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Theorem lgsdir2lem4 23726
Description: Lemma for lgsdir2 23728. (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 6324 . . 3  |-  ( A  mod  8 )  e. 
_V
21elpr 4050 . 2  |-  ( ( A  mod  8 )  e.  { 1 ,  7 }  <->  ( ( A  mod  8 )  =  1  \/  ( A  mod  8 )  =  7 ) )
3 zre 10889 . . . . . . 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 9628 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  1  e.  RR )
6 simplr 755 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  ZZ )
7 8re 10641 . . . . . . . 8  |-  8  e.  RR
8 8pos 10657 . . . . . . . 8  |-  0  <  8
97, 8elrpii 11248 . . . . . . 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 23723 . . . . . . . . 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 2516 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( A  mod  8 )  =  ( 1  mod  8 ) )
16 modmul1 12042 . . . . . 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 1239 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( ( 1  x.  B
)  mod  8 ) )
18 zcn 10890 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
1918ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  B  e.  CC )
2019mulid2d 9631 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( 1  x.  B )  =  B )
2120oveq1d 6311 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( (
1  x.  B )  mod  8 )  =  ( B  mod  8
) )
2217, 21eqtrd 2498 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  1 )  ->  ( ( A  x.  B )  mod  8 )  =  ( B  mod  8 ) )
2322eleq1d 2526 . . 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 10661 . . . . . . . 8  |-  -u 1  e.  RR
2625a1i 11 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  -u 1  e.  RR )
27 simplr 755 . . . . . . 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 2516 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( A  mod  8 )  =  (
-u 1  mod  8
) )
32 modmul1 12042 . . . . . . 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 1239 . . . . . 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 10029 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( -u 1  x.  B )  =  -u B )
3635oveq1d 6311 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( -u 1  x.  B )  mod  8 )  =  ( -u B  mod  8 ) )
3733, 36eqtrd 2498 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  mod  8 )  =  7 )  ->  ( ( A  x.  B )  mod  8 )  =  (
-u B  mod  8
) )
3837eleq1d 2526 . . . 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 10920 . . . . . . . 8  |-  ( B  e.  ZZ  ->  -u B  e.  ZZ )
40 oveq1 6303 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  (
x  mod  8 )  =  ( -u B  mod  8 ) )
4140eleq1d 2526 . . . . . . . . . 10  |-  ( x  =  -u B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( -u B  mod  8 )  e.  {
1 ,  7 } ) )
42 negeq 9831 . . . . . . . . . . . 12  |-  ( x  =  -u B  ->  -u x  =  -u -u B )
4342oveq1d 6311 . . . . . . . . . . 11  |-  ( x  =  -u B  ->  ( -u x  mod  8 )  =  ( -u -u B  mod  8 ) )
4443eleq1d 2526 . . . . . . . . . 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 10890 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  x  e.  CC )
47 neg1cn 10660 . . . . . . . . . . . . . . . . . . 19  |-  -u 1  e.  CC
48 mulcom 9595 . . . . . . . . . . . . . . . . . . 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 10019 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  CC  ->  ( -u 1  x.  x )  =  -u x )
5149, 50eqtrd 2498 . . . . . . . . . . . . . . . . 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 6311 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  (
-u x  mod  8
) )
55 zre 10889 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ZZ  ->  x  e.  RR )
5655adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  x  e.  RR )
57 1red 9628 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  1  e.  RR )
58 neg1z 10921 . . . . . . . . . . . . . . . 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 2516 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( x  mod  8 )  =  ( 1  mod  8 ) )
63 modmul1 12042 . . . . . . . . . . . . . . 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 1239 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6554, 64eqtr3d 2500 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  1 )  ->  ( -u x  mod  8 )  =  ( ( 1  x.  -u 1
)  mod  8 ) )
6647mulid2i 9616 . . . . . . . . . . . . . . 15  |-  ( 1  x.  -u 1 )  = 
-u 1
6766oveq1i 6306 . . . . . . . . . . . . . 14  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  ( -u 1  mod  8 )
6867, 30eqtri 2486 . . . . . . . . . . . . 13  |-  ( ( 1  x.  -u 1
)  mod  8 )  =  7
6965, 68syl6eq 2514 . . . . . . . . . . . 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 6311 . . . . . . . . . . . . . 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 2516 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( x  mod  8 )  =  (
-u 1  mod  8
) )
79 modmul1 12042 . . . . . . . . . . . . . . 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 1239 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( ( x  x.  -u 1 )  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
8172, 80eqtr3d 2500 . . . . . . . . . . . . 13  |-  ( ( x  e.  ZZ  /\  ( x  mod  8
)  =  7 )  ->  ( -u x  mod  8 )  =  ( ( -u 1  x.  -u 1 )  mod  8 ) )
82 neg1mulneg1e1 10774 . . . . . . . . . . . . . . 15  |-  ( -u
1  x.  -u 1
)  =  1
8382oveq1i 6306 . . . . . . . . . . . . . 14  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  ( 1  mod  8 )
8483, 14eqtri 2486 . . . . . . . . . . . . 13  |-  ( (
-u 1  x.  -u 1
)  mod  8 )  =  1
8581, 84syl6eq 2514 . . . . . . . . . . . 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 838 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  (
( ( x  mod  8 )  =  1  \/  ( x  mod  8 )  =  7 )  ->  ( ( -u x  mod  8 )  =  7  \/  ( -u x  mod  8 )  =  1 ) ) )
88 ovex 6324 . . . . . . . . . . 11  |-  ( x  mod  8 )  e. 
_V
8988elpr 4050 . . . . . . . . . 10  |-  ( ( x  mod  8 )  e.  { 1 ,  7 }  <->  ( (
x  mod  8 )  =  1  \/  (
x  mod  8 )  =  7 ) )
90 ovex 6324 . . . . . . . . . . . 12  |-  ( -u x  mod  8 )  e. 
_V
9190elpr 4050 . . . . . . . . . . 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 3173 . . . . . . . 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 9941 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  -u -u B  =  B )
9897oveq1d 6311 . . . . . . . 8  |-  ( B  e.  ZZ  ->  ( -u -u B  mod  8
)  =  ( B  mod  8 ) )
9998eleq1d 2526 . . . . . . 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 6303 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  mod  8 )  =  ( B  mod  8 ) )
102101eleq1d 2526 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  mod  8
)  e.  { 1 ,  7 }  <->  ( B  mod  8 )  e.  {
1 ,  7 } ) )
103 negeq 9831 . . . . . . . . . 10  |-  ( x  =  B  ->  -u x  =  -u B )
104103oveq1d 6311 . . . . . . . . 9  |-  ( x  =  B  ->  ( -u x  mod  8 )  =  ( -u B  mod  8 ) )
105104eleq1d 2526 . . . . . . . 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 3173 . . . . . 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 785 . 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 1395    e. wcel 1819   {cpr 4034  (class class class)co 6296   CCcc 9507   RRcr 9508   1c1 9510    x. cmul 9514   -ucneg 9825   3c3 10607   5c5 10609   7c7 10611   8c8 10612   ZZcz 10885   RR+crp 11245    mod cmo 11998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fl 11931  df-mod 11999
This theorem is referenced by:  lgsdir2  23728
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