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Theorem lgsdir2lem5 14100
Description: Lemma for lgsdir2 14101. (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
StepHypRef Expression
1 8nn 9075 . . . . . . . . 9  |-  8  e.  NN
2 zmodcl 10330 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  8  e.  NN )  ->  ( A  mod  8
)  e.  NN0 )
31, 2mpan2 425 . . . . . . . 8  |-  ( A  e.  ZZ  ->  ( A  mod  8 )  e. 
NN0 )
43adantr 276 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  mod  8
)  e.  NN0 )
5 elprg 3611 . . . . . . 7  |-  ( ( A  mod  8 )  e.  NN0  ->  ( ( A  mod  8 )  e.  { 3 ,  5 }  <->  ( ( A  mod  8 )  =  3  \/  ( A  mod  8 )  =  5 ) ) )
64, 5syl 14 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  mod  8 )  e.  {
3 ,  5 }  <-> 
( ( A  mod  8 )  =  3  \/  ( A  mod  8 )  =  5 ) ) )
7 zmodcl 10330 . . . . . . . . 9  |-  ( ( B  e.  ZZ  /\  8  e.  NN )  ->  ( B  mod  8
)  e.  NN0 )
81, 7mpan2 425 . . . . . . . 8  |-  ( B  e.  ZZ  ->  ( B  mod  8 )  e. 
NN0 )
98adantl 277 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  mod  8
)  e.  NN0 )
10 elprg 3611 . . . . . . 7  |-  ( ( B  mod  8 )  e.  NN0  ->  ( ( B  mod  8 )  e.  { 3 ,  5 }  <->  ( ( B  mod  8 )  =  3  \/  ( B  mod  8 )  =  5 ) ) )
119, 10syl 14 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( B  mod  8 )  e.  {
3 ,  5 }  <-> 
( ( B  mod  8 )  =  3  \/  ( B  mod  8 )  =  5 ) ) )
126, 11anbi12d 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 9271 . . . . . . . . . 10  |-  3  e.  ZZ
1514a1i 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 9622 . . . . . . . . . . 11  |-  ( 8  e.  NN  ->  8  e.  QQ )
181, 17ax-mp 5 . . . . . . . . . 10  |-  8  e.  QQ
1918a1i 9 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
8  e.  QQ )
20 8pos 9011 . . . . . . . . . 10  |-  0  <  8
2120a1i 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 14096 . . . . . . . . . . . 12  |-  ( ( ( 1  mod  8
)  =  1  /\  ( -u 1  mod  8 )  =  7 )  /\  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 ) )
2423simpri 113 . . . . . . . . . . 11  |-  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 )
2524simpli 111 . . . . . . . . . 10  |-  ( 3  mod  8 )  =  3
2622, 25eqtr4di 2228 . . . . . . . . 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 )
2827, 25eqtr4di 2228 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
( B  mod  8
)  =  ( 3  mod  8 ) )
2913, 15, 16, 15, 19, 21, 26, 28modqmul12d 10364 . . . . . . . 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 ) )
3029orcd 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
) ) )
3130ex 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 9273 . . . . . . . . . . 11  |-  ( 3  e.  ZZ  ->  -u 3  e.  ZZ )
3414, 33mp1i 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 )
3614a1i 9 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
3  e.  ZZ )
3718a1i 9 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
8  e.  QQ )
3820a1i 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 )
4024simpri 113 . . . . . . . . . . 11  |-  ( -u
3  mod  8 )  =  5
4139, 40eqtr4di 2228 . . . . . . . . . 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 )
4342, 25eqtr4di 2228 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
( B  mod  8
)  =  ( 3  mod  8 ) )
4432, 34, 35, 36, 37, 38, 41, 43modqmul12d 10364 . . . . . . . . 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 8983 . . . . . . . . . . 11  |-  3  e.  CC
4645, 45mulneg1i 8351 . . . . . . . . . 10  |-  ( -u
3  x.  3 )  =  -u ( 3  x.  3 )
4746oveq1i 5879 . . . . . . . . 9  |-  ( (
-u 3  x.  3 )  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
4844, 47eqtrdi 2226 . . . . . . . 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 ) )
4948olcd 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
) ) )
5049ex 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 )
5214a1i 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 )
5414, 33mp1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  ->  -u 3  e.  ZZ )
5518a1i 9 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
8  e.  QQ )
