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Theorem lgsdir2lem4 22791
Description: Lemma for lgsdir2 22793. (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 6218 . . 3  |-  ( A  mod  8 )  e. 
_V
21elpr 3996 . 2  |-  ( ( A  mod  8 )  e.  { 1 ,  7 }  <->  ( ( A  mod  8 )  =  1  \/  ( A  mod  8 )  =  7 ) )
3 zre 10754 . . . . . . 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 9505 . . . . . 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 10510 . . . . . . . 8  |-  8  e.  RR
8 8pos 10526 . . . . . . . 8  |-  0  <  8
97, 8elrpii 11098 . . . . . . 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 22788 . . . . . . . . 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 2510 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( A  mod  8 )  =  ( 1  mod  8 ) )
16 modmul1 11862 . . . . . 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 1230 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( ( 1  x.  B
)  mod  8 ) )
18 zcn 10755 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
1918ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  CC )
2019mulid2d 9508 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( 1  x.  B )  =  B )
2120oveq1d 6208 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( (
1  x.  B )  mod  8 )  =  ( B  mod  8
) )
2217, 21eqtrd 2492 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( B  mod  8 ) )
2322eleq1d 2520 . . 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 10530 . . . . . . . 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 2510 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( A  mod  8 )  =  (
-u 1  mod  8
) )
32 modmul1 11862 . . . . . . 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 1230 . . . . . 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 9900 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( -u 1  x.  B )  =  -u B )
3635oveq1d 6208 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( -u 1  x.  B )  mod  8 )  =  ( -u B  mod  8 ) )
3733, 36eqtrd 2492 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( A  x.  B )  mod  8 )  =  (
-u B  mod  8
) )
3837eleq1d 2520 . . . 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 10784 . . . . . . . 8  |-  ( B  e.  ZZ  ->  -u B  e.  ZZ )
40 oveq1 6200 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  (
x  mod  8 )  =  ( -u B  mod  8 ) )
4140eleq1d 2520 . . . . . . . . . 10  |-  ( x  =  -u B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
42 negeq 9706 . . . . . . . . . . . 12  |-  ( x  =  -u B  ->  -u x  =  -u -u B )
4342oveq1d 6208 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  ( -u x  mod  8 )  =  ( -u -u B  mod  8 ) )
4443eleq1d 2520 . . . . . . . . . 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 10755 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  x  e.  CC )
47 neg1cn 10529 . . . . . . . . . . . . . . . . . . 19  |-  -u 1  e.  CC
48 mulcom 9472 . . . . . . . . . . . . . . . . . . 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 9890 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  ( -u 1  x.  x )  =  -u x )
5149, 50eqtrd 2492 . . . . . . . . . . . . . . . . 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 6208 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  (
-u x  mod  8
) )
55 zre 10754 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  x  e.  RR )
5655adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  x  e.  RR )
57 1red 9505 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  1  e.  RR )
58 neg1z 10785 . . . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  mod  8 )  =  ( 1  mod  8 ) )
63 modmul1 11862 . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6554, 64eqtr3d 2494 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( -u x  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6647mulid2i 9493 . . . . . . . . . . . . . . 15  |-  ( 1  x.  -u 1 )  = 
-u 1
6766oveq1i 6203 . . . . . . . . . . . . . 14  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  ( -u 1  mod  8 )
6867, 30eqtri 2480 . . . . . . . . . . . . 13  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  7
6965, 68syl6eq 2508 . . . . . . . . . . . 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 6208 . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  mod  8 )  =  (
-u 1  mod  8
) )
79 modmul1 11862 . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
8172, 80eqtr3d 2494 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( -u x  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
82 neg1mulneg1e1 10643 . . . . . . . . . . . . . . 15  |-  ( -u
1  x.  -u 1
)  =  1
8382oveq1i 6203 . . . . . . . . . . . . . 14  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  ( 1  mod  8 )
8483, 14eqtri 2480 . . . . . . . . . . . . 13  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  1
8581, 84syl6eq 2508 . . . . . . . . . . . 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 6218 . . . . . . . . . . 11  |-  ( x  mod  8 )  e. 
_V
8988elpr 3996 . . . . . . . . . 10  |-  ( ( x  mod  8 )  e.  { 1 ,  7 }  <->  ( (
x  mod  8 )  =  1  \/  (
x  mod  8 )  =  7 ) )
90 ovex 6218 . . . . . . . . . . . 12  |-  ( -u x  mod  8 )  e. 
_V
9190elpr 3996 . . . . . . . . . . 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 3135 . . . . . . . 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 9814 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  -u -u B  =  B )
9897oveq1d 6208 . . . . . . . 8  |-  ( B  e.  ZZ  ->  ( -u -u B  mod  8
)  =  ( B  mod  8 ) )
9998eleq1d 2520 . . . . . . 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 6200 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  mod  8 )  =  ( B  mod  8 ) )
102101eleq1d 2520 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
103 negeq 9706 . . . . . . . . . 10  |-  ( x  =  B  ->  -u x  =  -u B )
104103oveq1d 6208 . . . . . . . . 9  |-  ( x  =  B  ->  ( -u x  mod  8 )  =  ( -u B  mod  8 ) )
105104eleq1d 2520 . . . . . . . 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 3135 . . . . . 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 1370    e. wcel 1758   {cpr 3980  (class class class)co 6193   CCcc 9384   RRcr 9385   1c1 9387    x. cmul 9391   -ucneg 9700   3c3 10476   5c5 10478   7c7 10480   8c8 10481   ZZcz 10750   RR+crp 11095    mod cmo 11818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fl 11752  df-mod 11819
This theorem is referenced by:  lgsdir2  22793
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