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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
StepHypRef Expression
1 8nn 9059 . . . . . . . . 9  |-  8  e.  NN
2 zmodcl 10314 . . . . . . . . 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 3609 . . . . . . 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 10314 . . . . . . . . 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 3609 . . . . . . 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 9255 . . . . . . . . . 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 9606 . . . . . . . . . . 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 8995 . . . . . . . . . 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 14000 . . . . . . . . . . . 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 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 )
2827, 25eqtr4di 2226 . . . . . . . . 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 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 ) )
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 9257 . . . . . . . . . . 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 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 )
4342, 25eqtr4di 2226 . . . . . . . . . 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 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
4645, 45mulneg1i 8335 . . . . . . . . . 10  |-  ( -u
3  x.  3 )  =  -u ( 3  x.  3 )
4746oveq1i 5875 . . . . . . . . 9  |-  ( (
-u 3  x.  3 )  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
4844, 47eqtrdi 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 ) )
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 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 )
6059, 40eqtr4di 2226 . . . . . . . . . 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 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 ) )
6245, 45mulneg2i 8336 . . . . . . . . . 10  |-  ( 3  x.  -u 3 )  = 
-u ( 3  x.  3 )
6362oveq1i 5875 . . . . . . . . 9  |-  ( ( 3  x.  -u 3
)  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
6461, 63eqtrdi 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 ) )
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 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 )
7574, 40eqtr4di 2226 . . . . . . . . . 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 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 ) )
7745, 45mul2negi 8337 . . . . . . . . . 10  |-  ( -u
3  x.  -u 3
)  =  ( 3  x.  3 )
7877oveq1i 5875 . . . . . . . . 9  |-  ( (
-u 3  x.  -u 3
)  mod  8 )  =  ( ( 3  x.  3 )  mod  8 )
7976, 78eqtrdi 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 ) )
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 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 )
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 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 ) ) ) )
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 8958 . . . . . . . 8  |-  9  =  ( 8  +  1 )
94 8cn 8978 . . . . . . . . 9  |-  8  e.  CC
95 ax-1cn 7879 . . . . . . . . 9  |-  1  e.  CC
9694, 95addcomi 8075 . . . . . . . 8  |-  ( 8  +  1 )  =  ( 1  +  8 )
9793, 96eqtri 2196 . . . . . . 7  |-  9  =  ( 1  +  8 )
98 3t3e9 9049 . . . . . . 7  |-  ( 3  x.  3 )  =  9
9994mulid2i 7935 . . . . . . . 8  |-  ( 1  x.  8 )  =  8
10099oveq2i 5876 . . . . . . 7  |-  ( 1  +  ( 1  x.  8 ) )  =  ( 1  +  8 )
10197, 98, 1003eqtr4i 2206 . . . . . 6  |-  ( 3  x.  3 )  =  ( 1  +  ( 1  x.  8 ) )
102101oveq1i 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 )
105103, 104ax-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 ) )
108105, 106, 18, 20, 107mp4an 427 . . . . 5  |-  ( ( 1  +  ( 1  x.  8 ) )  mod  8 )  =  ( 1  mod  8
)
109102, 108eqtri 2196 . . . 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 2196 . . 3  |-  ( ( 3  x.  3 )  mod  8 )  =  1
113 znegcl 9257 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
114106, 113mp1i 10 . . . . . . 7  |-  ( T. 
->  -u 1  e.  ZZ )
115 3nn 9054 . . . . . . . . . 10  |-  3  e.  NN
116115, 115nnmulcli 8914 . . . . . . . . 9  |-  ( 3  x.  3 )  e.  NN
117116nnzi 9247 . . . . . . . 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 2176 . . . . . . 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 10348 . . . . . 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 7937 . . . . . . 7  |-  ( 3  x.  3 )  e.  CC
127126mulm1i 8334 . . . . . 6  |-  ( -u
1  x.  ( 3  x.  3 ) )  =  -u ( 3  x.  3 )
128127oveq1i 5875 . . . . 5  |-  ( (
-u 1  x.  (
3  x.  3 ) )  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
12995mulm1i 8334 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
130129oveq1i 5875 . . . . 5  |-  ( (
-u 1  x.  1 )  mod  8 )  =  ( -u 1  mod  8 )
131125, 128, 1303eqtr3i 2204 . . . 4  |-  ( -u ( 3  x.  3 )  mod  8 )  =  ( -u 1  mod  8 )
132110simpri 113 . . . 4  |-  ( -u
1  mod  8 )  =  7
133131, 132eqtri 2196 . . 3  |-  ( -u ( 3  x.  3 )  mod  8 )  =  7
134112, 133preq12i 3671 . 2  |-  { ( ( 3  x.  3 )  mod  8 ) ,  ( -u (
3  x.  3 )  mod  8 ) }  =  { 1 ,  7 }
13592, 134eleqtrdi 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 } )
Colors of variables: wff set class
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
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