Theorem lgsdir2lem5 16151

Description: Lemma for lgsdir2 16152. (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 9472	. . . . . . . . 9 |- 8 e. NN
2		zmodcl 10781	. . . . . . . . 9 |- ( ( A e. ZZ /\ 8 e. NN ) -> ( A mod 8 ) e. NN0 )
3	1, 2	mpan2 429	. . . . . . . 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 3729	. . . . . . 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 10781	. . . . . . . . 9 |- ( ( B e. ZZ /\ 8 e. NN ) -> ( B mod 8 ) e. NN0 )
8	1, 7	mpan2 429	. . . . . . . 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 3729	. . . . . . 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 477	. . . . 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 531	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> A e. ZZ )
14		3z 9673	. . . . . . . . . 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 533	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> B e. ZZ )
17		nnq 10033	. . . . . . . . . . 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 9407	. . . . . . . . . 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 535	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = 3 )
23		lgsdir2lem1 16147	. . . . . . . . . . . 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 2289	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = ( 3 mod 8 ) )
27		simprr 537	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = 3 )
28	27, 25	eqtr4di 2289	. . . . . . . . 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 10815	. . . . . . . 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 745	. . . . . . 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 531	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> A e. ZZ )
33		znegcl 9675	. . . . . . . . . . 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 533	. . . . . . . . . 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 535	. . . . . . . . . . 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 2289	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( A mod 8 ) = ( -u 3 mod 8 ) )
42		simprr 537	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 3 ) ) -> ( B mod 8 ) = 3 )
43	42, 25	eqtr4di 2289	. . . . . . . . . 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 10815	. . . . . . . . 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 9379	. . . . . . . . . . 11 |- 3 e. CC
46	45, 45	mulneg1i 8731	. . . . . . . . . 10 |- ( -u 3 x. 3 ) = -u ( 3 x. 3 )
47	46	oveq1i 6095	. . . . . . . . 9 |- ( ( -u 3 x. 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
48	44, 47	eqtrdi 2287	. . . . . . . 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 746	. . . . . . 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 531	. . . . . . . . . 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 533	. . . . . . . . . 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 535	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = 3 )
58	57, 25	eqtr4di 2289	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = ( 3 mod 8 ) )
59		simprr 537	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 3 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = 5 )
60	59, 40	eqtr4di 2289	. . . . . . . . . 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 10815	. . . . . . . . 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 8732	. . . . . . . . . 10 |- ( 3 x. -u 3 ) = -u ( 3 x. 3 )
63	62	oveq1i 6095	. . . . . . . . 9 |- ( ( 3 x. -u 3 ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
64	61, 63	eqtrdi 2287	. . . . . . . 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 746	. . . . . . 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 531	. . . . . . . . . 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 533	. . . . . . . . . 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 535	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = 5 )
