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Theorem lgsdir2lem5 20566
Description: Lemma for lgsdir2 20567. (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 5883 . . . . . . 7  |-  ( A  mod  8 )  e. 
_V
21elpr 3658 . . . . . 6  |-  ( ( A  mod  8 )  e.  { 3 ,  5 }  <->  ( ( A  mod  8 )  =  3  \/  ( A  mod  8 )  =  5 ) )
3 ovex 5883 . . . . . . 7  |-  ( B  mod  8 )  e. 
_V
43elpr 3658 . . . . . 6  |-  ( ( B  mod  8 )  e.  { 3 ,  5 }  <->  ( ( B  mod  8 )  =  3  \/  ( B  mod  8 )  =  5 ) )
52, 4anbi12i 678 . . . . 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 730 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  ->  A  e.  ZZ )
7 3nn 9878 . . . . . . . . . . 11  |-  3  e.  NN
87nnzi 10047 . . . . . . . . . 10  |-  3  e.  ZZ
98a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
3  e.  ZZ )
10 simplr 731 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  ->  B  e.  ZZ )
11 8re 9824 . . . . . . . . . . 11  |-  8  e.  RR
12 8pos 9836 . . . . . . . . . . 11  |-  0  <  8
1311, 12elrpii 10357 . . . . . . . . . 10  |-  8  e.  RR+
1413a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
8  e.  RR+ )
15 simprl 732 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
( A  mod  8
)  =  3 )
16 lgsdir2lem1 20562 . . . . . . . . . . . 12  |-  ( ( ( 1  mod  8
)  =  1  /\  ( -u 1  mod  8 )  =  7 )  /\  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 ) )
1716simpri 448 . . . . . . . . . . 11  |-  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 )
1817simpli 444 . . . . . . . . . 10  |-  ( 3  mod  8 )  =  3
1915, 18syl6eqr 2333 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
( A  mod  8
)  =  ( 3  mod  8 ) )
20 simprr 733 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  3 ) )  -> 
( B  mod  8
)  =  3 )
2120, 18syl6eqr 2333 . . . . . . . . 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 11003 . . . . . . . 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 381 . . . . . . 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 423 . . . . . 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 730 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  ->  A  e.  ZZ )
26 znegcl 10055 . . . . . . . . . . 11  |-  ( 3  e.  ZZ  ->  -u 3  e.  ZZ )
278, 26mp1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  ->  -u 3  e.  ZZ )
28 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  ->  B  e.  ZZ )
298a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
3  e.  ZZ )
3013a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
8  e.  RR+ )
31 simprl 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
( A  mod  8
)  =  5 )
3217simpri 448 . . . . . . . . . . 11  |-  ( -u
3  mod  8 )  =  5
3331, 32syl6eqr 2333 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
( A  mod  8
)  =  ( -u
3  mod  8 ) )
34 simprr 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  3 ) )  -> 
( B  mod  8
)  =  3 )
3534, 18syl6eqr 2333 . . . . . . . . . 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 11003 . . . . . . . . 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 9818 . . . . . . . . . . 11  |-  3  e.  CC
3837, 37mulneg1i 9225 . . . . . . . . . 10  |-  ( -u
3  x.  3 )  =  -u ( 3  x.  3 )
3938oveq1i 5868 . . . . . . . . 9  |-  ( (
-u 3  x.  3 )  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
4036, 39syl6eq 2331 . . . . . . . 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 382 . . . . . . 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 423 . . . . . 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 730 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  ->  A  e.  ZZ )
448a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
3  e.  ZZ )
45 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  ->  B  e.  ZZ )
468, 26mp1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  ->  -u 3  e.  ZZ )
4713a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
8  e.  RR+ )
48 simprl 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
( A  mod  8
)  =  3 )
4948, 18syl6eqr 2333 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
( A  mod  8
)  =  ( 3  mod  8 ) )
50 simprr 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  3  /\  ( B  mod  8 )  =  5 ) )  -> 
( B  mod  8
)  =  5 )
5150, 32syl6eqr 2333 . . . . . . . . . 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 11003 . . . . . . . . 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 9226 . . . . . . . . . 10  |-  ( 3  x.  -u 3 )  = 
-u ( 3  x.  3 )
5453oveq1i 5868 . . . . . . . . 9  |-  ( ( 3  x.  -u 3
)  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
5552, 54syl6eq 2331 . . . . . . . 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 382 . . . . . . 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 423 . . . . . 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 730 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  ->  A  e.  ZZ )
598, 26mp1i 11 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  ->  -u 3  e.  ZZ )
60 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  ->  B  e.  ZZ )
6113a1i 10 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
8  e.  RR+ )
62 simprl 732 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
( A  mod  8
)  =  5 )
6362, 32syl6eqr 2333 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
( A  mod  8
)  =  ( -u
3  mod  8 ) )
64 simprr 733 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  5  /\  ( B  mod  8 )  =  5 ) )  -> 
( B  mod  8
)  =  5 )
6564, 32syl6eqr 2333 . . . . . . . . . 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 11003 . . . . . . . . 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 9227 . . . . . . . . . 10  |-  ( -u
3  x.  -u 3
)  =  ( 3  x.  3 )
6867oveq1i 5868 . . . . . . . . 9  |-  ( (
-u 3  x.  -u 3
)  mod  8 )  =  ( ( 3  x.  3 )  mod  8 )
6966, 68syl6eq 2331 . . . . . . . 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 381 . . . . . . 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 423 . . . . . 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 913 . . . . 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 208 . . . 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 418 . . 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 5883 . . . 4  |-  ( ( A  x.  B )  mod  8 )  e. 
