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Theorem lgsdir2lem5 21099
Description: Lemma for lgsdir2 21100. (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 ovex 6097 . . . . . . 7  |-  ( A  mod  8 )  e. 
_V
21elpr 3824 . . . . . 6  |-  ( ( A  mod  8 )  e.  { 3 ,  5 }  <->  ( ( A  mod  8 )  =  3  \/  ( A  mod  8 )  =  5 ) )
3 ovex 6097 . . . . . . 7  |-  ( B  mod  8 )  e. 
_V
43elpr 3824 . . . . . 6  |-  ( ( B  mod  8 )  e.  { 3 ,  5 }  <->  ( ( B  mod  8 )  =  3  \/  ( B  mod  8 )  =  5 ) )
52, 4anbi12i 679 . . . . 5  |-  ( ( ( 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 ) ) )
6 simpll 731 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  ->  A  e.  ZZ )
7 3nn 10123 . . . . . . . . . . 11  |-  3  e.  NN
87nnzi 10294 . . . . . . . . . 10  |-  3  e.  ZZ
98a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
3  e.  ZZ )
10 simplr 732 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  ->  B  e.  ZZ )
11 8re 10067 . . . . . . . . . . 11  |-  8  e.  RR
12 8pos 10079 . . . . . . . . . . 11  |-  0  <  8
1311, 12elrpii 10604 . . . . . . . . . 10  |-  8  e.  RR+
1413a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
8  e.  RR+ )
15 simprl 733 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
( A  mod  8
)  =  3 )
16 lgsdir2lem1 21095 . . . . . . . . . . . 12  |-  ( ( ( 1  mod  8
)  =  1  /\  ( -u 1  mod  8 )  =  7 )  /\  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 ) )
1716simpri 449 . . . . . . . . . . 11  |-  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 )
1817simpli 445 . . . . . . . . . 10  |-  ( 3  mod  8 )  =  3
1915, 18syl6eqr 2485 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
( A  mod  8
)  =  ( 3  mod  8 ) )
20 simprr 734 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
( B  mod  8
)  =  3 )
2120, 18syl6eqr 2485 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
( B  mod  8
)  =  ( 3  mod  8 ) )
226, 9, 10, 9, 14, 19, 21modmul12d 11268 . . . . . . . 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 ) )
2322orcd 382 . . . . . . 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
) ) )
2423ex 424 . . . . . 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 ) ) ) )
25 simpll 731 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  ->  A  e.  ZZ )
26 znegcl 10302 . . . . . . . . . . 11  |-  ( 3  e.  ZZ  ->  -u 3  e.  ZZ )
278, 26mp1i 12 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  ->  -u 3  e.  ZZ )
28 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  ->  B  e.  ZZ )
298a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
3  e.  ZZ )
3013a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
8  e.  RR+ )
31 simprl 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
( A  mod  8
)  =  5 )
3217simpri 449 . . . . . . . . . . 11  |-  ( -u
3  mod  8 )  =  5
3331, 32syl6eqr 2485 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
( A  mod  8
)  =  ( -u
3  mod  8 ) )
34 simprr 734 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
( B  mod  8
)  =  3 )
3534, 18syl6eqr 2485 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
( B  mod  8
)  =  ( 3  mod  8 ) )
3625, 27, 28, 29, 30, 33, 35modmul12d 11268 . . . . . . . . 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 ) )
37 3cn 10061 . . . . . . . . . . 11  |-  3  e.  CC
3837, 37mulneg1i 9468 . . . . . . . . . 10  |-  ( -u
3  x.  3 )  =  -u ( 3  x.  3 )
3938oveq1i 6082 . . . . . . . . 9  |-  ( (
-u 3  x.  3 )  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
4036, 39syl6eq 2483 . . . . . . . 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 ) )
4140olcd 383 . . . . . . 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
) ) )
4241ex 424 . . . . . 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 ) ) ) )
43 simpll 731 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  ->  A  e.  ZZ )
448a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
3  e.  ZZ )
45 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  ->  B  e.  ZZ )
468, 26mp1i 12 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  ->  -u 3  e.  ZZ )
4713a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
