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Theorem lgsdir2lem4 24310
Description: Lemma for lgsdir2 24312. (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 6348 . . 3  |-  ( A  mod  8 )  e. 
_V
21elpr 3998 . 2  |-  ( ( A  mod  8 )  e.  { 1 ,  7 }  <->  ( ( A  mod  8 )  =  1  \/  ( A  mod  8 )  =  7 ) )
3 zre 10975 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
43ad2antrr 737 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  A  e.  RR )
5 1red 9689 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  1  e.  RR )
6 simplr 767 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  ZZ )
7 8re 10727 . . . . . . . 8  |-  8  e.  RR
8 8pos 10743 . . . . . . . 8  |-  0  <  8
97, 8elrpii 11339 . . . . . . 7  |-  8  e.  RR+
109a1i 11 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  8  e.  RR+ )
11 simpr 467 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( A  mod  8 )  =  1 )
12 lgsdir2lem1 24307 . . . . . . . . 9  |-  ( ( ( 1  mod  8
)  =  1  /\  ( -u 1  mod  8 )  =  7 )  /\  ( ( 3  mod  8 )  =  3  /\  ( -u 3  mod  8 )  =  5 ) )
1312simpli 464 . . . . . . . 8  |-  ( ( 1  mod  8 )  =  1  /\  ( -u 1  mod  8 )  =  7 )
1413simpli 464 . . . . . . 7  |-  ( 1  mod  8 )  =  1
1511, 14syl6eqr 2514 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( A  mod  8 )  =  ( 1  mod  8 ) )
16 modmul1 12181 . . . . . 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 1287 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( ( 1  x.  B
)  mod  8 ) )
18 zcn 10976 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
1918ad2antlr 738 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  CC )
2019mulid2d 9692 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( 1  x.  B )  =  B )
2120oveq1d 6335 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( (
1  x.  B )  mod  8 )  =  ( B  mod  8
) )
2217, 21eqtrd 2496 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( B  mod  8 ) )
2322eleq1d 2524 . . 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 737 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  A  e.  RR )
25 neg1rr 10747 . . . . . . . 8  |-  -u 1  e.  RR
2625a1i 11 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  -u 1  e.  RR )
27 simplr 767 . . . . . . 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 467 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( A  mod  8 )  =  7 )
3013simpri 468 . . . . . . . 8  |-  ( -u
1  mod  8 )  =  7
3129, 30syl6eqr 2514 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( A  mod  8 )  =  (
-u 1  mod  8
) )
32 modmul1 12181 . . . . . . 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 1287 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( A  x.  B )  mod  8 )  =  ( ( -u 1  x.  B )  mod  8
) )
3418ad2antlr 738 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  B  e.  CC )
3534mulm1d 10103 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( -u 1  x.  B )  =  -u B )
3635oveq1d 6335 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( -u 1  x.  B )  mod  8 )  =  ( -u B  mod  8 ) )
3733, 36eqtrd 2496 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( A  x.  B )  mod  8 )  =  (
-u B  mod  8
) )
3837eleq1d 2524 . . . 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 11006 . . . . . . . 8  |-  ( B  e.  ZZ  ->  -u B  e.  ZZ )
40 oveq1 6327 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  (
x  mod  8 )  =  ( -u B  mod  8 ) )
4140eleq1d 2524 . . . . . . . . . 10  |-  ( x  =  -u B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
42 negeq 9898 . . . . . . . . . . . 12  |-  ( x  =  -u B  ->  -u x  =  -u -u B )
4342oveq1d 6335 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  ( -u x  mod  8 )  =  ( -u -u B  mod  8 ) )
4443eleq1d 2524 . . . . . . . . . 10  |-  ( x  =  -u B  ->  (
( -u x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u -u B  mod  8 )  e.  {
1 ,  7 } ) )
4541, 44imbi12d 326 . . . . . . . . 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 10976 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  x  e.  CC )
47 neg1cn 10746 . . . . . . . . . . . . . . . . . . 19  |-  -u 1  e.  CC
48 mulcom 9656 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  CC  /\  -u 1  e.  CC )  ->  ( x  x.  -u 1 )  =  ( -u 1  x.  x ) )
4947, 48mpan2 682 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  (
x  x.  -u 1
)  =  ( -u
1  x.  x ) )
50 mulm1 10093 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  ( -u 1  x.  x )  =  -u x )
5149, 50eqtrd 2496 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  CC  ->  (
x  x.  -u 1
)  =  -u x
)
5246, 51syl 17 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  (
x  x.  -u 1
)  =  -u x
)
5352adantr 471 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  x.  -u 1 )  = 
-u x )
5453oveq1d 6335 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  (
-u x  mod  8
) )
55 zre 10975 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  x  e.  RR )
5655adantr 471 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  x  e.  RR )
57 1red 9689 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  1  e.  RR )
58 neg1z 11007 . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  mod  8 )  =  1 )