_V
7675elpr 3658 . . 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 203 . 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 9811 . . . . . . . 8  |-  9  =  ( 8  +  1 )
7911recni 8849 . . . . . . . . 9  |-  8  e.  CC
80 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
8179, 80addcomi 9003 . . . . . . . 8  |-  ( 8  +  1 )  =  ( 1  +  8 )
8278, 81eqtri 2303 . . . . . . 7  |-  9  =  ( 1  +  8 )
83 3t3e9 9873 . . . . . . 7  |-  ( 3  x.  3 )  =  9
8479mulid2i 8840 . . . . . . . 8  |-  ( 1  x.  8 )  =  8
8584oveq2i 5869 . . . . . . 7  |-  ( 1  +  ( 1  x.  8 ) )  =  ( 1  +  8 )
8682, 83, 853eqtr4i 2313 . . . . . 6  |-  ( 3  x.  3 )  =  ( 1  +  ( 1  x.  8 ) )
8786oveq1i 5868 . . . . 5  |-  ( ( 3  x.  3 )  mod  8 )  =  ( ( 1  +  ( 1  x.  8 ) )  mod  8
)
88 1re 8837 . . . . . 6  |-  1  e.  RR
89 1z 10053 . . . . . 6  |-  1  e.  ZZ
90 modcyc 10999 . . . . . 6  |-  ( ( 1  e.  RR  /\  8  e.  RR+  /\  1  e.  ZZ )  ->  (
( 1  +  ( 1  x.  8 ) )  mod  8 )  =  ( 1  mod  8 ) )
9188, 13, 89, 90mp3an 1277 . . . . 5  |-  ( ( 1  +  ( 1  x.  8 ) )  mod  8 )  =  ( 1  mod  8
)
9287, 91eqtri 2303 . . . 4  |-  ( ( 3  x.  3 )  mod  8 )  =  ( 1  mod  8
)
9316simpli 444 . . . . 5  |-  ( ( 1  mod  8 )  =  1  /\  ( -u 1  mod  8 )  =  7 )
9493simpli 444 . . . 4  |-  ( 1  mod  8 )  =  1
9592, 94eqtri 2303 . . 3  |-  ( ( 3  x.  3 )  mod  8 )  =  1
96 znegcl 10055 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
9789, 96mp1i 11 . . . . . . 7  |-  (  T. 
->  -u 1  e.  ZZ )
987, 7nnmulcli 9770 . . . . . . . . 9  |-  ( 3  x.  3 )  e.  NN
9998nnzi 10047 . . . . . . . 8  |-  ( 3  x.  3 )  e.  ZZ
10099a1i 10 . . . . . . 7  |-  (  T. 
->  ( 3  x.  3 )  e.  ZZ )
10189a1i 10 . . . . . . 7  |-  (  T. 
->  1  e.  ZZ )
10213a1i 10 . . . . . . 7  |-  (  T. 
->  8  e.  RR+ )
103 eqidd 2284 . . . . . . 7  |-  (  T. 
->  ( -u 1  mod  8 )  =  (
-u 1  mod  8
) )
10492a1i 10 . . . . . . 7  |-  (  T. 
->  ( ( 3  x.  3 )  mod  8
)  =  ( 1  mod  8 ) )
10597, 97, 100, 101, 102, 103, 104modmul12d 11003 . . . . . 6  |-  (  T. 
->  ( ( -u 1  x.  ( 3  x.  3 ) )  mod  8
)  =  ( (
-u 1  x.  1 )  mod  8 ) )
106105trud 1314 . . . . 5  |-  ( (
-u 1  x.  (
3  x.  3 ) )  mod  8 )  =  ( ( -u
1  x.  1 )  mod  8 )
10737, 37mulcli 8842 . . . . . . 7  |-  ( 3  x.  3 )  e.  CC
108107mulm1i 9224 . . . . . 6  |-  ( -u
1  x.  ( 3  x.  3 ) )  =  -u ( 3  x.  3 )
109108oveq1i 5868 . . . . 5  |-  ( (
-u 1  x.  (
3  x.  3 ) )  mod  8 )  =  ( -u (
3  x.  3 )  mod  8 )
11080mulm1i 9224 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
111110oveq1i 5868 . . . . 5  |-  ( (
-u 1  x.  1 )  mod  8 )  =  ( -u 1  mod  8 )
112106, 109, 1113eqtr3i 2311 . . . 4  |-  ( -u ( 3  x.  3 )  mod  8 )  =  ( -u 1  mod  8 )
11393simpri 448 . . . 4  |-  ( -u
1  mod  8 )  =  7
114112, 113eqtri 2303 . . 3  |-  ( -u ( 3  x.  3 )  mod  8 )  =  7
11595, 114preq12i 3711 . 2  |-  { ( ( 3  x.  3 )  mod  8 ) ,  ( -u (
3  x.  3 )  mod  8 ) }  =  { 1 ,  7 }
11677, 115syl6eleq 2373 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 357    /\ wa 358    T. wtru 1307    = wceq 1623    e. wcel 1684   {cpr 3641  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   3c3 9796   5c5 9798   7c7 9800   8c8 9801   9c9 9802   ZZcz 10024   RR+crp 10354    mod cmo 10973
This theorem is referenced by:  lgsdir2  20567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974
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