8  e.  RR+ )
48 simprl 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
( A  mod  8
)  =  3 )
4948, 18syl6eqr 2485 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
( A  mod  8
)  =  ( 3  mod  8 ) )
50 simprr 734 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
( B  mod  8
)  =  5 )
5150, 32syl6eqr 2485 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
( B  mod  8
)  =  ( -u
3  mod  8 ) )
5243, 44, 45, 46, 47, 49, 51modmul12d 11268 . . . . . . . . 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 ) )
5337, 37mulneg2i 9469 . . . . . . . . . 10  |-  ( 3  x.  -u 3 )  = 
-u ( 3  x.  3 )
5453oveq1i 6082 . . . . . . . . 9  |-  ( ( 3  x.  -u 3
)  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
5552, 54syl6eq 2483 . . . . . . . 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 ) )
5655olcd 383 . . . . . . 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
) ) )
5756ex 424 . . . . . 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 ) ) ) )
58 simpll 731 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  ->  A  e.  ZZ )
598, 26mp1i 12 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  ->  -u 3  e.  ZZ )
60 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  ->  B  e.  ZZ )
6113a1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
8  e.  RR+ )
62 simprl 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
( A  mod  8
)  =  5 )
6362, 32syl6eqr 2485 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
( A  mod  8
)  =  ( -u
3  mod  8 ) )
64 simprr 734 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
( B  mod  8
)  =  5 )
6564, 32syl6eqr 2485 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
( B  mod  8
)  =  ( -u
3  mod  8 ) )
6658, 59, 60, 59, 61, 63, 65modmul12d 11268 . . . . . . . . 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 ) )
6737, 37mul2negi 9470 . . . . . . . . . 10  |-  ( -u
3  x.  -u 3
)  =  ( 3  x.  3 )
6867oveq1i 6082 . . . . . . . . 9  |-  ( (
-u 3  x.  -u 3
)  mod  8 )  =  ( ( 3  x.  3 )  mod  8 )
6966, 68syl6eq 2483 . . . . . . . 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 ) )
7069orcd 382 . . . . . . 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
) ) )
7170ex 424 . . . . . 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 ) ) ) )
7224, 42, 57, 71ccased 914 . . . . 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 ) ) ) )
735, 72syl5bi 209 . . . 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 ) ) ) )
7473imp 419 . . 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
) ) )
75 ovex 6097 . . . 4  |-  ( ( A  x.  B )  mod  8 )  e. 
_V
7675elpr 3824 . . 3  |-  ( ( ( 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 ) ) )
7774, 76sylibr 204 . 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 ) } )
78 df-9 10054 . . . . . . . 8  |-  9  =  ( 8  +  1 )
7911recni 9091 . . . . . . . . 9  |-  8  e.  CC
80 ax-1cn 9037 . . . . . . . . 9  |-  1  e.  CC
8179, 80addcomi 9246 . . . . . . . 8  |-  ( 8  +  1 )  =  ( 1  +  8 )
8278, 81eqtri 2455 . . . . . . 7  |-  9  =  ( 1  +  8 )
83 3t3e9 10118 . . . . . . 7  |-  ( 3  x.  3 )  =  9
8479mulid2i 9082 . . . . . . . 8  |-  ( 1  x.  8 )  =  8
8584oveq2i 6083 . . . . . . 7  |-  ( 1  +  ( 1  x.  8 ) )  =  ( 1  +  8 )
8682, 83, 853eqtr4i 2465 . . . . . 6  |-  ( 3  x.  3 )  =  ( 1  +  ( 1  x.  8 ) )
8786oveq1i 6082 . . . . 5  |-  ( ( 3  x.  3 )  mod  8 )  =  ( ( 1  +  ( 1  x.  8 ) )  mod  8
)
88 1re 9079 . . . . . 6  |-  1  e.  RR
89 1z 10300 . . . . . 6  |-  1  e.  ZZ
90 modcyc 11264 . . . . . 6  |-  ( ( 1  e.  RR  /\  8  e.  RR+  /\  1  e.  ZZ )  ->  (
( 1  +  ( 1  x.  8 ) )  mod  8 )  =  ( 1  mod  8 ) )
9188, 13, 89, 90mp3an 1279 . . . . 5  |-  ( ( 1  +  ( 1  x.  8 ) )  mod  8 )  =  ( 1  mod  8
)
9287, 91eqtri 2455 . . . 4  |-  ( ( 3  x.  3 )  mod  8 )  =  ( 1  mod  8
)
9316simpli 445 . . . . 5  |-  ( ( 1  mod  8 )  =  1  /\  ( -u 1  mod  8 )  =  7 )
9493simpli 445 . . . 4  |-  ( 1  mod  8 )  =  1
9592, 94eqtri 2455 . . 3  |-  ( ( 3  x.  3 )  mod  8 )  =  1
96 znegcl 10302 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
9789, 96mp1i 12 . . . . . . 7  |-  (  T. 
