31		oveq1 6657	. . . . . . . . . . 11 |- ( R = 1 -> ( R ^ 2 ) = ( 1 ^ 2 ) )
32	31	oveq1d 6665	. . . . . . . . . 10 |- ( R = 1 -> ( ( R ^ 2 ) - 1 ) = ( ( 1 ^ 2 ) - 1 ) )
33	32	oveq1d 6665	. . . . . . . . 9 |- ( R = 1 -> ( ( ( R ^ 2 ) - 1 ) / 8 ) = ( ( ( 1 ^ 2 ) - 1 ) / 8 ) )
34		2lgsoddprmlem3a 25135	. . . . . . . . 9 |- ( ( ( 1 ^ 2 ) - 1 ) / 8 ) = 0
35	33, 34	syl6eq 2672	. . . . . . . 8 |- ( R = 1 -> ( ( ( R ^ 2 ) - 1 ) / 8 ) = 0 )
36	30, 35	syl5breqr 4691	. . . . . . 7 |- ( R = 1 -> 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) )
37		2z 11409	. . . . . . . . 9 |- 2 e. ZZ
38		3z 11410	. . . . . . . . 9 |- 3 e. ZZ
39		dvdsmul1 15003	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ 3 e. ZZ ) -> 2 || ( 2 x. 3 ) )
40	37, 38, 39	mp2an 708	. . . . . . . 8 |- 2 || ( 2 x. 3 )
41		oveq1 6657	. . . . . . . . . . 11 |- ( R = 7 -> ( R ^ 2 ) = ( 7 ^ 2 ) )
42	41	oveq1d 6665	. . . . . . . . . 10 |- ( R = 7 -> ( ( R ^ 2 ) - 1 ) = ( ( 7 ^ 2 ) - 1 ) )
43	42	oveq1d 6665	. . . . . . . . 9 |- ( R = 7 -> ( ( ( R ^ 2 ) - 1 ) / 8 ) = ( ( ( 7 ^ 2 ) - 1 ) / 8 ) )
44		2lgsoddprmlem3d 25138	. . . . . . . . 9 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )
45	43, 44	syl6eq 2672	. . . . . . . 8 |- ( R = 7 -> ( ( ( R ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 ) )
46	40, 45	syl5breqr 4691	. . . . . . 7 |- ( R = 7 -> 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) )
47	36, 46	jaoi 394	. . . . . 6 |- ( ( R = 1 \/ R = 7 ) -> 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) )
48	29, 47	syl 17	. . . . 5 |- ( R e. { 1 , 7 } -> 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) )
49	28, 48	impbid1 215	. . . 4 |- ( R e. ( { 1 , 7 } u. { 3 , 5 } ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> R e. { 1 , 7 } ) )
50	3, 49	syl6bi 243	. . 3 |- ( R = ( N mod 8 ) -> ( ( N mod 8 ) e. ( { 1 , 7 } u. { 3 , 5 } ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> R e. { 1 , 7 } ) ) )
51	1, 50	syl5com 31	. 2 |- ( ( N e. ZZ /\ -. 2 || N ) -> ( R = ( N mod 8 ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> R e. { 1 , 7 } ) ) )
52	51	3impia 1261	1 |- ( ( N e. ZZ /\ -. 2 || N /\ R = ( N mod 8 ) ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> R e. { 1 , 7 } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   {cpr 4179   class class class wbr 4653  (class class class)co 6650   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   / cdiv 10684   2c2 11070   3c3 11071   5c5 11073   7c7 11075   8c8 11076   ZZcz 11377   mod cmo 12668   ^cexp 12860   || 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-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-dvds 14984

This theorem is referenced by:  2lgsoddprmlem4  25140