Theorem lgsdir2lem5 15451

Description: Lemma for lgsdir2 15452. (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 9203	. . . . . . . . 9 |- 8 e. NN
2		zmodcl 10487	. . . . . . . . 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 3652	. . . . . . 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 10487	. . . . . . . . 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 3652	. . . . . . 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 9400	. . . . . . . . . 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 9753	. . . . . . . . . . 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 9138	. . . . . . . . . 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 15447	. . . . . . . . . . . 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 2255	. . . . . . . . 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 2255	. . . . . . . . 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 10521	. . . . . . . 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 734	. . . . . . 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 9402	. . . . . . . . . . 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 2255	. . . . . . . . . 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 2255	. . . . . . . . . 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 10521	. . . . . . . . 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 9110	. . . . . . . . . . 11 |- 3 e. CC
46	45, 45	mulneg1i 8475	. . . . . . . . . 10 |- ( -u 3 x. 3 ) = -u ( 3 x. 3 )
47	46	oveq1i 5953	. . . . . . . . 9 |- ( ( -u 3 x. 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
48	44, 47	eqtrdi 2253	. . . . . . . 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 735	. . . . . . 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 2255	. . . . . . . . . 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 2255	. . . . . . . . . 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 10521	. . . . . . . . 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 8476	. . . . . . . . . 10 |- ( 3 x. -u 3 ) = -u ( 3 x. 3 )
63	62	oveq1i 5953	. . . . . . . . 9 |- ( ( 3 x. -u 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
64	61, 63	eqtrdi 2253	. . . . . . . 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 735	. . . . . . 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 2255	. . . . . . . . . 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 2255	. . . . . . . . . 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 10521	. . . . . . . . 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 8477	. . . . . . . . . 10 |- ( -u 3 x. -u 3 ) = ( 3 x. 3 )
78	77	oveq1i 5953	. . . . . . . . 9 |- ( ( -u 3 x. -u 3 ) mod 8 ) = ( ( 3 x. 3 ) mod 8 )
79	76, 78	eqtrdi 2253	. . . . . . . 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 734	. . . . . . 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 967	. . . . 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 9500	. . . . 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 10488	. . . 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 3652	. . . 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 9101	. . . . . . . 8 |- 9 = ( 8 + 1 )
94		8cn 9121	. . . . . . . . 9 |- 8 e. CC
95		ax-1cn 8017	. . . . . . . . 9 |- 1 e. CC
96	94, 95	addcomi 8215	. . . . . . . 8 |- ( 8 + 1 ) = ( 1 + 8 )
97	93, 96	eqtri 2225	. . . . . . 7 |- 9 = ( 1 + 8 )
98		3t3e9 9193	. . . . . . 7 |- ( 3 x. 3 ) = 9
99	94	mullidi 8074	. . . . . . . 8 |- ( 1 x. 8 ) = 8
100	99	oveq2i 5954	. . . . . . 7 |- ( 1 + ( 1 x. 8 ) ) = ( 1 + 8 )
101	97, 98, 100	3eqtr4i 2235	. . . . . 6 |- ( 3 x. 3 ) = ( 1 + ( 1 x. 8 ) )
102	101	oveq1i 5953	. . . . 5 |- ( ( 3 x. 3 ) mod 8 ) = ( ( 1 + ( 1 x. 8 ) ) mod 8 )
103		1nn 9046	. . . . . . 7 |- 1 e. NN
104		nnq 9753	. . . . . . 7 |- ( 1 e. NN -> 1 e. QQ )
105	103, 104	ax-mp 5	. . . . . 6 |- 1 e. QQ
106		1z 9397	. . . . . 6 |- 1 e. ZZ
107		modqcyc 10502	. . . . . 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 2225	. . . 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 2225	. . 3 |- ( ( 3 x. 3 ) mod 8 ) = 1
113		znegcl 9402	. . . . . . . 8 |- ( 1 e. ZZ -> -u 1 e. ZZ )
114	106, 113	mp1i 10	. . . . . . 7 |- ( T. -> -u 1 e. ZZ )
115		3nn 9198	. . . . . . . . . 10 |- 3 e. NN
116	115, 115	nnmulcli 9057	. . . . . . . . 9 |- ( 3 x. 3 ) e. NN
117	116	nnzi 9392	. . . . . . . 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 2205	. . . . . . 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 10521	. . . . . 6 |- ( T. -> ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 ) )
125	124	mptru 1381	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 )
126	45, 45	mulcli 8076	. . . . . . 7 |- ( 3 x. 3 ) e. CC
127	126	mulm1i 8474	. . . . . 6 |- ( -u 1 x. ( 3 x. 3 ) ) = -u ( 3 x. 3 )
128	127	oveq1i 5953	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
129	95	mulm1i 8474	. . . . . 6 |- ( -u 1 x. 1 ) = -u 1
130	129	oveq1i 5953	. . . . 5 |- ( ( -u 1 x. 1 ) mod 8 ) = ( -u 1 mod 8 )
131	125, 128, 130	3eqtr3i 2233	. . . 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 2225	. . 3 |- ( -u ( 3 x. 3 ) mod 8 ) = 7
134	112, 133	preq12i 3714	. 2 |- { ( ( 3 x. 3 ) mod 8 ) , ( -u ( 3 x. 3 ) mod 8 ) } = { 1 , 7 }
135	92, 134	eleqtrdi 2297	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 709   = wceq 1372   T. wtru 1373   e. wcel 2175   {cpr 3633   class class class wbr 4043  (class class class)co 5943   0cc0 7924   1c1 7925   + caddc 7927   x. cmul 7929   < clt 8106   -ucneg 8243   NNcn 9035   3c3 9087   5c5 9089   7c7 9091   8c8 9092   9c9 9093   NN0cn0 9294   ZZcz 9371   QQcq 9739   mod cmo 10465

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-po 4342  df-iso 4343  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-6 9098  df-7 9099  df-8 9100  df-9 9101  df-n0 9295  df-z 9372  df-q 9740  df-rp 9775  df-fl 10411  df-mod 10466

This theorem is referenced by:  lgsdir2  15452