5620a1i 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 )
5857, 25eqtr4di 2228 . . . . . . . . . 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 )
6059, 40eqtr4di 2228 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
( B  mod  8
)  =  ( -u
3  mod  8 ) )
6151, 52, 53, 54, 55, 56, 58, 60modqmul12d 10364 . . . . . . . . 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 ) )
6245, 45mulneg2i 8352 . . . . . . . . . 10  |-  ( 3  x.  -u 3 )  = 
-u ( 3  x.  3 )
6362oveq1i 5879 . . . . . . . . 9  |-  ( ( 3  x.  -u 3
)  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
6461, 63eqtrdi 2226 . . . . . . . 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 ) )
6564olcd 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
) ) )
6665ex 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 )
6814, 33mp1i 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 )
7018a1i 9 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
8  e.  QQ )
7120a1i 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 )
7372, 40eqtr4di 2228 . . . . . . . . . 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 )
7574, 40eqtr4di 2228 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
( B  mod  8
)  =  ( -u
3  mod  8 ) )
7667, 68, 69, 68, 70, 71, 73, 75modqmul12d 10364 . . . . . . . . 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 ) )
7745, 45mul2negi 8353 . . . . . . . . . 10  |-  ( -u
3  x.  -u 3
)  =  ( 3  x.  3 )
7877oveq1i 5879 . . . . . . . . 9  |-  ( (
-u 3  x.  -u 3
)  mod  8 )  =  ( ( 3  x.  3 )  mod  8 )
7976, 78eqtrdi 2226 . . . . . . . 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 ) )
8079orcd 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
) ) )
8180ex 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 ) ) ) )
8231, 50, 66, 81ccased 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 ) ) ) )
8312, 82sylbid 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 ) ) ) )
8483imp 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 )
8785, 86zmulcld 9370 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  e. 
{ 3 ,  5 }  /\  ( B  mod  8 )  e. 
{ 3 ,  5 } ) )  -> 
( A  x.  B
)  e.  ZZ )
881a1i 9 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  e. 
{ 3 ,  5 }  /\  ( B  mod  8 )  e. 
{ 3 ,  5 } ) )  -> 
8  e.  NN )
8987, 88zmodcld 10331 . . . 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 3611 . . . 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 ) ) ) )
9189, 90syl 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 ) ) ) )
9284, 91mpbird 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 8974 . . . . . . . 8  |-  9  =  ( 8  +  1 )
94 8cn 8994 . . . . . . . . 9  |-  8  e.  CC
95 ax-1cn 7895 . . . . . . . . 9  |-  1  e.  CC
9694, 95addcomi 8091 . . . . . . . 8  |-  ( 8  +  1 )  =  ( 1  +  8 )
9793, 96eqtri 2198 . . . . . . 7  |-  9  =  ( 1  +  8 )
98 3t3e9 9065 . . . . . . 7  |-  ( 3  x.  3 )  =  9
9994mulid2i 7951 . . . . . . . 8  |-  ( 1  x.  8 )  =  8
10099oveq2i 5880 . . . . . . 7  |-  ( 1  +  ( 1  x.  8 ) )  =  ( 1  +  8 )
10197, 98, 1003eqtr4i 2208 . . . . . 6  |-  ( 3  x.  3 )  =  ( 1  +  ( 1  x.  8 ) )
102101oveq1i 5879 . . . . 5  |-  ( ( 3  x.  3 )  mod  8 )  =  ( ( 1  +  ( 1  x.  8 ) )  mod  8
)
103 1nn 8919 . . . . . . 7  |-  1  e.  NN
104 nnq 9622 . . . . . . 7  |-  ( 1  e.  NN  ->  1  e.  QQ )
105103, 104ax-mp 5 . . . . . 6  |-  1  e.  QQ
106 1z 9268 . . . . . 6  |-  1  e.  ZZ
107 modqcyc 10345 . . . . . 6  |-  ( ( ( 1  e.  QQ  /\  1  e.  ZZ )  /\  ( 8  e.  QQ  /\  0  <  8 ) )  -> 
( ( 1  +  ( 1  x.  8 ) )  mod  8
)  =  ( 1  mod  8 ) )
108105, 106, 18, 20, 107mp4an 427 . . . . 5  |-  ( ( 1  +  ( 1  x.  8 ) )  mod  8 )  =  ( 1  mod  8
)
109102, 108eqtri 2198 . . . 4  |-  ( ( 3  x.  3 )  mod  8 )  =  ( 1  mod  8
)
11023simpli 111 . . . . 5  |-  ( ( 1  mod  8 )  =  1  /\  ( -u 1  mod  8 )  =  7 )
111110simpli 111 . . . 4  |-  ( 1  mod  8 )  =  1
112109, 111eqtri 2198 . . 3  |-  ( ( 3  x.  3 )  mod  8 )  =  1
113 znegcl 9273 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
114106, 113mp1i 10 . . . . . . 7  |-  ( T. 