73	72, 40	eqtr4di 2289	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( A mod 8 ) = ( -u 3 mod 8 ) )
74		simprr 537	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) = 5 /\ ( B mod 8 ) = 5 ) ) -> ( B mod 8 ) = 5 )
75	74, 40	eqtr4di 2289	. . . . . . . . . 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 10815	. . . . . . . . 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 8733	. . . . . . . . . 10 |- ( -u 3 x. -u 3 ) = ( 3 x. 3 )
78	77	oveq1i 6095	. . . . . . . . 9 |- ( ( -u 3 x. -u 3 ) mod 8 ) = ( ( 3 x. 3 ) mod 8 )
79	76, 78	eqtrdi 2287	. . . . . . . 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 745	. . . . . . 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 978	. . . . 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 531	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( ( A mod 8 ) e. { 3 , 5 } /\ ( B mod 8 ) e. { 3 , 5 } ) ) -> A e. ZZ )
86		simplr 533	. . . . . 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 9774	. . . . 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 10782	. . . 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 3729	. . . 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 9370	. . . . . . . 8 |- 9 = ( 8 + 1 )
94		8cn 9390	. . . . . . . . 9 |- 8 e. CC
95		ax-1cn 8272	. . . . . . . . 9 |- 1 e. CC
96	94, 95	addcomi 8470	. . . . . . . 8 |- ( 8 + 1 ) = ( 1 + 8 )
97	93, 96	eqtri 2259	. . . . . . 7 |- 9 = ( 1 + 8 )
98		3t3e9 9462	. . . . . . 7 |- ( 3 x. 3 ) = 9
99	94	mullidi 8329	. . . . . . . 8 |- ( 1 x. 8 ) = 8
100	99	oveq2i 6096	. . . . . . 7 |- ( 1 + ( 1 x. 8 ) ) = ( 1 + 8 )
101	97, 98, 100	3eqtr4i 2269	. . . . . 6 |- ( 3 x. 3 ) = ( 1 + ( 1 x. 8 ) )
102	101	oveq1i 6095	. . . . 5 |- ( ( 3 x. 3 ) mod 8 ) = ( ( 1 + ( 1 x. 8 ) ) mod 8 )
103		1nn 9315	. . . . . . 7 |- 1 e. NN
104		nnq 10033	. . . . . . 7 |- ( 1 e. NN -> 1 e. QQ )
105	103, 104	ax-mp 5	. . . . . 6 |- 1 e. QQ
106		1z 9670	. . . . . 6 |- 1 e. ZZ
107		modqcyc 10796	. . . . . 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 431	. . . . 5 |- ( ( 1 + ( 1 x. 8 ) ) mod 8 ) = ( 1 mod 8 )
109	102, 108	eqtri 2259	. . . 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 2259	. . 3 |- ( ( 3 x. 3 ) mod 8 ) = 1
113		znegcl 9675	. . . . . . . 8 |- ( 1 e. ZZ -> -u 1 e. ZZ )
114	106, 113	mp1i 10	. . . . . . 7 |- ( T. -> -u 1 e. ZZ )
115		3nn 9467	. . . . . . . . . 10 |- 3 e. NN
116	115, 115	nnmulcli 9326	. . . . . . . . 9 |- ( 3 x. 3 ) e. NN
117	116	nnzi 9665	. . . . . . . 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 2239	. . . . . . 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 10815	. . . . . 6 |- ( T. -> ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 ) )
125	124	mptru 1411	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( ( -u 1 x. 1 ) mod 8 )
126	45, 45	mulcli 8331	. . . . . . 7 |- ( 3 x. 3 ) e. CC
127	126	mulm1i 8730	. . . . . 6 |- ( -u 1 x. ( 3 x. 3 ) ) = -u ( 3 x. 3 )
128	127	oveq1i 6095	. . . . 5 |- ( ( -u 1 x. ( 3 x. 3 ) ) mod 8 ) = ( -u ( 3 x. 3 ) mod 8 )
129	95	mulm1i 8730	. . . . . 6 |- ( -u 1 x. 1 ) = -u 1
130	129	oveq1i 6095	. . . . 5 |- ( ( -u 1 x. 1 ) mod 8 ) = ( -u 1 mod 8 )
131	125, 128, 130	3eqtr3i 2267	. . . 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 2259	. . 3 |- ( -u ( 3 x. 3 ) mod 8 ) = 7
134	112, 133	preq12i 3793	. 2 |- { ( ( 3 x. 3 ) mod 8 ) , ( -u ( 3 x. 3 ) mod 8 ) } = { 1 , 7 }
135	92, 134	eleqtrdi 2331	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 } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   T. wtru 1403   e. wcel 2209   {cpr 3710   class class class wbr 4130  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   -ucneg 8498   NNcn 9304   3c3 9356   5c5 9358   7c7 9360   8c8 9361   9c9 9362   NN0cn0 9563   ZZcz 9644   QQcq 10019   mod cmo 10759

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-q 10020  df-rp 10055  df-fl 10705  df-mod 10760

This theorem is used by:  lgsdir2  16152