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Theorem lgsdir2lem4 22640
Description: Lemma for lgsdir2 22642. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
lgsdir2lem4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  e.  {
1 ,  7 } )  ->  ( (
( A  x.  B
)  mod  8 )  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )

Proof of Theorem lgsdir2lem4
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ovex 6111 . . 3  |-  ( A  mod  8 )  e. 
_V
21elpr 3890 . 2  |-  ( ( A  mod  8 )  e.  { 1 ,  7 }  <->  ( ( A  mod  8 )  =  1  \/  ( A  mod  8 )  =  7 ) )
3 zre 10642 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
43ad2antrr 725 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  A  e.  RR )
5 1red 9393 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  1  e.  RR )
6 simplr 754 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  ZZ )
7 8re 10398 . . . . . . . 8  |-  8  e.  RR
8 8pos 10414 . . . . . . . 8  |-  0  <  8
97, 8elrpii 10986 . . . . . . 7  |-  8  e.  RR+
109a1i 11 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  8  e.  RR+ )
11 simpr 461 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( A  mod  8 )  =  1 )
12 lgsdir2lem1 22637 . . . . . . . . 9  |-  ( ( ( 1  mod  8
)  =  1  /\  ( -u 1  mod  8 )  =  7 )  /\  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 ) )
1312simpli 458 . . . . . . . 8  |-  ( ( 1  mod  8 )  =  1  /\  ( -u 1  mod  8 )  =  7 )
1413simpli 458 . . . . . . 7  |-  ( 1  mod  8 )  =  1
1511, 14syl6eqr 2488 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( A  mod  8 )  =  ( 1  mod  8 ) )
16 modmul1 11744 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  e.  RR )  /\  ( B  e.  ZZ  /\  8  e.  RR+ )  /\  ( A  mod  8 )  =  ( 1  mod  8
) )  ->  (
( A  x.  B
)  mod  8 )  =  ( ( 1  x.  B )  mod  8 ) )
174, 5, 6, 10, 15, 16syl221anc 1229 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( ( 1  x.  B
)  mod  8 ) )
18 zcn 10643 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
1918ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  CC )
2019mulid2d 9396 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( 1  x.  B )  =  B )
2120oveq1d 6101 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( (
1  x.  B )  mod  8 )  =  ( B  mod  8
) )
2217, 21eqtrd 2470 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( B  mod  8 ) )
2322eleq1d 2504 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( (
( A  x.  B
)  mod  8 )  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
243ad2antrr 725 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  A  e.  RR )
25 neg1rr 10418 . . . . . . . 8  |-  -u 1  e.  RR
2625a1i 11 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  -u 1  e.  RR )
27 simplr 754 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  B  e.  ZZ )
289a1i 11 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  8  e.  RR+ )
29 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( A  mod  8 )  =  7 )
3013simpri 462 . . . . . . . 8  |-  ( -u
1  mod  8 )  =  7
3129, 30syl6eqr 2488 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( A  mod  8 )  =  (
-u 1  mod  8
) )
32 modmul1 11744 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  -u 1  e.  RR )  /\  ( B  e.  ZZ  /\  8  e.  RR+ )  /\  ( A  mod  8 )  =  ( -u 1  mod  8 ) )  -> 
( ( A  x.  B )  mod  8
)  =  ( (
-u 1  x.  B
)  mod  8 ) )
3324, 26, 27, 28, 31, 32syl221anc 1229 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( A  x.  B )  mod  8 )  =  ( ( -u 1  x.  B )  mod  8
) )
3418ad2antlr 726 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  B  e.  CC )
3534mulm1d 9788 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( -u 1  x.  B )  =  -u B )
3635oveq1d 6101 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( -u 1  x.  B )  mod  8 )  =  ( -u B  mod  8 ) )
3733, 36eqtrd 2470 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( A  x.  B )  mod  8 )  =  (
-u B  mod  8
) )
3837eleq1d 2504 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( (
( A  x.  B
)  mod  8 )  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
39 znegcl 10672 . . . . . . . 8  |-  ( B  e.  ZZ  ->  -u B  e.  ZZ )
40 oveq1 6093 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  (
x  mod  8 )  =  ( -u B  mod  8 ) )
4140eleq1d 2504 . . . . . . . . . 10  |-  ( x  =  -u B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
42 negeq 9594 . . . . . . . . . . . 12  |-  ( x  =  -u B  ->  -u x  =  -u -u B )
4342oveq1d 6101 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  ( -u x  mod  8 )  =  ( -u -u B  mod  8 ) )
4443eleq1d 2504 . . . . . . . . . 10  |-  ( x  =  -u B  ->  (
( -u x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u -u B  mod  8 )  e.  {
1 ,  7 } ) )
4541, 44imbi12d 320 . . . . . . . . 9  |-  ( x  =  -u B  ->  (
( ( x  mod  8 )  e.  {
1 ,  7 }  ->  ( -u x  mod  8 )  e.  {
1 ,  7 } )  <->  ( ( -u B  mod  8 )  e. 
