Theorem lgsdir2lem5 14004

Description: Lemma for lgsdir2 14005. (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		8nn 9059	. . . . . . . . 9 |- 8 e. NN
2		zmodcl 10314	. . . . . . . . 9 |- ( ( A e. ZZ /\ 8 e. NN ) -> ( A mod 8 ) e. NN0 )
3	1, 2	mpan2 425	. . . . . . . 8 |- ( A e. ZZ -> ( A mod 8 ) e. NN0 )
4	3	adantr 276	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A mod 8 ) e. NN0 )
5		elprg 3609	. . . . . . 7 |- ( ( A mod 8 ) e. NN0 -> ( ( A mod 8 ) e. { 3 , 5 } <-> ( ( A mod 8 ) = 3 \/ ( A mod 8 ) = 5 ) ) )
6	4, 5	syl 14	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A mod 8 ) e. { 3 , 5 } <-> ( ( A mod 8 ) = 3 \/ ( A mod 8 ) = 5 ) ) )
7		zmodcl 10314	. . . . . . . . 9 |- ( ( B e. ZZ /\ 8 e. NN ) -> ( B mod 8 ) e. NN0 )
8	1, 7	mpan2 425	. . . . . . . 8 |- ( B e. ZZ -> ( B mod 8 ) e. NN0 )
9	8	adantl 277	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( B mod 8 ) e. NN0 )
10		elprg 3609	. . . . . . 7 |- ( ( B mod 8 ) e. NN0 -> ( ( B mod 8 ) e. { 3 , 5 } <-> ( ( B mod 8 ) = 3 \/ ( B mod 8 ) = 5 ) ) )
11	9, 10	syl 14	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( B mod 8 ) e. { 3 , 5 } <-> ( ( B mod 8 ) = 3 \/ ( B mod 8 ) = 5 ) ) )
12	6, 11	anbi12d 473	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 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 ) ) ) )
13		simpll 527	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> A e. ZZ )
14		3z 9255	. . . . . . . . . 10 |- 3 e. ZZ
15	14	a1i 9	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> 3 e. ZZ )
16		simplr 528	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> B e. ZZ )
17		nnq 9606	. . . . . . . . . . 11 |- ( 8 e. NN -> 8 e. QQ )
18	1, 17	ax-mp 5	. . . . . . . . . 10 |- 8 e. QQ
19	18	a1i 9	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> 8 e. QQ )
20		8pos 8995	. . . . . . . . . 10 |- 0 < 8
21	20	a1i 9	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> 0 < 8 )
22		simprl 529	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = 3 )
23		lgsdir2lem1 14000	. . . . . . . . . . . 12 |- ( ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 ) /\ ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 ) )
24	23	simpri 113	. . . . . . . . . . 11 |- ( ( 3 mod 8 ) = 3 /\ ( -u 3 mod 8 ) = 5 )
25	24	simpli 111	. . . . . . . . . 10 |- ( 3 mod 8 ) = 3
26	22, 25	eqtr4di 2226	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = ( 3 mod 8 ) )
27		simprr 531	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = 3 )
28	27, 25	eqtr4di 2226	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = ( 3 mod 8 ) )
29	13, 15, 16, 15, 19, 21, 26, 28	modqmul12d 10348	. . . . . . . 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 ) )
30	29	orcd 733	. . . . . . 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 ) ) )
31	30	ex 115	. . . . . 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 ) ) ) )
32		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> A e. ZZ )
33		znegcl 9257	. . . . . . . . . . 11 |- ( 3 e. ZZ -> -u 3 e. ZZ )
34	14, 33	mp1i 10	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> -u 3 e. ZZ )
35		simplr 528	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> B e. ZZ )
36	14	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> 3 e. ZZ )
37	18	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> 8 e. QQ )
38	20	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> 0 < 8 )
39		simprl 529	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = 5 )
40	24	simpri 113	. . . . . . . . . . 11 |- ( -u 3 mod 8 ) = 5
41	39, 40	eqtr4di 2226	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = ( -u 3 mod 8 ) )
42		simprr 531	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = 3 )
43	42, 25	eqtr4di 2226	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = ( 3 mod 8 ) )
44	32, 34, 35, 36, 37, 38, 41, 43	modqmul12d 10348	. . . . . . . . 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 ) )