->  -u 1  e.  ZZ )
987, 7nnmulcli 10013 . . . . . . . . 9  |-  ( 3  x.  3 )  e.  NN
9998nnzi 10294 . . . . . . . 8  |-  ( 3  x.  3 )  e.  ZZ
10099a1i 11 . . . . . . 7  |-  (  T. 
->  ( 3  x.  3 )  e.  ZZ )
10189a1i 11 . . . . . . 7  |-  (  T. 
->  1  e.  ZZ )
10213a1i 11 . . . . . . 7  |-  (  T. 
->  8  e.  RR+ )
103 eqidd 2436 . . . . . . 7  |-  (  T. 
->  ( -u 1  mod  8 )  =  (
-u 1  mod  8
) )
10492a1i 11 . . . . . . 7  |-  (  T. 
->  ( ( 3  x.  3 )  mod  8
)  =  ( 1  mod  8 ) )
10597, 97, 100, 101, 102, 103, 104modmul12d 11268 . . . . . 6  |-  (  T. 
->  ( ( -u 1  x.  ( 3  x.  3 ) )  mod  8
)  =  ( (
-u 1  x.  1 )  mod  8 ) )
106105trud 1332 . . . . 5  |-  ( (
-u 1  x.  (
3  x.  3 ) )  mod  8 )  =  ( ( -u
1  x.  1 )  mod  8 )
10737, 37mulcli 9084 . . . . . . 7  |-  ( 3  x.  3 )  e.  CC
108107mulm1i 9467 . . . . . 6  |-  ( -u
1  x.  ( 3  x.  3 ) )  =  -u ( 3  x.  3 )
109108oveq1i 6082 . . . . 5  |-  ( (
-u 1  x.  (
3  x.  3 ) )  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
11080mulm1i 9467 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
111110oveq1i 6082 . . . . 5  |-  ( (
-u 1  x.  1 )  mod  8 )  =  ( -u 1  mod  8 )
112106, 109, 1113eqtr3i 2463 . . . 4  |-  ( -u ( 3  x.  3 )  mod  8 )  =  ( -u 1  mod  8 )
11393simpri 449 . . . 4  |-  ( -u
1  mod  8 )  =  7
114112, 113eqtri 2455 . . 3  |-  ( -u ( 3  x.  3 )  mod  8 )  =  7
11595, 114preq12i 3880 . 2  |-  { ( ( 3  x.  3 )  mod  8 ) ,  ( -u (
3  x.  3 )  mod  8 ) }  =  { 1 ,  7 }
11677, 115syl6eleq 2525 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    \/ wo 358    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725   {cpr 3807  (class class class)co 6072   RRcr 8978   1c1 8980    + caddc 8982    x. cmul 8984   -ucneg 9281   3c3 10039   5c5 10041   7c7 10043   8c8 10044   9c9 10045   ZZcz 10271   RR+crp 10601    mod cmo 11238
This theorem is referenced by:  lgsdir2  21100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-fl 11190  df-mod 11239
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