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Theorem lgsdilem 14095
Description: Lemma for lgsdi 14105 and lgsdir 14103: 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 536 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  =/=  0 )
21biantrud 304 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <_  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
3 0z 9253 . . . . . . . . . . 11  |-  0  e.  ZZ
4 simpl2 1001 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
54adantr 276 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  ZZ )
6 zltlen 9320 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  B  e.  ZZ )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
73, 5, 6sylancr 414 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
8 simpl1 1000 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
98zred 9364 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  RR )
109adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  A  e.  RR )
1110renegcld 8327 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  -u A  e.  RR )
1211recnd 7976 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  -u A  e.  CC )
1312mul01d 8340 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( -u A  x.  0 )  =  0 )
1410recnd 7976 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  A  e.  CC )
154zred 9364 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  RR )
1615adantr 276 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  RR )
1716recnd 7976 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  CC )
1814, 17mulneg1d 8358 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( -u A  x.  B
)  =  -u ( A  x.  B )
)
1913, 18breq12d 4013 . . . . . . . . . . 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 )
) )
20 0red 7949 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
0  e.  RR )
219lt0neg1d 8462 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  <  0  <->  0  <  -u A ) )
2221biimpa 296 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
0  <  -u A )
23 ltmul2 8802 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  B  e.  RR  /\  ( -u A  e.  RR  /\  0  <  -u A ) )  ->  ( 0  < 
B  <->  ( -u A  x.  0 )  <  ( -u A  x.  B ) ) )
2420, 16, 11, 22, 23syl112anc 1242 . . . . . . . . . . 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 )
) )
259, 15remulcld 7978 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  RR )
2625adantr 276 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( A  x.  B
)  e.  RR )
2726lt0neg1d 8462 . . . . . . . . . . 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 ) ) )
2819, 24, 273bitr4d 220 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <  B  <->  ( A  x.  B )  <  0 ) )
292, 7, 283bitr2rd 217 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( ( A  x.  B )  <  0  <->  0  <_  B ) )
30 0re 7948 . . . . . . . . . 10  |-  0  e.  RR
31 lenlt 8023 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
3230, 16, 31sylancr 414 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <_  B  <->  -.  B  <  0 ) )
3329, 32bitrd 188 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( ( A  x.  B )  <  0  <->  -.  B  <  0 ) )
3433ifbid 3555 . . . . . . 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 ) )
35 zdclt 9319 . . . . . . . . . . 11  |-  ( ( B  e.  ZZ  /\  0  e.  ZZ )  -> DECID  B  <  0 )
363, 35mpan2 425 . . . . . . . . . 10  |-  ( B  e.  ZZ  -> DECID  B  <  0
)
37 oveq2 5877 . . . . . . . . . . . . 13  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  ( -u 1  x.  -u 1 ) )
38 neg1mulneg1e1 9120 . . . . . . . . . . . . 13  |-  ( -u
1  x.  -u 1
)  =  1
3937, 38eqtrdi 2226 . . . . . . . . . . . 12  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  1 )
40 oveq2 5877 . . . . . . . . . . . . 13  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
41 ax-1cn 7895 . . . . . . . . . . . . . 14  |-  1  e.  CC
4241mulm1i 8350 . . . . . . . . . . . . 13  |-  ( -u
1  x.  1 )  =  -u 1
4340, 42eqtrdi 2226 . . . . . . . . . . . 12  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
4439, 43ifsbdc 3546 . . . . . . . . . . 11  |-  (DECID  B  <  0  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  if ( B  <  0 ,  1 ,  -u 1 ) )
45 ifnotdc 3570 . . . . . . . . . . 11  |-  (DECID  B  <  0  ->  if ( -.  B  <  0 ,  -u 1 ,  1 )  =  if ( B  <  0 ,  1 ,  -u 1
) )
4644, 45eqtr4d 2213 . . . . . . . . . 10  |-  (DECID  B  <  0  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  if ( -.  B  <  0 , 
-u 1 ,  1 ) )
4736, 46syl 14 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  ( -u 1  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  if ( -.  B  <  0 ,  -u 1 ,  1 ) )
48473ad2ant2 1019 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  ->  ( -u 1  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  if ( -.  B  <  0 ,  -u 1 ,  1 ) )
4948ad2antrr 488 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  if ( -.  B  <  0 ,  -u 1 ,  1 ) )
5034, 49eqtr4d 2213 . . . . . 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 ) ) )
51 iftrue 3539 . . . . . . . 8  |-  ( A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
5251adantl 277 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
5352oveq1d 5884 . . . . . 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 ) ) )
5450, 53eqtr4d 2213 . . . . 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 ) ) )
55 iffalse 3542 . . . . . . . 8  |-  ( -.  A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  1 )
5655adantl 277 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  if ( A  <  0 ,  -u
1 ,  1 )  =  1 )
5756oveq1d 5884 . . . . . 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 ) ) )
58 neg1cn 9013 . . . . . . . . 9  |-  -u 1  e.  CC
5958a1i 9 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  -u 1  e.  CC )
6041a1i 9 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  1  e.  CC )