45		3cn 8967	. . . . . . . . . . 11 |- 3 e. CC
46	45, 45	mulneg1i 8335	. . . . . . . . . 10 |- ( -u 3 x. 3 ) = -u ( 3 x. 3 )
47	46	oveq1i 5875	. . . . . . . . 9 |- ( ( -u 3 x. 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
48	44, 47	eqtrdi 2224	. . . . . . . 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 ) )
49	48	olcd 734	. . . . . . 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 ) ) )
50	49	ex 115	. . . . . 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 ) ) ) )
51		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> A e. ZZ )
52	14	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> 3 e. ZZ )
53		simplr 528	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> B e. ZZ )
54	14, 33	mp1i 10	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> -u 3 e. ZZ )
55	18	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> 8 e. QQ )
56	20	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> 0 < 8 )
57		simprl 529	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = 3 )
58	57, 25	eqtr4di 2226	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = ( 3 mod 8 ) )
59		simprr 531	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = 5 )
60	59, 40	eqtr4di 2226	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = ( -u 3 mod 8 ) )
61	51, 52, 53, 54, 55, 56, 58, 60	modqmul12d 10348	. . . . . . . . 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 ) )
62	45, 45	mulneg2i 8336	. . . . . . . . . 10 |- ( 3 x. -u 3 ) = -u ( 3 x. 3 )
63	62	oveq1i 5875	. . . . . . . . 9 |- ( ( 3 x. -u 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
64	61, 63	eqtrdi 2224	. . . . . . . 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 ) )
65	64	olcd 734	. . . . . . 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 ) ) )
66	65	ex 115	. . . . . 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 ) ) ) )
67		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> A e. ZZ )
68	14, 33	mp1i 10	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> -u 3 e. ZZ )
69		simplr 528	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> B e. ZZ )
70	18	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> 8 e. QQ )
71	20	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> 0 < 8 )
72		simprl 529	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = 5 )
73	72, 40	eqtr4di 2226	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = ( -u 3 mod 8 ) )
74		simprr 531	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = 5 )
75	74, 40	eqtr4di 2226	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = ( -u 3 mod 8 ) )
76	67, 68, 69, 68, 70, 71, 73, 75	modqmul12d 10348	. . . . . . . . 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 ) )
77	45, 45	mul2negi 8337	. . . . . . . . . 10 |- ( -u 3 x. -u 3 ) = ( 3 x. 3 )
78	77	oveq1i 5875	. . . . . . . . 9 |- ( ( -u 3 x. -u 3 ) mod 8 ) = ( ( 3 x. 3 ) mod 8 )
79	76, 78	eqtrdi 2224	. . . . . . . 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 ) )
80	79	orcd 733	. . . . . . 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 ) ) )
81	80	ex 115	. . . . . 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 ) ) ) )
82	31, 50, 66, 81	ccased 965	. . . . 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 ) ) ) )
83	12, 82	sylbid 150	. . . 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 ) ) ) )
84	83	imp 124	. . 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 ) ) )
85		simpll 527	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> A e. ZZ )
86		simplr 528	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> B e. ZZ )
87	85, 86	zmulcld 9354	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> ( A x. B ) e. ZZ )
88	1	a1i 9	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> 8 e. NN )
89	87, 88	zmodcld 10315	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> ( ( A x. B ) mod 8 ) e. NN0 )
