MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgsdilem Structured version   Unicode version

Theorem lgsdilem 23795
Description: Lemma for lgsdi 23805 and lgsdir 23803: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
lgsdilem  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  if ( ( N  <  0  /\  ( A  x.  B )  <  0 ) ,  -u
1 ,  1 )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 ) ) )

Proof of Theorem lgsdilem
StepHypRef Expression
1 simplrr 760 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  =/=  0 )
21biantrud 505 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <_  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
3 0re 9585 . . . . . . . . . . 11  |-  0  e.  RR
4 simpl2 998 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
54zred 10965 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  RR )
65adantr 463 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  RR )
7 ltlen 9675 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
83, 6, 7sylancr 661 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
9 simpl1 997 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
109zred 10965 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  RR )
1110adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  A  e.  RR )
1211renegcld 9982 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  -u A  e.  RR )
1312recnd 9611 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  -u A  e.  CC )
1413mul01d 9768 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( -u A  x.  0 )  =  0 )
1511recnd 9611 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  A  e.  CC )
166recnd 9611 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  CC )
1715, 16mulneg1d 10005 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( -u A  x.  B
)  =  -u ( A  x.  B )
)
1814, 17breq12d 4452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( ( -u A  x.  0 )  <  ( -u A  x.  B )  <->  0  <  -u ( A  x.  B )
) )
19 0red 9586 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
0  e.  RR )
2010lt0neg1d 10118 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  <  0  <->  0  <  -u A ) )
2120biimpa 482 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
0  <  -u A )
22 ltmul2 10389 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  ( -u A  e.  RR  /\  0  <  -u A ) )  ->  ( 0  < 
B  <->  ( -u A  x.  0 )  <  ( -u A  x.  B ) ) )
2319, 6, 12, 21, 22syl112anc 1230 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <  B  <->  (
-u A  x.  0 )  <  ( -u A  x.  B )
) )
2410, 5remulcld 9613 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  RR )
2524adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( A  x.  B
)  e.  RR )
2625lt0neg1d 10118 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( ( A  x.  B )  <  0  <->  0  <  -u ( A  x.  B ) ) )
2718, 23, 263bitr4d 285 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <  B  <->  ( A  x.  B )  <  0 ) )
282, 8, 273bitr2rd 282 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( ( A  x.  B )  <  0  <->  0  <_  B ) )
29 lenlt 9652 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
303, 6, 29sylancr 661 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <_  B  <->  -.  B  <  0 ) )
3128, 30bitrd 253 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( ( A  x.  B )  <  0  <->  -.  B  <  0 ) )
3231ifbid 3951 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  if ( ( A  x.  B )  <  0 ,  -u 1 ,  1 )  =  if ( -.  B  <  0 ,  -u 1 ,  1 ) )
33 oveq2 6278 . . . . . . . . . 10  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  ( -u 1  x.  -u 1 ) )
34 neg1mulneg1e1 10749 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
3533, 34syl6eq 2511 . . . . . . . . 9  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  1 )
36 oveq2 6278 . . . . . . . . . 10  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
37 ax-1cn 9539 . . . . . . . . . . 11  |-  1  e.  CC
3837mulm1i 9997 . . . . . . . . . 10  |-  ( -u
1  x.  1 )  =  -u 1
3936, 38syl6eq 2511 . . . . . . . . 9  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
4035, 39ifsb 3942 . . . . . . . 8  |-  ( -u
1  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  if ( B  <  0 ,  1 ,  -u
1 )
41 ifnot 3974 . . . . . . . 8  |-  if ( -.  B  <  0 ,  -u 1 ,  1 )  =  if ( B  <  0 ,  1 ,  -u 1
)
4240, 41eqtr4i 2486 . . . . . . 7  |-  ( -u
1  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  if ( -.  B  <  0 ,  -u 1 ,  1 )
4332, 42syl6eqr 2513 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  if ( ( A  x.  B )  <  0 ,  -u 1 ,  1 )  =  ( -u
1  x.  if ( B  <  0 , 
-u 1 ,  1 ) ) )
44 iftrue 3935 . . . . . . . 8  |-  ( A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
4544adantl 464 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
4645oveq1d 6285 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( if ( A  <  0 ,  -u
1 ,  1 )  x.  if ( B  <  0 ,  -u
1 ,  1 ) )  =  ( -u
1  x.  if ( B  <  0 , 
-u 1 ,  1 ) ) )
4743, 46eqtr4d 2498 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  if ( ( A  x.  B )  <  0 ,  -u 1 ,  1 )  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  if ( B  <  0 , 
-u 1 ,  1 ) ) )
48 iffalse 3938 . . . . . . . 8  |-  ( -.  A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  1 )
4948adantl 464 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  if ( A  <  0 ,  -u
1 ,  1 )  =  1 )
5049oveq1d 6285 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  ( 1  x.  if ( B  <  0 , 
-u 1 ,  1 ) ) )
51 neg1cn 10635 . . . . . . . . 9  |-  -u 1  e.  CC
