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Theorem lgsdir2lem4 24191
Description: Lemma for lgsdir2 24193. (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 6272 . . 3  |-  ( A  mod  8 )  e. 
_V
21elpr 3954 . 2  |-  ( ( A  mod  8 )  e.  { 1 ,  7 }  <->  ( ( A  mod  8 )  =  1  \/  ( A  mod  8 )  =  7 ) )
3 zre 10887 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
43ad2antrr 730 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  A  e.  RR )
5 1red 9604 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  1  e.  RR )
6 simplr 760 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  ZZ )
7 8re 10640 . . . . . . . 8  |-  8  e.  RR
8 8pos 10656 . . . . . . . 8  |-  0  <  8
97, 8elrpii 11251 . . . . . . 7  |-  8  e.  RR+
109a1i 11 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  8  e.  RR+ )
11 simpr 462 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( A  mod  8 )  =  1 )
12 lgsdir2lem1 24188 . . . . . . . . 9  |-  ( ( ( 1  mod  8
)  =  1  /\  ( -u 1  mod  8 )  =  7 )  /\  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 ) )
1312simpli 459 . . . . . . . 8  |-  ( ( 1  mod  8 )  =  1  /\  ( -u 1  mod  8 )  =  7 )
1413simpli 459 . . . . . . 7  |-  ( 1  mod  8 )  =  1
1511, 14syl6eqr 2475 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( A  mod  8 )  =  ( 1  mod  8 ) )
16 modmul1 12088 . . . . . 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 1275 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( ( 1  x.  B
)  mod  8 ) )
18 zcn 10888 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
1918ad2antlr 731 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  CC )
2019mulid2d 9607 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( 1  x.  B )  =  B )
2120oveq1d 6259 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( (
1  x.  B )  mod  8 )  =  ( B  mod  8
) )
2217, 21eqtrd 2457 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( B  mod  8 ) )
2322eleq1d 2485 . . 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 730 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  A  e.  RR )
25 neg1rr 10660 . . . . . . . 8  |-  -u 1  e.  RR
2625a1i 11 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  -u 1  e.  RR )
27 simplr 760 . . . . . . 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 462 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( A  mod  8 )  =  7 )
3013simpri 463 . . . . . . . 8  |-  ( -u
1  mod  8 )  =  7
3129, 30syl6eqr 2475 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( A  mod  8 )  =  (
-u 1  mod  8
) )
32 modmul1 12088 . . . . . . 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 1275 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( A  x.  B )  mod  8 )  =  ( ( -u 1  x.  B )  mod  8
) )
3418ad2antlr 731 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  B  e.  CC )
3534mulm1d 10016 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( -u 1  x.  B )  =  -u B )
3635oveq1d 6259 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( -u 1  x.  B )  mod  8 )  =  ( -u B  mod  8 ) )
3733, 36eqtrd 2457 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( A  x.  B )  mod  8 )  =  (
-u B  mod  8
) )
3837eleq1d 2485 . . . 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 10918 . . . . . . . 8  |-  ( B  e.  ZZ  ->  -u B  e.  ZZ )
40 oveq1 6251 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  (
x  mod  8 )  =  ( -u B  mod  8 ) )
4140eleq1d 2485 . . . . . . . . . 10  |-  ( x  =  -u B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
42 negeq 9813 . . . . . . . . . . . 12  |-  ( x  =  -u B  ->  -u x  =  -u -u B )
4342oveq1d 6259 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  ( -u x  mod  8 )  =  ( -u -u B  mod  8 ) )
4443eleq1d 2485 . . . . . . . . . 10  |-  ( x  =  -u B  ->  (
( -u x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u -u B  mod  8 )  e.  {
1 ,  7 } ) )
4541, 44imbi12d 321 . . . . . . . . 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 10888 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  x  e.  CC )
47 neg1cn 10659 . . . . . . . . . . . . . . . . . . 19  |-  -u 1  e.  CC
48 mulcom 9571 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  CC  /\  -u 1  e.  CC )  ->  ( x  x.  -u 1 )  =  ( -u 1  x.  x ) )
4947, 48mpan2 675 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  (
x  x.  -u 1
)  =  ( -u
1  x.  x ) )
50 mulm1 10006 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  ( -u 1  x.  x )  =  -u x )
5149, 50eqtrd 2457 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  CC  ->  (
x  x.  -u 1
)  =  -u x
)
5246, 51syl 17 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  x.  -u 1
)  =  -u x
)
5352adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  x.  -u 1 )  = 
-u x )
5453oveq1d 6259 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  (
-u x  mod  8
) )
55 zre 10887 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  x  e.  RR )
5655adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  x  e.  RR )
57 1red 9604 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  1  e.  RR )
58 neg1z 10919 . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  mod  8 )  =  1 )
6261, 14syl6eqr 2475 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  mod  8 )  =  ( 1  mod  8 ) )
63 modmul1 12088 . . . . . . . . . . . . . . 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 1275 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6554, 64eqtr3d 2459 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( -u x  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6647mulid2i 9592 . . . . . . . . . . . . . . 15  |-  ( 1  x.  -u 1 )  = 
-u 1
6766oveq1i 6254 . . . . . . . . . . . . . 14  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  ( -u 1  mod  8 )
6867, 30eqtri 2445 . . . . . . . . . . . . 13  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  7
6965, 68syl6eq 2473 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( -u x  mod  8 )  =  7 )
7069ex 435 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  =  1  -> 
( -u x  mod  8
)  =  7 ) )
7152adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  x.  -u 1 )  = 
-u x )
7271oveq1d 6259 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  (
-u x  mod  8
) )
7355adantr 466 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  mod  8 )  =  7 )
7877, 30syl6eqr 2475 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  mod  8 )  =  (
-u 1  mod  8
) )
79 modmul1 12088 . . . . . . . . . . . . . . 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 1275 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
8172, 80eqtr3d 2459 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( -u x  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
82 neg1mulneg1e1 10773 . . . . . . . . . . . . . . 15  |-  ( -u
1  x.  -u 1
)  =  1
8382oveq1i 6254 . . . . . . . . . . . . . 14  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  ( 1  mod  8 )
8483, 14eqtri 2445 . . . . . . . . . . . . 13  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  1
8581, 84syl6eq 2473 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( -u x  mod  8 )  =  1 )
8685ex 435 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  =  7  -> 
( -u x  mod  8
)  =  1 ) )
8770, 86orim12d 846 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  (
( ( x  mod  8 )  =  1  \/  ( x  mod  8 )  =  7 )  ->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) ) )
88 ovex 6272 . . . . . . . . . . 11  |-  ( x  mod  8 )  e. 
