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Theorem lgsdilem 23353
Description: Lemma for lgsdi 23363 and lgsdir 23361: 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 507 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <_  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
3 0re 9596 . . . . . . . . . . 11  |-  0  e.  RR
4 simpl2 1000 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  ZZ )
54zred 10966 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  B  e.  RR )
65adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  RR )
7 ltlen 9686 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
83, 6, 7sylancr 663 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
9 simpl1 999 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  ZZ )
109zred 10966 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  A  e.  RR )
1110adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  A  e.  RR )
1211renegcld 9986 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  -u A  e.  RR )
1312recnd 9622 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  -u A  e.  CC )
1413mul01d 9778 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( -u A  x.  0 )  =  0 )
1511recnd 9622 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  A  e.  CC )
166recnd 9622 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  ->  B  e.  CC )
1715, 16mulneg1d 10009 . . . . . . . . . . . 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 4460 . . . . . . . . . . 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 9597 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
0  e.  RR )
2010lt0neg1d 10122 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  <  0  <->  0  <  -u A ) )
2120biimpa 484 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
0  <  -u A )
22 ltmul2 10393 . . . . . . . . . . . 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 1232 . . . . . . . . . . 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 9624 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  ( A  x.  B )  e.  RR )
2524adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  A  <  0 )  -> 
( A  x.  B
)  e.  RR )
2625lt0neg1d 10122 . . . . . . . . . . 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 9663 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
303, 6, 29sylancr 663 . . . . . . . . 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 3961 . . . . . . 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 6292 . . . . . . . . . 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 10753 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
3533, 34syl6eq 2524 . . . . . . . . 9  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  -u 1  ->  ( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  1 )
36 oveq2 6292 . . . . . . . . . 10  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  (
-u 1  x.  1 ) )
37 ax-1cn 9550 . . . . . . . . . . 11  |-  1  e.  CC
3837mulm1i 10001 . . . . . . . . . 10  |-  ( -u
1  x.  1 )  =  -u 1
3936, 38syl6eq 2524 . . . . . . . . 9  |-  ( if ( B  <  0 ,  -u 1 ,  1 )  =  1  -> 
( -u 1  x.  if ( B  <  0 ,  -u 1 ,  1 ) )  =  -u
1 )
4035, 39ifsb 3952 . . . . . . . 8  |-  ( -u
1  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  if ( B  <  0 ,  1 ,  -u
1 )
41 ifnot 3984 . . . . . . . 8  |-  if ( -.  B  <  0 ,  -u 1 ,  1 )  =  if ( B  <  0 ,  1 ,  -u 1
)
4240, 41eqtr4i 2499 . . . . . . 7  |-  ( -u
1  x.  if ( B  <  0 , 
-u 1 ,  1 ) )  =  if ( -.  B  <  0 ,  -u 1 ,  1 )
4332, 42syl6eqr 2526 . . . . . 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 3945 . . . . . . . 8  |-  ( A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  -u 1
)
4544adantl 466 . . . . . . 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 6299 . . . . . 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 2511 . . . . 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 3948 . . . . . . . 8  |-  ( -.  A  <  0  ->  if ( A  <  0 ,  -u 1 ,  1 )  =  1 )
4948adantl 466 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  if ( A  <  0 ,  -u
1 ,  1 )  =  1 )
5049oveq1d 6299 . . . . . 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 10639 . . . . . . . . 9  |-  -u 1  e.  CC
5251, 37keepel 4007 . . . . . . . 8  |-  if ( B  <  0 , 
-u 1 ,  1 )  e.  CC
5352mulid2i 9599 . . . . . . 7  |-  ( 1  x.  if ( B  <  0 ,  -u
1 ,  1 ) )  =  if ( B  <  0 , 
-u 1 ,  1 )
545adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  B  e.  RR )
55 0red 9597 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  e.  RR )
5610adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  RR )
57 lenlt 9663 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
583, 10, 57sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
0  <_  A  <->  -.  A  <  0 ) )
5958biimpar 485 . . . . . . . . . . 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 9721 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  0  <  A )
62 ltmul2 10393 . . . . . . . . . 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 1232 . . . . . . . . 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 9622 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  A  e.  CC )
6564mul01d 9778 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  A  <  0
)  ->  ( A  x.  0 )  =  0 )
6665breq2d 4459 . . . . . . . . 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 3961 . . . . . . 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 2520 . . . . . 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 2509 . . . . 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 465 . . 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 461 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  ->  N  <  0 )
7473biantrurd 508 . . . 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 3961 . . 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 508 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( A  <  0  <->  ( N  <  0  /\  A  <  0 ) ) )
7776ifbid 3961 . . . 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 508 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  N  <  0 )  -> 
( B  <  0  <->  ( N  <  0  /\  B  <  0 ) ) )
7978ifbid 3961 . . . 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 6302 . . 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 2516 . 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 461 . . . . . 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 ) )
84 iffalse 3948 . . . . 5  |-  ( -.  ( N  <  0  /\  ( A  x.  B
)  <  0 )  ->  if ( ( N  <  0  /\  ( A  x.  B
)  <  0 ) ,  -u 1 ,  1 )  =  1 )
8583, 84syl 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 )
86 1t1e1 10683 . . . 4  |-  ( 1  x.  1 )  =  1
8785, 86syl6eqr 2526 . . 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 ) )
8882intnanrd 915 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  A  <  0 ) )
89 iffalse 3948 . . . . 5  |-  ( -.  ( N  <  0  /\  A  <  0
)  ->  if (
( N  <  0  /\  A  <  0
) ,  -u 1 ,  1 )  =  1 )
9088, 89syl 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 )
9182intnanrd 915 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0 ) )  /\  -.  N  <  0
)  ->  -.  ( N  <  0  /\  B  <  0 ) )
92 iffalse 3948 . . . . 5  |-  ( -.  ( N  <  0  /\  B  <  0
)  ->  if (
( N  <  0  /\  B  <  0
) ,  -u 1 ,  1 )  =  1 )
9391, 92syl 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 )
9490, 93oveq12d 6302 . . 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 ) )
9587, 94eqtr4d 2511 . 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 ) ) )
9681, 95pm2.61dan 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3939   class class class wbr 4447  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    x. cmul 9497    < clt 9628    <_ cle 9629   -ucneg 9806   ZZcz 10864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-z 10865
This theorem is referenced by:  lgsdir  23361  lgsdi  23363
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