5251, 37keepel 3996 . . . . . . . 8  |-  if ( B  <  0 , 
-u 1 ,  1 )  e.  CC
5352mulid2i 9588 . . . . . . 7  |-  ( 1  x.  if ( B  <  0 ,  -u
1 ,  1 ) )  =  if ( B  <  0 , 
-u 1 ,  1 )
545adantr 463 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  B  e.  RR )
55 0red 9586 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  e.  RR )
5610adantr 463 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  RR )
57 lenlt 9652 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
583, 10, 57sylancr 661 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
0  <_  A  <->  -.  A  <  0 ) )
5958biimpar 483 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  <_  A )
60 simplrl 759 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  =/=  0 )
6156, 59, 60ne0gt0d 9711 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  <  A )
62 ltmul2 10389 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  0  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( B  <  0  <->  ( A  x.  B )  <  ( A  x.  0 ) ) )
6354, 55, 56, 61, 62syl112anc 1230 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( B  <  0  <->  ( A  x.  B )  <  ( A  x.  0 ) ) )
6456recnd 9611 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  CC )
6564mul01d 9768 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( A  x.  0 )  =  0 )
6665breq2d 4451 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( ( A  x.  B )  <  ( A  x.  0 )  <->  ( A  x.  B )  <  0
) )
6763, 66bitrd 253 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( B  <  0  <->  ( A  x.  B )  <  0
) )
6867ifbid 3951 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  if ( B  <  0 ,  -u
1 ,  1 )  =  if ( ( A  x.  B )  <  0 ,  -u
1 ,  1 ) )
6953, 68syl5eq 2507 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( 1  x.  if ( B  <  0 ,  -u
1 ,  1 ) )  =  if ( ( A  x.  B
)  <  0 ,  -u 1 ,  1 ) )
7050, 69eqtr2d 2496 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  if (
( A  x.  B
)  <  0 ,  -u 1 ,  1 )  =  ( if ( A  <  0 , 
-u 1 ,  1 )  x.  if ( B  <  0 , 
-u 1 ,  1 ) ) )
7147, 70pm2.61dan 789 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  if ( ( A  x.  B )  <  0 ,  -u 1 ,  1 )  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  if ( B  <  0 , 
-u 1 ,  1 ) ) )
7271adantr 463 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  if ( ( A  x.  B )  <  0 ,  -u 1 ,  1 )  =  ( if ( A  <  0 ,  -u 1 ,  1 )  x.  if ( B  <  0 , 
-u 1 ,  1 ) ) )
73 simpr 459 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  N  <  0 )
7473biantrurd 506 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( ( A  x.  B )  <  0  <->  ( N  <  0  /\  ( A  x.  B
)  <  0 ) ) )
7574ifbid 3951 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  if ( ( A  x.  B )  <  0 ,  -u 1 ,  1 )  =  if ( ( N  <  0  /\  ( A  x.  B
)  <  0 ) ,  -u 1 ,  1 ) )
7673biantrurd 506 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( A  <  0  <->  ( N  <  0  /\  A  <  0 ) ) )
7776ifbid 3951 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 ) )
7873biantrurd 506 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( B  <  0  <->  ( N  <  0  /\  B  <  0 ) ) )
7978ifbid 3951 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  if ( B  <  0 ,  -u 1 ,  1 )  =  if ( ( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 ) )
8077, 79oveq12d 6288 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( if ( A  <  0 ,  -u
1 ,  1 )  x.  if ( B  <  0 ,  -u
1 ,  1 ) )  =  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u 1 ,  1 ) ) )
8172, 75, 803eqtr3d 2503 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  if ( ( N  <  0  /\  ( A  x.  B )  <  0 ) ,  -u
1 ,  1 )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 ) ) )
82 simpr 459 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  N  <  0 )
8382intnanrd 915 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  ( A  x.  B )  <  0 ) )
8483iffalsed 3940 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  if (
( N  <  0  /\  ( A  x.  B
)  <  0 ) ,  -u 1 ,  1 )  =  1 )
85 1t1e1 10679 . . . 4  |-  ( 1  x.  1 )  =  1
8684, 85syl6eqr 2513 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  if (
( N  <  0  /\  ( A  x.  B
)  <  0 ) ,  -u 1 ,  1 )  =  ( 1  x.  1 ) )
8782intnanrd 915 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
8887iffalsed 3940 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
8982intnanrd 915 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  B  <  0 ) )
9089iffalsed 3940 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  if (
( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 )  =  1 )
9188, 90oveq12d 6288 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u 1 ,  1 ) )  =  ( 1  x.  1 ) )
9286, 91eqtr4d 2498 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  if (
( N  <  0  /\  ( A  x.  B
)  <  0 ) ,  -u 1 ,  1 )  =  ( if ( ( N  <  0  /\  A  <  0 ) ,  -u
1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u 1 ,  1 ) ) )
9381, 92pm2.61dan 789 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  if ( ( N  <  0  /\  ( A  x.  B )  <  0 ) ,  -u
1 ,  1 )  =  ( if ( ( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  x.  if ( ( N  <  0  /\  B  <  0 ) ,  -u
1 ,  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   ifcif 3929   class class class wbr 4439  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486    < clt 9617    <_ cle 9618   -ucneg 9797   ZZcz 10860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-z 10861
This theorem is referenced by:  lgsdir  23803  lgsdi  23805
  Copyright terms: Public domain W3C validator