Theorem lgsdir2lem5 25054

Description: Lemma for lgsdir2 25055. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
lgsdir2lem5	|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> ( ( A x. B ) mod 8 ) e. { 1 , 7 } )

Proof of Theorem lgsdir2lem5

Step	Hyp	Ref	Expression
1		ovex 6678	. . . . . . 7 |- ( A mod 8 ) e. _V
2	1	elpr 4198	. . . . . 6 |- ( ( A mod 8 ) e. { 3 , 5 } <-> ( ( A mod 8 ) = 3 \/ ( A mod 8 ) = 5 ) )
3		ovex 6678	. . . . . . 7 |- ( B mod 8 ) e. _V
4	3	elpr 4198	. . . . . 6 |- ( ( B mod 8 ) e. { 3 , 5 } <-> ( ( B mod 8 ) = 3 \/ ( B mod 8 ) = 5 ) )
5	2, 4	anbi12i 733	. . . . 5 |- ( ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) <-> ( ( ( A mod 8 ) = 3 \/ ( A mod 8 ) = 5 ) /\ ( ( B mod 8 ) = 3 \/ ( B mod 8 ) = 5 ) ) )
6		simpll 790	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> A e. ZZ )
7		3z 11410	. . . . . . . . . 10 |- 3 e. ZZ
8	7	a1i 11	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> 3 e. ZZ )
9		simplr 792	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> B e. ZZ )
10		8re 11105	. . . . . . . . . . 11 |- 8 e. RR
11		8pos 11121	. . . . . . . . . . 11 |- 0 < 8
12	10, 11	elrpii 11835	. . . . . . . . . 10 |- 8 e. RR+
13	12	a1i 11	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> 8 e. RR+ )
14		simprl 794	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = 3 )
15		lgsdir2lem1 25050	. . . . . . . . . . . 12 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )
16	15	simpri 478	. . . . . . . . . . 11 |- ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 )
17	16	simpli 474	. . . . . . . . . 10 |- ( 3 mod 8 ) = 3
18	14, 17	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = ( 3 mod 8 ) )
19		simprr 796	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = 3 )
20	19, 17	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = ( 3 mod 8 ) )
21	6, 8, 9, 8, 13, 18, 20	modmul12d 12724	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) )
22	21	orcd 407	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) )
23	22	ex 450	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) ) )
24		simpll 790	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> A e. ZZ )
25		znegcl 11412	. . . . . . . . . . 11 |- ( 3 e. ZZ -> -u 3 e. ZZ )
26	7, 25	mp1i 13	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> -u 3 e. ZZ )
27		simplr 792	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> B e. ZZ )
28	7	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> 3 e. ZZ )
29	12	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> 8 e. RR+ )
30		simprl 794	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = 5 )
31	16	simpri 478	. . . . . . . . . . 11 |- ( -u 3 mod 8 ) = 5
32	30, 31	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = ( -u 3 mod 8 ) )
33		simprr 796	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = 3 )
34	33, 17	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = ( 3 mod 8 ) )
35	24, 26, 27, 28, 29, 32, 34	modmul12d 12724	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( ( A x. B ) mod 8 ) = ( ( -u 3 x. 3 ) mod 8 ) )
36		3cn 11095	. . . . . . . . . . 11 |- 3 e. CC
37	36, 36	mulneg1i 10476	. . . . . . . . . 10 |- ( -u 3 x. 3 ) = -u ( 3 x. 3 )
38	37	oveq1i 6660	. . . . . . . . 9 |- ( ( -u 3 x. 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
39	35, 38	syl6eq 2672	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) )
40	39	olcd 408	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) )
41	40	ex 450	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) ) )
42		simpll 790	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> A e. ZZ )
43	7	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> 3 e. ZZ )
44		simplr 792	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> B e. ZZ )
45	7, 25	mp1i 13	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> -u 3 e. ZZ )
46	12	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> 8 e. RR+ )
47		simprl 794	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = 3 )
48	47, 17	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = ( 3 mod 8 ) )
49		simprr 796	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = 5 )
50	49, 31	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = ( -u 3 mod 8 ) )
51	42, 43, 44, 45, 46, 48, 50	modmul12d 12724	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( ( A x. B ) mod 8 ) = ( ( 3 x. -u 3 ) mod 8 ) )
52	36, 36	mulneg2i 10477	. . . . . . . . . 10 |- ( 3 x. -u 3 ) = -u ( 3 x. 3 )
53	52	oveq1i 6660	. . . . . . . . 9 |- ( ( 3 x. -u 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
54	51, 53	syl6eq 2672	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) )
55	54	olcd 408	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) )
56	55	ex 450	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) ) )
57		simpll 790	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> A e. ZZ )