90		elprg 3609	. . . 4 |- ( ( ( A x. B ) mod 8 ) e. NN0 -> ( ( ( 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 ) ) ) )
91	89, 90	syl 14	. . 3 |- ( ( ( 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 ) } <-> ( ( ( A x. B ) mod 8 ) = ( ( 3 x. 3 ) mod 8 ) \/ ( ( A x. B ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 ) ) ) )
92	84, 91	mpbird 167	. 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 ) } )
93		df-9 8958	. . . . . . . 8 |- 9 = ( 8 + 1 )
94		8cn 8978	. . . . . . . . 9 |- 8 e. CC
95		ax-1cn 7879	. . . . . . . . 9 |- 1 e. CC
96	94, 95	addcomi 8075	. . . . . . . 8 |- ( 8 + 1 ) = ( 1 + 8 )
97	93, 96	eqtri 2196	. . . . . . 7 |- 9 = ( 1 + 8 )
98		3t3e9 9049	. . . . . . 7 |- ( 3 x. 3 ) = 9
99	94	mulid2i 7935	. . . . . . . 8 |- ( 1 x. 8 ) = 8
100	99	oveq2i 5876	. . . . . . 7 |- ( 1 + ( 1 x. 8 ) ) = ( 1 + 8 )
101	97, 98, 100	3eqtr4i 2206	. . . . . 6 |- ( 3 x. 3 ) = ( 1 + ( 1 x. 8 ) )
102	101	oveq1i 5875	. . . . 5 |- ( ( 3 x. 3 ) mod 8 ) = ( ( 1 + ( 1 x. 8 ) ) mod 8 )
103		1nn 8903	. . . . . . 7 |- 1 e. NN
104		nnq 9606	. . . . . . 7 |- ( 1 e. NN -> 1 e. QQ )
105	103, 104	ax-mp 5	. . . . . 6 |- 1 e. QQ
106		1z 9252	. . . . . 6 |- 1 e. ZZ
107		modqcyc 10329	. . . . . 6 |- ( ( ( 1 e. QQ /\ 1 e. ZZ ) /\ ( 8 e. QQ /\ 0 < 8 ) ) -> ( ( 1 + ( 1 x. 8 ) ) mod 8 ) = ( 1 mod 8 ) )
108	105, 106, 18, 20, 107	mp4an 427	. . . . 5 |- ( ( 1 + ( 1 x. 8 ) ) mod 8 ) = ( 1 mod 8 )
109	102, 108	eqtri 2196	. . . 4 |- ( ( 3 x. 3 ) mod 8 ) = ( 1 mod 8 )
110	23	simpli 111	. . . . 5 |- ( ( 1 mod 8 ) = 1 /\ ( -u 1 mod 8 ) = 7 )
111	110	simpli 111	. . . 4 |- ( 1 mod 8 ) = 1
112	109, 111	eqtri 2196	. . 3 |- ( ( 3 x. 3 ) mod 8 ) = 1
113		znegcl 9257	. . . . . . . 8 |- ( 1 e. ZZ -> -u 1 e. ZZ )
114	106, 113	mp1i 10	. . . . . . 7 |- ( T. -> -u 1 e. ZZ )
115		3nn 9054	. . . . . . . . . 10 |- 3 e. NN
116	115, 115	nnmulcli 8914	. . . . . . . . 9 |- ( 3 x. 3 ) e. NN
117	116	nnzi 9247	. . . . . . . 8 |- ( 3 x. 3 ) e. ZZ
118	117	a1i 9	. . . . . . 7 |- ( T. -> ( 3 x. 3 ) e. ZZ )
119	106	a1i 9	. . . . . . 7 |- ( T. -> 1 e. ZZ )
120	18	a1i 9	. . . . . . 7 |- ( T. -> 8 e. QQ )
121	20	a1i 9	. . . . . . 7 |- ( T. -> 0 < 8 )
122		eqidd 2176	. . . . . . 7 |- ( T. -> ( -u 1 mod 8 ) = ( -u 1 mod 8 ) )
123	109	a1i 9	. . . . . . 7 |- ( T. -> ( ( 3 x. 3 ) mod 8 ) = ( 1 mod 8 ) )
124	114, 114, 118, 119, 120, 121, 122, 123	modqmul12d 10348	. . . . . 6 |- ( T. -> ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 ) )
125	124	mptru 1362	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 )
126	45, 45	mulcli 7937	. . . . . . 7 |- ( 3 x. 3 ) e. CC
127	126	mulm1i 8334	. . . . . 6 |- ( -u 1 x. ( 3 x. 3 ) ) = -u ( 3 x. 3 )
128	127	oveq1i 5875	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
129	95	mulm1i 8334	. . . . . 6 |- ( -u 1 x. 1 ) = -u 1
130	129	oveq1i 5875	. . . . 5 |- ( ( -u 1 x. 1 ) mod 8 ) = ( -u 1 mod 8 )
131	125, 128, 130	3eqtr3i 2204	. . . 4 |- ( -u ( 3 x. 3 ) mod 8 ) = ( -u 1 mod 8 )
132	110	simpri 113	. . . 4 |- ( -u 1 mod 8 ) = 7
133	131, 132	eqtri 2196	. . 3 |- ( -u ( 3 x. 3 ) mod 8 ) = 7
134	112, 133	preq12i 3671	. 2 |- { ( ( 3 x. 3 ) mod 8 ) , ( -u ( 3 x. 3 ) mod 8 ) } = { 1 , 7 }
135	92, 134	eleqtrdi 2268	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   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   T. wtru 1354   e. wcel 2146   {cpr 3590   class class class wbr 3998  (class class class)co 5865   0cc0 7786   1c1 7787   + caddc 7789   x. cmul 7791   < clt 7966   -ucneg 8103   NNcn 8892   3c3 8944   5c5 8946   7c7 8948   8c8 8949   9c9 8950   NN0cn0 9149   ZZcz 9226   QQcq 9592   mod cmo 10292

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-5 8954  df-6 8955  df-7 8956  df-8 8957  df-9 8958  df-n0 9150  df-z 9227  df-q 9593  df-rp 9625  df-fl 10240  df-mod 10293

This theorem is referenced by:  lgsdir2  14005