->  -u 1  e.  ZZ )
115 3nn 9070 . . . . . . . . . 10  |-  3  e.  NN
116115, 115nnmulcli 8930 . . . . . . . . 9  |-  ( 3  x.  3 )  e.  NN
117116nnzi 9263 . . . . . . . 8  |-  ( 3  x.  3 )  e.  ZZ
118117a1i 9 . . . . . . 7  |-  ( T. 
->  ( 3  x.  3 )  e.  ZZ )
119106a1i 9 . . . . . . 7  |-  ( T. 
->  1  e.  ZZ )
12018a1i 9 . . . . . . 7  |-  ( T. 
->  8  e.  QQ )
12120a1i 9 . . . . . . 7  |-  ( T. 
->  0  <  8
)
122 eqidd 2178 . . . . . . 7  |-  ( T. 
->  ( -u 1  mod  8 )  =  (
-u 1  mod  8
) )
123109a1i 9 . . . . . . 7  |-  ( T. 
->  ( ( 3  x.  3 )  mod  8
)  =  ( 1  mod  8 ) )
124114, 114, 118, 119, 120, 121, 122, 123modqmul12d 10364 . . . . . 6  |-  ( T. 
->  ( ( -u 1  x.  ( 3  x.  3 ) )  mod  8
)  =  ( (
-u 1  x.  1 )  mod  8 ) )
125124mptru 1362 . . . . 5  |-  ( (
-u 1  x.  (
3  x.  3 ) )  mod  8 )  =  ( ( -u
1  x.  1 )  mod  8 )
12645, 45mulcli 7953 . . . . . . 7  |-  ( 3  x.  3 )  e.  CC
127126mulm1i 8350 . . . . . 6  |-  ( -u
1  x.  ( 3  x.  3 ) )  =  -u ( 3  x.  3 )
128127oveq1i 5879 . . . . 5  |-  ( (
-u 1  x.  (
3  x.  3 ) )  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
12995mulm1i 8350 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
130129oveq1i 5879 . . . . 5  |-  ( (
-u 1  x.  1 )  mod  8 )  =  ( -u 1  mod  8 )
131125, 128, 1303eqtr3i 2206 . . . 4  |-  ( -u ( 3  x.  3 )  mod  8 )  =  ( -u 1  mod  8 )
132110simpri 113 . . . 4  |-  ( -u
1  mod  8 )  =  7
133131, 132eqtri 2198 . . 3  |-  ( -u ( 3  x.  3 )  mod  8 )  =  7
134112, 133preq12i 3673 . 2  |-  { ( ( 3  x.  3 )  mod  8 ) ,  ( -u (
3  x.  3 )  mod  8 ) }  =  { 1 ,  7 }
13592, 134eleqtrdi 2270 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 } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353   T. wtru 1354    e. wcel 2148   {cpr 3592   class class class wbr 4000  (class class class)co 5869   0cc0 7802   1c1 7803    + caddc 7805    x. cmul 7807    < clt 7982   -ucneg 8119   NNcn 8908   3c3 8960   5c5 8962   7c7 8964   8c8 8965   9c9 8966   NN0cn0 9165   ZZcz 9242   QQcq 9608    mod cmo 10308
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-n0 9166  df-z 9243  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309
This theorem is referenced by:  lgsdir2  14101
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