6261, 14syl6eqr 2514 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  mod  8 )  =  ( 1  mod  8 ) )
63 modmul1 12181 . . . . . . . . . . . . . . 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 1287 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6554, 64eqtr3d 2498 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( -u x  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6647mulid2i 9677 . . . . . . . . . . . . . . 15  |-  ( 1  x.  -u 1 )  = 
-u 1
6766oveq1i 6330 . . . . . . . . . . . . . 14  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  ( -u 1  mod  8 )
6867, 30eqtri 2484 . . . . . . . . . . . . 13  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  7
6965, 68syl6eq 2512 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( -u x  mod  8 )  =  7 )
7069ex 440 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  =  1  -> 
( -u x  mod  8
)  =  7 ) )
7152adantr 471 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  x.  -u 1 )  = 
-u x )
7271oveq1d 6335 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  (
-u x  mod  8
) )
7355adantr 471 . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  mod  8 )  =  7 )
7877, 30syl6eqr 2514 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  mod  8 )  =  (
-u 1  mod  8
) )
79 modmul1 12181 . . . . . . . . . . . . . . 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 1287 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
8172, 80eqtr3d 2498 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( -u x  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
82 neg1mulneg1e1 10861 . . . . . . . . . . . . . . 15  |-  ( -u
1  x.  -u 1
)  =  1
8382oveq1i 6330 . . . . . . . . . . . . . 14  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  ( 1  mod  8 )
8483, 14eqtri 2484 . . . . . . . . . . . . 13  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  1
8581, 84syl6eq 2512 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( -u x  mod  8 )  =  1 )
8685ex 440 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  =  7  -> 
( -u x  mod  8
)  =  1 ) )
8770, 86orim12d 854 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  (
( ( x  mod  8 )  =  1  \/  ( x  mod  8 )  =  7 )  ->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) ) )
88 ovex 6348 . . . . . . . . . . 11  |-  ( x  mod  8 )  e. 
_V
8988elpr 3998 . . . . . . . . . 10  |-  ( ( x  mod  8 )  e.  { 1 ,  7 }  <->  ( (
x  mod  8 )  =  1  \/  (
x  mod  8 )  =  7 ) )
90 ovex 6348 . . . . . . . . . . . 12  |-  ( -u x  mod  8 )  e. 
_V
9190elpr 3998 . . . . . . . . . . 11  |-  ( (
-u x  mod  8
)  e.  { 1 ,  7 }  <->  ( ( -u x  mod  8 )  =  1  \/  ( -u x  mod  8 )  =  7 ) )
92 orcom 393 . . . . . . . . . . 11  |-  ( ( ( -u x  mod  8 )  =  1  \/  ( -u x  mod  8 )  =  7 )  <->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) )
9391, 92bitri 257 . . . . . . . . . 10  |-  ( (
-u x  mod  8
)  e.  { 1 ,  7 }  <->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) )
9487, 89, 933imtr4g 278 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  ->  (
-u x  mod  8
)  e.  { 1 ,  7 } ) )
9545, 94vtoclga 3125 . . . . . . . 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 10008 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  -u -u B  =  B )
9897oveq1d 6335 . . . . . . . 8  |-  ( B  e.  ZZ  ->  ( -u -u B  mod  8
)  =  ( B  mod  8 ) )
9998eleq1d 2524 . . . . . . 7  |-  ( B  e.  ZZ  ->  (
( -u -u B  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
10096, 99sylibd 222 . . . . . 6  |-  ( B  e.  ZZ  ->  (
( -u B  mod  8
)  e.  { 1 ,  7 }  ->  ( B  mod  8 )  e.  { 1 ,  7 } ) )
101 oveq1 6327 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  mod  8 )  =  ( B  mod  8 ) )
102101eleq1d 2524 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
103 negeq 9898 . . . . . . . . . 10  |-  ( x  =  B  ->  -u x  =  -u B )
104103oveq1d 6335 . . . . . . . . 9  |-  ( x  =  B  ->  ( -u x  mod  8 )  =  ( -u B  mod  8 ) )
105104eleq1d 2524 . . . . . . . 8  |-  ( x  =  B  ->  (
( -u x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
106102, 105imbi12d 326 . . . . . . 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 3125 . . . . . 6  |-  ( B  e.  ZZ  ->  (
( B  mod  8
)  e.  { 1 ,  7 }  ->  (
-u B  mod  8
)  e.  { 1 ,  7 } ) )
108100, 107impbid 195 . . . . 5  |-  ( B  e.  ZZ  ->  (
( -u B  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
109108ad2antlr 738 . . . 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 261 . . 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 799 . 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 482 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 189    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898   {cpr 3982  (class class class)co 6320   CCcc 9568   RRcr 9569   1c1 9571    x. cmul 9575   -ucneg 9892   3c3 10693   5c5 10695   7c7 10697   8c8 10698   ZZcz 10971   RR+crp 11336    mod cmo 12134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-sup 7987  df-inf 7988  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-7 10706  df-8 10707  df-n0 10904  df-z 10972  df-uz 11194  df-rp 11337  df-fl 12066  df-mod 12135
This theorem is referenced by:  lgsdir2  24312
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