_V
8988elpr 3954 . . . . . . . . . 10  |-  ( ( x  mod  8 )  e.  { 1 ,  7 }  <->  ( (
x  mod  8 )  =  1  \/  (
x  mod  8 )  =  7 ) )
90 ovex 6272 . . . . . . . . . . . 12  |-  ( -u x  mod  8 )  e. 
_V
9190elpr 3954 . . . . . . . . . . 11  |-  ( (
-u x  mod  8
)  e.  { 1 ,  7 }  <->  ( ( -u x  mod  8 )  =  1  \/  ( -u x  mod  8 )  =  7 ) )
92 orcom 388 . . . . . . . . . . 11  |-  ( ( ( -u x  mod  8 )  =  1  \/  ( -u x  mod  8 )  =  7 )  <->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) )
9391, 92bitri 252 . . . . . . . . . 10  |-  ( (
-u x  mod  8
)  e.  { 1 ,  7 }  <->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) )
9487, 89, 933imtr4g 273 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  ->  (
-u x  mod  8
)  e.  { 1 ,  7 } ) )
9545, 94vtoclga 3083 . . . . . . . 8  |-  ( -u B  e.  ZZ  ->  ( ( -u B  mod  8 )  e.  {
1 ,  7 }  ->  ( -u -u B  mod  8 )  e.  {
1 ,  7 } ) )
9639, 95syl 17 . . . . . . 7  |-  ( B  e.  ZZ  ->  (
( -u B  mod  8
)  e.  { 1 ,  7 }  ->  (
-u -u B  mod  8
)  e.  { 1 ,  7 } ) )
9718negnegd 9923 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  -u -u B  =  B )
9897oveq1d 6259 . . . . . . . 8  |-  ( B  e.  ZZ  ->  ( -u -u B  mod  8
)  =  ( B  mod  8 ) )
9998eleq1d 2485 . . . . . . 7  |-  ( B  e.  ZZ  ->  (
( -u -u B  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
10096, 99sylibd 217 . . . . . 6  |-  ( B  e.  ZZ  ->  (
( -u B  mod  8
)  e.  { 1 ,  7 }  ->  ( B  mod  8 )  e.  { 1 ,  7 } ) )
101 oveq1 6251 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  mod  8 )  =  ( B  mod  8 ) )
102101eleq1d 2485 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
103 negeq 9813 . . . . . . . . . 10  |-  ( x  =  B  ->  -u x  =  -u B )
104103oveq1d 6259 . . . . . . . . 9  |-  ( x  =  B  ->  ( -u x  mod  8 )  =  ( -u B  mod  8 ) )
105104eleq1d 2485 . . . . . . . 8  |-  ( x  =  B  ->  (
( -u x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
106102, 105imbi12d 321 . . . . . . 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 3083 . . . . . 6  |-  ( B  e.  ZZ  ->  (
( B  mod  8
)  e.  { 1 ,  7 }  ->  (
-u B  mod  8
)  e.  { 1 ,  7 } ) )
108100, 107impbid 193 . . . . 5  |-  ( B  e.  ZZ  ->  (
( -u B  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
109108ad2antlr 731 . . . 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 256 . . 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 792 . 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 477 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   {cpr 3938  (class class class)co 6244   CCcc 9483   RRcr 9484   1c1 9486    x. cmul 9490   -ucneg 9807   3c3 10606   5c5 10608   7c7 10610   8c8 10611   ZZcz 10883   RR+crp 11248    mod cmo 12041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-sup 7904  df-inf 7905  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-4 10616  df-5 10617  df-6 10618  df-7 10619  df-8 10620  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-fl 11973  df-mod 12042
This theorem is referenced by:  lgsdir2  24193
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