Theorem lgsdir2lem5 15148

Description: Lemma for lgsdir2 15149. (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 9149	. . . . . . . . 9 |- 8 e. NN
2		zmodcl 10415	. . . . . . . . 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 3638	. . . . . . 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 10415	. . . . . . . . 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 3638	. . . . . . 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 9346	. . . . . . . . . 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 9698	. . . . . . . . . . 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 9085	. . . . . . . . . 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 15144	. . . . . . . . . . . 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 2244	. . . . . . . . 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 2244	. . . . . . . . 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 10449	. . . . . . . 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 9348	. . . . . . . . . . 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 2244	. . . . . . . . . 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 2244	. . . . . . . . . 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 10449	. . . . . . . . 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 9057	. . . . . . . . . . 11 |- 3 e. CC
46	45, 45	mulneg1i 8423	. . . . . . . . . 10 |- ( -u 3 x. 3 ) = -u ( 3 x. 3 )
47	46	oveq1i 5928	. . . . . . . . 9 |- ( ( -u 3 x. 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
48	44, 47	eqtrdi 2242	. . . . . . . 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 2244	. . . . . . . . . 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 2244	. . . . . . . . . 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 10449	. . . . . . . . 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 8424	. . . . . . . . . 10 |- ( 3 x. -u 3 ) = -u ( 3 x. 3 )
63	62	oveq1i 5928	. . . . . . . . 9 |- ( ( 3 x. -u 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
64	61, 63	eqtrdi 2242	. . . . . . . 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 2244	. . . . . . . . . 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 2244	. . . . . . . . . 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 10449	. . . . . . . . 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 8425	. . . . . . . . . 10 |- ( -u 3 x. -u 3 ) = ( 3 x. 3 )
78	77	oveq1i 5928	. . . . . . . . 9 |- ( ( -u 3 x. -u 3 ) mod 8 ) = ( ( 3 x. 3 ) mod 8 )
79	76, 78	eqtrdi 2242	. . . . . . . 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 9445	. . . . 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 10416	. . . 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 3638	. . . 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 9048	. . . . . . . 8 |- 9 = ( 8 + 1 )
94		8cn 9068	. . . . . . . . 9 |- 8 e. CC
95		ax-1cn 7965	. . . . . . . . 9 |- 1 e. CC
96	94, 95	addcomi 8163	. . . . . . . 8 |- ( 8 + 1 ) = ( 1 + 8 )
97	93, 96	eqtri 2214	. . . . . . 7 |- 9 = ( 1 + 8 )
98		3t3e9 9139	. . . . . . 7 |- ( 3 x. 3 ) = 9
99	94	mullidi 8022	. . . . . . . 8 |- ( 1 x. 8 ) = 8
100	99	oveq2i 5929	. . . . . . 7 |- ( 1 + ( 1 x. 8 ) ) = ( 1 + 8 )
101	97, 98, 100	3eqtr4i 2224	. . . . . 6 |- ( 3 x. 3 ) = ( 1 + ( 1 x. 8 ) )
102	101	oveq1i 5928	. . . . 5 |- ( ( 3 x. 3 ) mod 8 ) = ( ( 1 + ( 1 x. 8 ) ) mod 8 )
103		1nn 8993	. . . . . . 7 |- 1 e. NN
104		nnq 9698	. . . . . . 7 |- ( 1 e. NN -> 1 e. QQ )
105	103, 104	ax-mp 5	. . . . . 6 |- 1 e. QQ
106		1z 9343	. . . . . 6 |- 1 e. ZZ
107		modqcyc 10430	. . . . . 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 2214	. . . 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 2214	. . 3 |- ( ( 3 x. 3 ) mod 8 ) = 1
113		znegcl 9348	. . . . . . . 8 |- ( 1 e. ZZ -> -u 1 e. ZZ )
114	106, 113	mp1i 10	. . . . . . 7 |- ( T. -> -u 1 e. ZZ )
115		3nn 9144	. . . . . . . . . 10 |- 3 e. NN
116	115, 115	nnmulcli 9004	. . . . . . . . 9 |- ( 3 x. 3 ) e. NN
117	116	nnzi 9338	. . . . . . . 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 2194	. . . . . . 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 10449	. . . . . 6 |- ( T. -> ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 ) )
125	124	mptru 1373	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 )
126	45, 45	mulcli 8024	. . . . . . 7 |- ( 3 x. 3 ) e. CC
127	126	mulm1i 8422	. . . . . 6 |- ( -u 1 x. ( 3 x. 3 ) ) = -u ( 3 x. 3 )
128	127	oveq1i 5928	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
129	95	mulm1i 8422	. . . . . 6 |- ( -u 1 x. 1 ) = -u 1
130	129	oveq1i 5928	. . . . 5 |- ( ( -u 1 x. 1 ) mod 8 ) = ( -u 1 mod 8 )
131	125, 128, 130	3eqtr3i 2222	. . . 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 2214	. . 3 |- ( -u ( 3 x. 3 ) mod 8 ) = 7
134	112, 133	preq12i 3700	. 2 |- { ( ( 3 x. 3 ) mod 8 ) , ( -u ( 3 x. 3 ) mod 8 ) } = { 1 , 7 }
135	92, 134	eleqtrdi 2286	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 1364   T. wtru 1365   e. wcel 2164   {cpr 3619   class class class wbr 4029  (class class class)co 5918   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   -ucneg 8191   NNcn 8982   3c3 9034   5c5 9036   7c7 9038   8c8 9039   9c9 9040   NN0cn0 9240   ZZcz 9317   QQcq 9684   mod cmo 10393

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-n0 9241  df-z 9318  df-q 9685  df-rp 9720  df-fl 10339  df-mod 10394

This theorem is referenced by:  lgsdir2  15149