58	7, 25	mp1i 13	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> -u 3 e. ZZ )
59		simplr 792	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> B e. ZZ )
60	12	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> 8 e. RR+ )
61		simprl 794	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = 5 )
62	61, 31	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = ( -u 3 mod 8 ) )
63		simprr 796	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = 5 )
64	63, 31	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = ( -u 3 mod 8 ) )
65	57, 58, 59, 58, 60, 62, 64	modmul12d 12724	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( ( A x. B ) mod 8 ) = ( ( -u 3 x. -u 3 ) mod 8 ) )
66	36, 36	mul2negi 10478	. . . . . . . . . 10 |- ( -u 3 x. -u 3 ) = ( 3 x. 3 )
67	66	oveq1i 6660	. . . . . . . . 9 |- ( ( -u 3 x. -u 3 ) mod 8 ) = ( ( 3 x. 3 ) mod 8 )
68	65, 67	syl6eq 2672	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) )
69	68	orcd 407	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) )
70	69	ex 450	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) ) )
71	23, 41, 56, 70	ccased 988	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( ( A mod 8 ) = 3 \/ ( A mod 8 ) = 5 ) /\ ( ( B mod 8 ) = 3 \/ ( B mod 8 ) = 5 ) ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) ) )
72	5, 71	syl5bi 232	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) ) )
73	72	imp 445	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) )
74		ovex 6678	. . . 4 |- ( ( A x. B ) mod 8 ) e. _V
75	74	elpr 4198	. . 3 |- ( ( ( A x. B ) mod 8 ) e. { ( ( 3 x. 3 ) mod 8 ) , ( -u ( 3 x. 3 ) mod 8 ) } <-> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) )
76	73, 75	sylibr 224	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> ( ( A x. B ) mod 8 ) e. { ( ( 3 x. 3 ) mod 8 ) , ( -u ( 3 x. 3 ) mod 8 ) } )
77		df-9 11086	. . . . . . . 8 |- 9 = ( 8 + 1 )
78		8cn 11106	. . . . . . . . 9 |- 8 e. CC
79		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
80	78, 79	addcomi 10227	. . . . . . . 8 |- ( 8 + 1 ) = ( 1 + 8 )
81	77, 80	eqtri 2644	. . . . . . 7 |- 9 = ( 1 + 8 )
82		3t3e9 11180	. . . . . . 7 |- ( 3 x. 3 ) = 9
83	78	mulid2i 10043	. . . . . . . 8 |- ( 1 x. 8 ) = 8
84	83	oveq2i 6661	. . . . . . 7 |- ( 1 + ( 1 x. 8 ) ) = ( 1 + 8 )
85	81, 82, 84	3eqtr4i 2654	. . . . . 6 |- ( 3 x. 3 ) = ( 1 + ( 1 x. 8 ) )
86	85	oveq1i 6660	. . . . 5 |- ( ( 3 x. 3 ) mod 8 ) = ( ( 1 + ( 1 x. 8 ) ) mod 8 )
87		1re 10039	. . . . . 6 |- 1 e. RR
88		1z 11407	. . . . . 6 |- 1 e. ZZ
89		modcyc 12705	. . . . . 6 |- ( ( 1 e. RR /\ 8 e. RR+ /\ 1 e. ZZ ) -> ( ( 1 + ( 1 x. 8 ) ) mod 8 ) = ( 1 mod 8 ) )
90	87, 12, 88, 89	mp3an 1424	. . . . 5 |- ( ( 1 + ( 1 x. 8 ) ) mod 8 ) = ( 1 mod 8 )
91	86, 90	eqtri 2644	. . . 4 |- ( ( 3 x. 3 ) mod 8 ) = ( 1 mod 8 )
92	15	simpli 474	. . . . 5 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
93	92	simpli 474	. . . 4 |- ( 1 mod 8 ) = 1
94	91, 93	eqtri 2644	. . 3 |- ( ( 3 x. 3 ) mod 8 ) = 1
95		znegcl 11412	. . . . . . . 8 |- ( 1 e. ZZ -> -u 1 e. ZZ )
96	88, 95	mp1i 13	. . . . . . 7 |- ( T. -> -u 1 e. ZZ )
97		3nn 11186	. . . . . . . . . 10 |- 3 e. NN
98	97, 97	nnmulcli 11044	. . . . . . . . 9 |- ( 3 x. 3 ) e. NN
99	98	nnzi 11401	. . . . . . . 8 |- ( 3 x. 3 ) e. ZZ
100	99	a1i 11	. . . . . . 7 |- ( T. -> ( 3 x. 3 ) e. ZZ )
101	88	a1i 11	. . . . . . 7 |- ( T. -> 1 e. ZZ )
102	12	a1i 11	. . . . . . 7 |- ( T. -> 8 e. RR+ )
103		eqidd 2623	. . . . . . 7 |- ( T. -> ( -u 1 mod 8 ) = ( -u 1 mod 8 ) )
104	91	a1i 11	. . . . . . 7 |- ( T. -> ( ( 3 x. 3 ) mod 8 ) = ( 1 mod 8 ) )
105	96, 96, 100, 101, 102, 103, 104	modmul12d 12724	. . . . . 6 |- ( T. -> ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 ) )
106	105	trud 1493	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 )
107	36, 36	mulcli 10045	. . . . . . 7 |- ( 3 x. 3 ) e. CC
108	107	mulm1i 10475	. . . . . 6 |- ( -u 1 x. ( 3 x. 3 ) ) = -u ( 3 x. 3 )
109	108	oveq1i 6660	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
110	79	mulm1i 10475	. . . . . 6 |- ( -u 1 x. 1 ) = -u 1
111	110	oveq1i 6660	. . . . 5 |- ( ( -u 1 x. 1 ) mod 8 ) = ( -u 1 mod 8 )
112	106, 109, 111	3eqtr3i 2652	. . . 4 |- ( -u ( 3 x. 3 ) mod 8 ) = ( -u 1 mod 8 )
113	92	simpri 478	. . . 4 |- ( -u 1 mod 8 ) = 7
114	112, 113	eqtri 2644	. . 3 |- ( -u ( 3 x. 3 ) mod 8 ) = 7
115	94, 114	preq12i 4273	. 2 |- { ( ( 3 x. 3 ) mod 8 ) , ( -u ( 3 x. 3 ) mod 8 ) } = { 1 , 7 }
116	76, 115	syl6eleq 2711	1 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> ( ( A x. B ) mod 8 ) e. { 1 , 7 } )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   {cpr 4179  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   -ucneg 10267   3c3 11071   5c5 11073   7c7 11075   8c8 11076   9c9 11077   ZZcz 11377   RR+crp 11832   mod cmo 12668

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-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-fl 12593  df-mod 12669

This theorem is referenced by:  lgsdir2  25055