{ 1 ,  7 }  ->  ( -u -u B  mod  8 )  e.  {
1 ,  7 } ) ) )
46 zcn 10643 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  x  e.  CC )
47 neg1cn 10417 . . . . . . . . . . . . . . . . . . 19  |-  -u 1  e.  CC
48 mulcom 9360 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  CC  /\  -u 1  e.  CC )  ->  ( x  x.  -u 1 )  =  ( -u 1  x.  x ) )
4947, 48mpan2 671 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  (
x  x.  -u 1
)  =  ( -u
1  x.  x ) )
50 mulm1 9778 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  ( -u 1  x.  x )  =  -u x )
5149, 50eqtrd 2470 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  CC  ->  (
x  x.  -u 1
)  =  -u x
)
5246, 51syl 16 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  x.  -u 1
)  =  -u x
)
5352adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  x.  -u 1 )  = 
-u x )
5453oveq1d 6101 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  (
-u x  mod  8
) )
55 zre 10642 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  x  e.  RR )
5655adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  x  e.  RR )
57 1red 9393 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  1  e.  RR )
58 neg1z 10673 . . . . . . . . . . . . . . . 16  |-  -u 1  e.  ZZ
5958a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  -u 1  e.  ZZ )
609a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  8  e.  RR+ )
61 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  mod  8 )  =  1 )
6261, 14syl6eqr 2488 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  mod  8 )  =  ( 1  mod  8 ) )
63 modmul1 11744 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  1  e.  RR )  /\  ( -u 1  e.  ZZ  /\  8  e.  RR+ )  /\  (
x  mod  8 )  =  ( 1  mod  8 ) )  -> 
( ( x  x.  -u 1 )  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6456, 57, 59, 60, 62, 63syl221anc 1229 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6554, 64eqtr3d 2472 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( -u x  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6647mulid2i 9381 . . . . . . . . . . . . . . 15  |-  ( 1  x.  -u 1 )  = 
-u 1
6766oveq1i 6096 . . . . . . . . . . . . . 14  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  ( -u 1  mod  8 )
6867, 30eqtri 2458 . . . . . . . . . . . . 13  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  7
6965, 68syl6eq 2486 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( -u x  mod  8 )  =  7 )
7069ex 434 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  =  1  -> 
( -u x  mod  8
)  =  7 ) )
7152adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  x.  -u 1 )  = 
-u x )
7271oveq1d 6101 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  (
-u x  mod  8
) )
7355adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  x  e.  RR )
7425a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  -u 1  e.  RR )
7558a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  -u 1  e.  ZZ )
769a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  8  e.  RR+ )
77 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  mod  8 )  =  7 )
7877, 30syl6eqr 2488 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  mod  8 )  =  (
-u 1  mod  8
) )
79 modmul1 11744 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  -u 1  e.  RR )  /\  ( -u 1  e.  ZZ  /\  8  e.  RR+ )  /\  (
x  mod  8 )  =  ( -u 1  mod  8 ) )  -> 
( ( x  x.  -u 1 )  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
8073, 74, 75, 76, 78, 79syl221anc 1229 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
8172, 80eqtr3d 2472 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( -u x  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
82 neg1mulneg1e1 10531 . . . . . . . . . . . . . . 15  |-  ( -u
1  x.  -u 1
)  =  1
8382oveq1i 6096 . . . . . . . . . . . . . 14  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  ( 1  mod  8 )
8483, 14eqtri 2458 . . . . . . . . . . . . 13  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  1
8581, 84syl6eq 2486 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( -u x  mod  8 )  =  1 )
8685ex 434 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  =  7  -> 
( -u x  mod  8
)  =  1 ) )
8770, 86orim12d 834 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  (
( ( x  mod  8 )  =  1  \/  ( x  mod  8 )  =  7 )  ->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) ) )
88 ovex 6111 . . . . . . . . . . 11  |-  ( x  mod  8 )  e. 