614adantr 276 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  B  e.  ZZ )
6261, 3, 35sylancl 413 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  -> DECID  B  <  0
)
6359, 60, 62ifcldcd 3569 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  if ( B  <  0 ,  -u
1 ,  1 )  e.  CC )
6463mulid2d 7966 . . . . . 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 ( B  <  0 , 
-u 1 ,  1 ) )
6515adantr 276 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  B  e.  RR )
66 0red 7949 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  e.  RR )
679adantr 276 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  RR )
68 simplrl 535 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  =/=  0 )
6968neneqd 2368 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  -.  A  =  0 )
70 simpr 110 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  -.  A  <  0 )
718adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  ZZ )
72 ztri3or 9285 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  ->  ( A  <  0  \/  A  =  0  \/  0  <  A ) )
7371, 3, 72sylancl 413 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( A  <  0  \/  A  =  0  \/  0  < 
A ) )
74 3orass 981 . . . . . . . . . . . . . 14  |-  ( ( A  <  0  \/  A  =  0  \/  0  <  A )  <-> 
( A  <  0  \/  ( A  =  0  \/  0  <  A
) ) )
7573, 74sylib 122 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( A  <  0  \/  ( A  =  0  \/  0  <  A ) ) )
7675orcomd 729 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( ( A  =  0  \/  0  <  A )  \/  A  <  0 ) )
7770, 76ecased 1349 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( A  =  0  \/  0  <  A ) )
7877orcomd 729 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( 0  <  A  \/  A  =  0 ) )
7969, 78ecased 1349 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  <  A )
80 ltmul2 8802 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( B  <  0  <->  ( A  x.  B )  <  ( A  x.  0 ) ) )
8165, 66, 67, 79, 80syl112anc 1242 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( B  <  0  <->  ( A  x.  B )  <  ( A  x.  0 ) ) )
8267recnd 7976 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  CC )
8382mul01d 8340 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( A  x.  0 )  =  0 )
8483breq2d 4012 . . . . . . . 8  |-  ( ( ( ( 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
) )
8581, 84bitrd 188 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( B  <  0  <->  ( A  x.  B )  <  0
) )
8685ifbid 3555 . . . . . 6  |-  ( ( ( ( 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 ) )
8757, 64, 863eqtrrd 2215 . . . . 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 ) ) )
88 zdclt 9319 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  -> DECID  A  <  0 )
898, 3, 88sylancl 413 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  -> DECID  A  <  0
)
90 exmiddc 836 . . . . . 6  |-  (DECID  A  <  0  ->  ( A  <  0  \/  -.  A  <  0 ) )
9189, 90syl 14 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  <  0  \/  -.  A  <  0 ) )
9254, 87, 91mpjaodan 798 . . . 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 ) ) )
9392adantr 276 . . 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 ) ) )
94 simpr 110 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  N  <  0 )
9594biantrurd 305 . . . 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 ) ) )
9695ifbid 3555 . . 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 ) )
9794biantrurd 305 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( A  <  0  <->  ( N  <  0  /\  A  <  0 ) ) )
9897ifbid 3555 . . . 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 ) )
9994biantrurd 305 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( B  <  0  <->  ( N  <  0  /\  B  <  0 ) ) )
10099ifbid 3555 . . . 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 ) )
10198, 100oveq12d 5887 . . 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 ) ) )
10293, 96, 1013eqtr3d 2218 . 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 ) ) )
103 simpr 110 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  N  <  0 )
104103intnanrd 932 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  ( A  x.  B )  <  0 ) )
105104iffalsed 3544 . . . 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 )
106 1t1e1 9060 . . . 4  |-  ( 1  x.  1 )  =  1
107105, 106eqtr4di 2228 . . 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 ) )
108103intnanrd 932 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
109108iffalsed 3544 . . . 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 )
110103intnanrd 932 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  B  <  0 ) )
111110iffalsed 3544 . . . 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 )
112109, 111oveq12d 5887 . . 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 ) )
113107, 112eqtr4d 2213 . 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 ) ) )
114 simpl3 1002 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  N  e.  ZZ )
115 zdclt 9319 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  <  0 )
116114, 3, 115sylancl 413 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  -> DECID  N  <  0
)
117 exmiddc 836 . . 3  |-  (DECID  N  <  0  ->  ( N  <  0  \/  -.  N  <  0 ) )
118116, 117syl 14 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( N  <  0  \/  -.  N  <  0 ) )
119102, 113, 118mpjaodan 798 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    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    \/ w3o 977    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   ifcif 3534   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803    x. cmul 7807    < clt 7982    <_ cle 7983   -ucneg 8119   ZZcz 9242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-inn 8909  df-n0 9166  df-z 9243
This theorem is referenced by:  lgsdir  14103  lgsdi  14105
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