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Theorem lgsdilem 21098
Description: Lemma for lgsdi 21108 and lgsdir 21106: 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 738 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  =/=  0 )
21biantrud 494 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <_  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
3 0re 9083 . . . . . . . . . . 11  |-  0  e.  RR
4 simpl2 961 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
54zred 10367 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  RR )
65adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  RR )
7 ltlen 9167 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
83, 6, 7sylancr 645 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
9 simpl1 960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
109zred 10367 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  RR )
1110adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  A  e.  RR )
1211renegcld 9456 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  -u A  e.  RR )
1312recnd 9106 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  -u A  e.  CC )
1413mul01d 9257 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( -u A  x.  0 )  =  0 )
1511recnd 9106 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  A  e.  CC )
166recnd 9106 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  CC )
1715, 16mulneg1d 9478 . . . . . . . . . . . 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 4217 . . . . . . . . . . 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 )
) )
193a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
0  e.  RR )
2010lt0neg1d 9588 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  <  0  <->  0  <  -u A ) )
2120biimpa 471 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
0  <  -u A )
22 ltmul2 9853 . . . . . . . . . . . 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 1188 . . . . . . . . . . 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 9108 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  RR )
2524adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( A  x.  B
)  e.  RR )
2625lt0neg1d 9588 . . . . . . . . . . 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 277 . . . . . . . . . 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 274 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( ( A  x.  B )  <  0  <->  0  <_  B ) )
29 lenlt 9146 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
303, 6, 29sylancr 645 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <_  B  <->  -.  B  <  0 ) )
3128, 30bitrd 245 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( ( A  x.  B )  <  0  <->  -.  B  <  0 ) )
3231ifbid 3749 . . . . . . 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 6081 . . . . . . . . . 10  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  ( -u 1  x.  -u 1 ) )
34 neg1cn 10059 . . . . . . . . . . . 12  |-  -u 1  e.  CC
3534mulm1i 9470 . . . . . . . . . . 11  |-  ( -u
1  x.  -u 1
)  =  -u -u 1
36 ax-1cn 9040 . . . . . . . . . . . 12  |-  1  e.  CC
3736negnegi 9362 . . . . . . . . . . 11  |-  -u -u 1  =  1
3835, 37eqtri 2455 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
3933, 38syl6eq 2483 . . . . . . . . 9  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  1 )
40 oveq2 6081 . . . . . . . . . 10  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
4136mulm1i 9470 . . . . . . . . . 10  |-  ( -u
1  x.  1 )  =  -u 1
4240, 41syl6eq 2483 . . . . . . . . 9  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
4339, 42ifsb 3740 . . . . . . . 8  |-  ( -u
1  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  if ( B  <  0 ,  1 ,  -u
1 )
44 ifnot 3769 . . . . . . . 8  |-  if ( -.  B  <  0 ,  -u 1 ,  1 )  =  if ( B  <  0 ,  1 ,  -u 1
)
4543, 44eqtr4i 2458 . . . . . . 7  |-  ( -u
1  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  if ( -.  B  <  0 ,  -u 1 ,  1 )
4632, 45syl6eqr 2485 . . . . . 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 ) ) )
47 iftrue 3737 . . . . . . . 8  |-  ( A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
4847adantl 453 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
4948oveq1d 6088 . . . . . 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 ) ) )
5046, 49eqtr4d 2470 . . . . 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 ) ) )
51 iffalse 3738 . . . . . . . 8  |-  ( -.  A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  1 )
5251adantl 453 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  if ( A  <  0 ,  -u
1 ,  1 )  =  1 )
5352oveq1d 6088 . . . . . 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 ) ) )
5434, 36keepel 3788 . . . . . . . 8  |-  if ( B  <  0 , 
-u 1 ,  1 )  e.  CC
5554mulid2i 9085 . . . . . . 7  |-  ( 1  x.  if ( B  <  0 ,  -u
1 ,  1 ) )  =  if ( B  <  0 , 
-u 1 ,  1 )
565adantr 452 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  B  e.  RR )
573a1i 11 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  e.  RR )
5810adantr 452 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  RR )
59 lenlt 9146 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
603, 10, 59sylancr 645 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
0  <_  A  <->  -.  A  <  0 ) )
6160biimpar 472 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  <_  A )
62 simplrl 737 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  =/=  0 )
6358, 61, 62ne0gt0d 9202 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  <  A )
64 ltmul2 9853 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  0  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( B  <  0  <->  ( A  x.  B )  <  ( A  x.  0 ) ) )
6556, 57, 58, 63, 64syl112anc 1188 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( B  <  0  <->  ( A  x.  B )  <  ( A  x.  0 ) ) )
6658recnd 9106 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  CC )
6766mul01d 9257 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( A  x.  0 )  =  0 )
6867breq2d 4216 . . . . . . . . 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
) )
6965, 68bitrd 245 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( B  <  0  <->  ( A  x.  B )  <  0
) )
7069ifbid 3749 . . . . . . 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 ) )
7155, 70syl5eq 2479 . . . . . 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 ) )
7253, 71eqtr2d 2468 . . . . 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 ) ) )
7350, 72pm2.61dan 767 . . . 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 ) ) )
7473adantr 452 . . 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 ) ) )
75 simpr 448 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  N  <  0 )
7675biantrurd 495 . . . 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 ) ) )
7776ifbid 3749 . . 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 ) )
7875biantrurd 495 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( A  <  0  <->  ( N  <  0  /\  A  <  0 ) ) )
7978ifbid 3749 . . . 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 ) )
8075biantrurd 495 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( B  <  0  <->  ( N  <  0  /\  B  <  0 ) ) )
8180ifbid 3749 . . . 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 ) )
8279, 81oveq12d 6091 . . 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 ) ) )
8374, 77, 823eqtr3d 2475 . 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 ) ) )
84 simpr 448 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  N  <  0 )
8584intnanrd 884 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  ( A  x.  B )  <  0 ) )
86 iffalse 3738 . . . . 5  |-  ( -.  ( N  <  0  /\  ( A  x.  B
)  <  0 )  ->  if ( ( N  <  0  /\  ( A  x.  B
)  <  0 ) ,  -u 1 ,  1 )  =  1 )
8785, 86syl 16 . . . 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 )
88 1t1e1 10118 . . . 4  |-  ( 1  x.  1 )  =  1
8987, 88syl6eqr 2485 . . 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 ) )
9084intnanrd 884 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
91 iffalse 3738 . . . . 5  |-  ( -.  ( N  <  0  /\  A  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
9290, 91syl 16 . . . 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 )
9384intnanrd 884 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  B  <  0 ) )
94 iffalse 3738 . . . . 5  |-  ( -.  ( N  <  0  /\  B  <  0
)  ->  if (
( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 )  =  1 )
9593, 94syl 16 . . . 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 )
9692, 95oveq12d 6091 . . 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 ) )
9789, 96eqtr4d 2470 . 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 ) ) )
9883, 97pm2.61dan 767 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ifcif 3731   class class class wbr 4204  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    < clt 9112    <_ cle 9113   -ucneg 9284   ZZcz 10274
This theorem is referenced by:  lgsdir  21106  lgsdi  21108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-z 10275
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