_V
8988elpr 3890 . . . . . . . . . 10  |-  ( ( x  mod  8 )  e.  { 1 ,  7 }  <->  ( (
x  mod  8 )  =  1  \/  (
x  mod  8 )  =  7 ) )
90 ovex 6111 . . . . . . . . . . . 12  |-  ( -u x  mod  8 )  e. 
_V
9190elpr 3890 . . . . . . . . . . 11  |-  ( (
-u x  mod  8
)  e.  { 1 ,  7 }  <->  ( ( -u x  mod  8 )  =  1  \/  ( -u x  mod  8 )  =  7 ) )
92 orcom 387 . . . . . . . . . . 11  |-  ( ( ( -u x  mod  8 )  =  1  \/  ( -u x  mod  8 )  =  7 )  <->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) )
9391, 92bitri 249 . . . . . . . . . 10  |-  ( (
-u x  mod  8
)  e.  { 1 ,  7 }  <->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) )
9487, 89, 933imtr4g 270 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  ->  (
-u x  mod  8
)  e.  { 1 ,  7 } ) )
9545, 94vtoclga 3031 . . . . . . . 8  |-  ( -u B  e.  ZZ  ->  ( ( -u B  mod  8 )  e.  {
1 ,  7 }  ->  ( -u -u B  mod  8 )  e.  {
1 ,  7 } ) )
9639, 95syl 16 . . . . . . 7  |-  ( B  e.  ZZ  ->  (
( -u B  mod  8
)  e.  { 1 ,  7 }  ->  (
-u -u B  mod  8
)  e.  { 1 ,  7 } ) )
9718negnegd 9702 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  -u -u B  =  B )
9897oveq1d 6101 . . . . . . . 8  |-  ( B  e.  ZZ  ->  ( -u -u B  mod  8
)  =  ( B  mod  8 ) )
9998eleq1d 2504 . . . . . . 7  |-  ( B  e.  ZZ  ->  (
( -u -u B  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
10096, 99sylibd 214 . . . . . 6  |-  ( B  e.  ZZ  ->  (
( -u B  mod  8
)  e.  { 1 ,  7 }  ->  ( B  mod  8 )  e.  { 1 ,  7 } ) )
101 oveq1 6093 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  mod  8 )  =  ( B  mod  8 ) )
102101eleq1d 2504 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
103 negeq 9594 . . . . . . . . . 10  |-  ( x  =  B  ->  -u x  =  -u B )
104103oveq1d 6101 . . . . . . . . 9  |-  ( x  =  B  ->  ( -u x  mod  8 )  =  ( -u B  mod  8 ) )
105104eleq1d 2504 . . . . . . . 8  |-  ( x  =  B  ->  (
( -u x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
106102, 105imbi12d 320 . . . . . . 7  |-  ( x  =  B  ->  (
( ( x  mod  8 )  e.  {
1 ,  7 }  ->  ( -u x  mod  8 )  e.  {
1 ,  7 } )  <->  ( ( B  mod  8 )  e. 
{ 1 ,  7 }  ->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) ) )
107106, 94vtoclga 3031 . . . . . 6  |-  ( B  e.  ZZ  ->  (
( B  mod  8
)  e.  { 1 ,  7 }  ->  (
-u B  mod  8
)  e.  { 1 ,  7 } ) )
108100, 107impbid 191 . . . . 5  |-  ( B  e.  ZZ  ->  (
( -u B  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
109108ad2antlr 726 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( -u B  mod  8 )  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
11038, 109bitrd 253 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( (
( A  x.  B
)  mod  8 )  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
11123, 110jaodan 783 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  mod  8 )  =  1  \/  ( A  mod  8 )  =  7 ) )  -> 
( ( ( A  x.  B )  mod  8 )  e.  {
1 ,  7 }  <-> 
( B  mod  8
)  e.  { 1 ,  7 } ) )
1122, 111sylan2b 475 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  e.  {
1 ,  7 } )  ->  ( (
( A  x.  B
)  mod  8 )  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   {cpr 3874  (class class class)co 6086   CCcc 9272   RRcr 9273   1c1 9275    x. cmul 9279   -ucneg 9588   3c3 10364   5c5 10366   7c7 10368   8c8 10369   ZZcz 10638   RR+crp 10983    mod cmo 11700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fl 11634  df-mod 11701
This theorem is referenced by:  lgsdir2  22642
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